\begin{equation}
		\label{eq:orth_decomp}
		\{f\in\ell^2(E)\colon \partial f=0\}= \ell_{0,c}^2(E)\oplus_{\ell^2(E)} \{du\mid u\colon G\to \R, \, \Delta_0 u = 0, \, du\in \ell^2(E)\}.
	\end{equation}

\label{sec:Intro}

Let $G=(V,E)$ be countable oriented graph with degrees of vertices bounded from the above. Let $\tilde G$ be the non-oriented graph obtained from $G$ by forgetting the orientation of edges. Pick $D\in\mathbb N$ large enough. Consider all cycles in $\tilde G$ having lengths $\le D$. In $G$, glue each such cycle with a polygon. Choose any orientation of the latter polygons. We arrive to oriented $2$-dimensional complex, denote it by $G^{(2)}$ with implicit dependence on~$D$. Denote by $F$ the set of $2$-dimensional faces in $G^{(2)}$ which are polygons. Sometimes we write $F=FG$ and also $E=EG$ to indicate the dependence of these sets on $G$. Any of sets $V, E, F$ is endowed with counting measure which we denote by $\card$. In graph $G$, we define graph metric $\dist_G$ at $V\cup E$ along edges in $E$ so that any edge has length $1$.

If $\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\Cay(\Gamma,S)$, Cayley graph of $\Gamma$; then $V=\Gamma$. If $\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\Cay^{(2)}(\Gamma,S)$ is $\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that
\begin{equation*}\mbox{$G^{(2)}$ is simply connected}
\end{equation*}
(and that such $D$ does exist).

A \emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\mathbb R}$. We often understand  cochains as chains.
Discrete differentials (coboundaries)
$$
\{0\mbox{-cochains}\}\xrightarrow{d}\{1\mbox{-cochains}\}\xrightarrow{d}\{2\mbox{-cochains}\}
$$
and boundary operators
$$
\{2\mbox{-cochains}\}\xrightarrow{\partial}\{1\mbox{-cochains}\}\xrightarrow{\partial}\{0\mbox{-cochains}\}
$$
are introduced in the standard way with respect to the orientation of edges and faces.  Since valencies of vertices are bounded, all these operators are also bounded with respect to $\ell^2$-norms on cochains. We have $(d|_{\ell^2(V)})^*=\partial|_{\ell^2(E)}$, $(d|_{\ell^2(E)})^*=\partial|_{\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for  $\ell^2$-cochains by $\ell^2$-approximation.
If $\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\gamma$: for $e\in E$, let $f_\gamma(e)$ be the number of passes of $\gamma$ through $e$ in its direction minus number of passes of $\gamma$ over $e$ in its reversed direction. Then we have $\partial f_\gamma=0$.

Our space of interest is
\begin{equation*}
	\ell_{0,c}^2(E) := \clos_{\ell^2(E)}\{f\colon E\to {\mathbb R}\mid \partial f=0, \, \supp f
	\,\mbox{is finite}\}.\end{equation*}
Any of $1$-cochains at the right-hand side can be (convexly) decomposed into simple finite loops. Thus, $\ell_{0,c}^2(E)$ is $\ell^2$-closed linear span of (co)chains of the form $f_\gamma$ with $\gamma$ a finite loop in $G$.

Laplace operator on $0$-cochains is
$$
-\Delta_0=\partial d\colon \left(\mbox{functions on }V\right)\to \left(\mbox{functions on }V\right).
$$
A discrete integration by parts leads to the following Hodge-type decomposition:

\begin{predl}

	We have
	\begin{equation}
		\label{eq:orth_decomp}
		\{f\in\ell^2(E)\colon \partial f=0\}= \ell_{0,c}^2(E)\oplus_{\ell^2(E)} \{du\mid u\colon G\to {\mathbb R}, \, \Delta_0 u = 0, \, du\in \ell^2(E)\}.
	\end{equation}
\end{predl}

The second summand in the right-hand side of the latter relation is \emph{$\ell^2$-cohomology} of $G$. It is known to be invariant with respect to change of generating system in a group: the factorspace nature of cohomology allows to implement "discrete change of variables" from one to another set of generators. Cohomology is invariant with respect to more general quasiisometries.
Now, we pass to spectral estimates for $1$-cochains. Define non-negative Laplacian operator $\Delta_1:=\partial d+d\partial \colon \ell^2(E)\to \ell^2(E)$. On $\ell^2_{0,c}(E)$, our space of interest, this reduces to $\partial d$.

We have one more Hodge-type decomposition:
$$
\ell^2(E) = \clos_{\ell^2(E)}\{du\mid u \colon V\to {\mathbb R}, \, \supp u \mbox{ is finite}\} \oplus_{\ell^2(E)} \{f\in\ell^2(E)\colon \partial f=0\}.$$
Spectral questions for $\Delta_1$ on the first summand are generally reduced to the same for $\Delta_0$ on $\ell^2(V)$.   What concerns decomposition (\ref{eq:orth_decomp}) for $\{f\in\ell^2(E)\colon \partial f=0\}$, operator $\Delta_1$ vanishes at the second summand of its right-hand side, $\ell^2$-cohomology. Also, by the definition of $\ell^2_{0,c}(E)$ and by $\ell^2$-approximation, we see that $\Delta_1(\ell^2_{0,c}(E))\subset\ell^2_{0,c}(E)$.

\begin{define}
	We say that $\Delta_1$ \emph{has a spectral gap at $\ell^2_{0,c}(E)$} \emph{(}or just that graph $G$ has \emph{{coexact $1$-Laplacian spectral gap}}\emph{)} if $$\spec\left(\Delta_1|_{\ell^2_{0,c}(E)}\right)\cap [0,\varepsilon)=\varnothing$$ for some $\varepsilon>0$ small enough.
\end{define}

Applying discrete integration by parts, we conclude that this is equivalent to the estimate
\begin{equation}
	\label{eq:rot_estim}
\langle f, f \rangle_{\ell^2(E)}	\le 1/\varepsilon\cdot \langle df, df\rangle_{\ell^2(F)}
\end{equation}
for $f\in \ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.
 We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.
 \begin{predl}
 	\label{predl:general_invariance}
 	Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\subset E_2$.  	Assume the following:
 	\begin{enumerate}
 		\item $G_1$ is connected;

 		\item degrees of vertices in $G_2$ are bounded from the above;

 		\item $\sup\limits_{e\in E_2\setminus E_1} \dist_{G_1}(\beg e, \ennd e) < +\infty$ with the obvious notation.
 	\end{enumerate}
	\emph{(}In other words, metrics $\dist_{G_1}$ and $\dist_{G_2}$ on $V$ are bilipshitz equivalent.\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap  \emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\emph{)}
	if and only if  $G_2$ has such a spectral gap \emph{(}with some $D'$ for $G_2^{(2)}$\emph{)}.
 \end{predl}

 \noindent Proof of this quasiinvariance  is similar to the proof of quasiiinvariance of $\ell^2$-cohomology; both are based on orthogonal projection. We give a detailed argument in the Appendix. Notice also that the proof is constructive: we may estimate $D'$ via $D$ and the supremum from the third assumption of Proposition \ref{predl:general_invariance}, and vice versa.

 \begin{sled}
 	For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them  are equivalent.
 \end{sled}

Our main result is

If $\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\Cay(\Gamma,S)$, Cayley graph of $\Gamma$; then $V=\Gamma$. If $\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\Cay^{(2)}(\Gamma,S)$ is $\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that
\begin{equation*}\mbox{$G^{(2)}$ is simply connected}
\end{equation*}
(and that such $D$ does exist).

A \emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\mathbb R}$. We often understand cochains as chains.
Discrete differentials (coboundaries)
$$
\{0\mbox{-cochains}\}\xrightarrow{d}\{1\mbox{-cochains}\}\xrightarrow{d}\{2\mbox{-cochains}\}
$$
and boundary operators
$$
\{2\mbox{-cochains}\}\xrightarrow{\partial}\{1\mbox{-cochains}\}\xrightarrow{\partial}\{0\mbox{-cochains}\}
$$
are introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\ell^2$-norms on cochains. We have $(d|_{\ell^2(V)})^*=\partial|_{\ell^2(E)}$, $(d|_{\ell^2(E)})^*=\partial|_{\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\ell^2$-cochains by $\ell^2$-approximation.
If $\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\gamma$: for $e\in E$, let $f_\gamma(e)$ be the number of passes of $\gamma$ through $e$ in its direction minus number of passes of $\gamma$ over $e$ in its reversed direction. Then we have $\partial f_\gamma=0$.

\begin{define}
 We say that $\Delta_1$ \emph{has a spectral gap at $\ell^2_{0,c}(E)$} \emph{(}or just that graph $G$ has \emph{{coexact $1$-Laplacian spectral gap}}\emph{)} if $$\spec\left(\Delta_1|_{\ell^2_{0,c}(E)}\right)\cap [0,\varepsilon)=\varnothing$$ for some $\varepsilon>0$ small enough.
\end{define}

Applying discrete integration by parts, we conclude that this is equivalent to the estimate
\begin{equation}
 \label{eq:rot_estim}
\langle f, f \rangle_{\ell^2(E)} \le 1/\varepsilon\cdot \langle df, df\rangle_{\ell^2(F)}
\end{equation}
for $f\in \ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.
 We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.
 \begin{predl}
 \label{predl:general_invariance}
 Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\subset E_2$. Assume the following:
 \begin{enumerate}
 \item $G_1$ is connected;

\begin{sled}
 For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.
 \end{sled}

Our main result is

\renewcommand{\abstractname}{}
\begin{abstract}
Let $\Gamma$ be a discrete finitely presented group. Pick any system $S$ of generators in $\Gamma$. In Cayley graph $\Cay(\Gamma)=\Cay(\Gamma, S)$ with edge set~$E$, glue with oriented polygons all the group relations translated to all the points of $\Gamma$; denote the obtained simply connected complex by $\Cay^{(2)}(\Gamma)$.
We study non-negative \emph{Hodge--Laplace operator $\Delta_1$} on edge functions which is defined via complex $\Cay^{(2)}(\Gamma)$; $\Delta_1$ acts on
$$
\ell^2_{0,c}(E):= \clos_{\ell^2(E)}\left\{\mbox{finitely supported closed $1$-(co)chains in }\Cay^{}(\Gamma)\right\}.
$$

We prove the following implication in the spirit of Kesten Theorem: \emph{if $\Delta_1|_{\ell_{0,c}^2(E)}$ has a spectral gap then $\Gamma$ either has exponential growth or is virtually $\mathbb Z$}. \end{abstract}

Our main result is

Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:

\begin{theorem}[\cite{K59}] \label{th:Kesten}
 Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$, $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

If we assume the contrary to Theorem \ref{th:exp_growth}, then, first, $\Gamma$ cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic, see, e.g., \cite{Me08}. Second, $\Gamma$ cannot have infinitely many ends because then $\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end.

Now, let us briefly recall the proof of Kesten Theorem \ref{th:Kesten} to compare it to our argument. Non-amenability of $\Gamma$ means that
\begin{equation}
\label{eq:isoperimetr}
\|du\|_{\ell^1(E)} \gtrsim \|u\|_{\ell^1(\Gamma)}
\end{equation}
for $u=\mathds 1_E$, $E$ ranges all finite subsets in $\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\colon \Gamma\to\R$.
The spectral gap condition $0\notin \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$ means that
$$
\langle u, u\rangle_{\ell^2(\Gamma)} \lesssim \langle du, du\rangle_{\ell^2(E)}, ~~~ u\colon\Gamma\to\R \mbox{ is finitely supported}.
$$
To obtain this from (\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.

This paper is organized as follows. In Section \ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \ref{th:exp_growth}. In Section~\ref{sec:loop}, we prove Lemma \ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \ref{sec:spectra_homology}, we conclude the proof of Theorem~\ref{th:exp_growth}. This is done by approximating the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$ provided that $0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\gamma$ from Lemma \ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\Cay(\Gamma)$ and also make use of homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$. Finally, in Section \ref{sec:examples} we check the most natural examples of Cayley graphs.

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

\begin{theorem}[\cite{K59}]	\label{th:Kesten}
	Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$,  $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

\begin{theorem}
	\label{th:exp_growth}
	Let $\Gamma$ be a countable finitely presented group.
	If $\Delta_1$ has a spectral gap on 	$\ell_{0,c}^2(E)$ 	then either $\Gamma$ has exponential growth, 	 or $\Gamma$ is virtually infinite cyclic. \end{theorem}

Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:

\begin{theorem}[\cite{K59}]	\label{th:Kesten}
	Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$,  $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

Non-amenability of a group, that is, the existence of a spectral gap for $-\Delta_0$, easily implies exponential growth. The reverse is not true, in general. Thus, it is natural to ask, for example, whether Baumslag--Solitar groups $\langle\mathscr a, \mathscr b\mid \mathscr b^{-1}\mathscr a\mathscr b=\mathscr a^n\rangle$, $n\in\mathbb N$, have a spectral gap for $1$-Laplacian. These groups are \emph{non-elementary amenable} but have exponential growth. Such groups are not covered by Theorem \ref{th:exp_growth}, and it is still unclear for the author whether $1$-Laplacian has a spectral gap on them.

If we assume the contrary to Theorem \ref{th:exp_growth}, then, first, $\Gamma$ cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic, see, e.g., \cite{Me08}. Second, $\Gamma$ cannot have infinitely many ends because then $\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end.

For $\mathscr L>0$, denote by $\mathbb T_{\mathscr{L}}$ a circle of length $\mathscr L$. On $\mathbb T_{\mathscr L}$, one may measure distances along this loop. To prove Theorem \ref{th:exp_growth}, we need the following

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

In fact, we are able to make $\mathscr L$ arbitrarily large with fixed $x$.

Now, let us briefly recall the proof of Kesten Theorem \ref{th:Kesten} to compare it to our argument. Non-amenability of $\Gamma$ means that
\begin{equation}
\label{eq:isoperimetr}
\|du\|_{\ell^1(E)} \gtrsim \|u\|_{\ell^1(\Gamma)}
\end{equation}
for $u=\mathds 1_E$, $E$ ranges all finite subsets in $\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\colon \Gamma\to{\mathbb R}$.
The spectral gap condition $0\notin \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$ means that
$$
\langle u, u\rangle_{\ell^2(\Gamma)} \lesssim \langle du, du\rangle_{\ell^2(E)}, ~~~ u\colon\Gamma\to{\mathbb R} \mbox{ is finitely supported}.
$$
To obtain this from (\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.

In the first step of the latter argument, we assemble a function $u\colon \Gamma\to{\mathbb R}$, say, non-negative one, from its super-level sets $\mathds 1_{\{u\ge \mathscr t\}}$, $\mathscr t$ ranges $[0,+\infty)$; we also assemble $du$ from $d\mathds 1_{\{u\ge \mathscr t\}}$. (Both decompositions are $\ell^1$-convex.) Thus, in Kesten Theorem, we deal with "sets of codimensions $0$ and $1$". At least, we will have such genuine codimensions in the case of a manifold instead of a group, the corresponding result linking spectra and isoperimetry is known as Cheeger--Yau inequality, see \cite{Cheeger}, \cite{Yau}.

Unlike this, in our argument we work with dimension $1$ sets --- loops, in particular, as in Lemma \ref{lemma:embed}. Also, in Section \ref{sec:spectra_homology} we bound $1$-cycles with $2$-dimensional surfaces.

Notice also that an analogue of Cheeger--Yau inequality for $1$-forms was obtained in \cite{BC22} in the case of manifolds. Coexact $1$-Laplacian spectrum is indeed related to appropriate isoperimetric ratio, namely, to $\sup_{\gamma}\inf_h|h|/\length\gamma$ with $\gamma$ ranging homologicaly trivial loops at a manifold $\mathscr M$ and $h$ be a $2$-dimensional chain in $\mathscr M$ bounding $\gamma$; here, $|h|$ is area of $h$. Some Poincar\'e-type estimates for operator $d$ on coclosed $1$-forms are  possible if isoperimetric ratios as above are bounded from the below. But, in \cite{BC22}, authors impose the condition of finite diameter of $\mathscr M$  which is not our case; also, \cite{BC22} does not deal with effects of negative curvature.

What concerns spaces with negative curvature, let us mention recent works \cite{AAGLZ24}, \cite{R23} devoted to $3$-dimensional hyperbolic manifolds. It turns out that, first, $1$-coexact spectral gap is related to exponential growth of torsion $1$-homology of the manifolds; second, there are relations  between the spectral gap and isoperimetric ratios. The latter estimates from \cite{R23} are also volume-dependent, as in \cite{BC22}.

\medskip

This paper is organized as follows. In Section \ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \ref{th:exp_growth}. In Section~\ref{sec:loop}, we prove Lemma \ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \ref{sec:spectra_homology}, we conclude the proof of Theorem~\ref{th:exp_growth}. This is done by approximating the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$ provided that $0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\gamma$ from Lemma \ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\Cay(\Gamma)$ and also  make use of homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$.  Finally, in Section \ref{sec:examples} we check the most natural examples of Cayley graphs.

\medskip

\noindent {\bf Some notation.} For a set $A$ we denote by $\card A$ the number of its elements.
If $v_1, v_2$ are vertices of some oriented graph then we denote by $\edge(v_1, v_2)$ the oriented edge in the graph under consideration provided that the edge exists. If $e$ is an edge in a oriented graph or $\gamma$ is an oriented path in a metric space then we write $\beg e$, $\beg\gamma$ for their beginnings and $\ennd e$, $\ennd\gamma$ for their endpoints, respectively. The notation $\length\gamma$ is obvious.

We write $\mathcal B_X(x, \rho)$ for the open ball in a metric space $X$ centered in a point $x\in X$ and having radius $\rho\ge 0$.

Assuming the contrary to Theorem~\ref{th:exp_growth}, one first reduces the possibilities for the number of ends: $\Gamma$ "cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic"; it also "cannot have infinitely many ends because then $\Gamma$ has exponential growth". Hence, "by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end."

Under the hypothesis that $\Gamma$ has subexponential growth and one end, one proves Lemma~\ref{lemma:embed} ("on loop embedding"), producing for given $C>20$, $x\ge1$ an injective loop $\gamma\colon\mathbb T_{\mathscr L}\to\Cay(\Gamma)$ with $\mathscr L>2DCx$ and the metric control (\ref{eq:x_bilip}): if $\dist_{\Cay(\Gamma)}(\gamma(t_1),\gamma(t_2))\le x$ then $\dist_{\mathbb T_{\mathscr L}}(t_1,t_2)\le Cx$ (and "we are able to make $\mathscr L$ arbitrarily large with fixed $x$"). The lemma is proved "by dropping lots of geodesic perpendiculars in a branching way."

The proof of Theorem~\ref{th:exp_growth} is then completed in Section~\ref{sec:spectra_homology}: assuming a spectral gap, i.e. "$0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$", one "approximat[es] the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$" and applies this to the "$1$-cochain given by curve $\gamma$ from Lemma~\ref{lemma:embed}$." One also "put[s] some metric control on the approximating process" in $\Cay(\Gamma)$ and uses the "homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$", together with the fact that "we bound $1$-cycles with $2$-dimensional surfaces."

Assuming the contrary to the main theorem, one first reduces the possibilities for the number of ends: $\Gamma$ "cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic"; it also "cannot have infinitely many ends because then $\Gamma$ has exponential growth". Hence, "by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end."

Under the hypothesis that $\Gamma$ has subexponential growth and one end, one proves the following lemma.

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

This produces for given $C>20$, $x\ge1$ an injective loop $\gamma\colon\mathbb T_{\mathscr L}\to\Cay(\Gamma)$ with $\mathscr L>2DCx$ and the metric control given by the displayed equation above (and "we are able to make $\mathscr L$ arbitrarily large with fixed $x$"). The lemma is proved "by dropping lots of geodesic perpendiculars in a branching way."

In completing the proof of the main theorem, assuming a spectral gap, i.e. "$0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$", one "approximat[es] the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$" and applies this to the "$1$-cochain given by curve $\gamma$ from the lemma above." One also "put[s] some metric control on the approximating process" in $\Cay(\Gamma)$ and uses the "homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$", together with the fact that "we bound $1$-cycles with $2$-dimensional surfaces."

\label{th:exp_growth}
	Let $\Gamma$ be a countable finitely presented group.
	If $\Delta_1$ has a spectral gap on 	$\ell_{0,c}^2(E)$ 	then either $\Gamma$ has exponential growth,	 or $\Gamma$ is virtually infinite cyclic.,

Question: Let Γ be a countable finitely presented group, choose a generating set S, and form its Cayley graph Cay(Γ,S) with edge set E. Glue oriented polygons along all translates of the defining relations to obtain the associated simply connected 2-complex, and let Δ1 be the corresponding Hodge–Laplace operator on 1-cochains. Write ℓ^2_{0,c}(E) for the ℓ^2-closure of the finitely supported closed 1-cochains on E. If Δ1 has a spectral gap on ℓ^2_{0,c}(E), meaning spec(Δ1|_{ℓ^2_{0,c}(E)}) avoids some interval [0, ε) with ε > 0, what must be true of Γ?

Γ either has exponential growth or is virtually infinite cyclic.

Choices:
B. Γ either is non-amenable or is virtually infinite cyclic.
C. Γ is not a torsion group; in particular, Γ has an element of infinite order.
D. Γ either has exponential growth or is virtually cyclic.
E. If Δ_1 has a spectral gap on \ell^2_{0,c}(E), then for every choice of generating set S the group Γ has exactly one end unless it is virtually infinite cyclic.

replace growth conclusion by amenability-type conclusion in Kesten spirit

dropped the dichotomy 'exponential growth or virtually infinite cyclic' to the weaker consequence 'contains an infinite order element'

strength of the exceptional case: 'virtually infinite cyclic' broadened to 'virtually cyclic'

one-end reduction used only under contradiction turned into a global conclusion for every generating set


\label{thmintro:delocalised-trace-identification-simplified}
  Let $\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$.  Let $\cO \subseteq W$ be a conjugacy class.  Then there is a unique trace $\vphi_\cO$ on $\cH$ that satisfies $\vphi_\cO(T_w) = \mathbb{1}_\cO(w)$ for every element $w \in W$ that is of minimal length in its conjugacy class.

\begin{remark}
  \label{rem:finite-order-element-cliques}
  Let $\Gamma$ be the commutation graph of a right-angled Coxeter system $(W, S)$.  Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques in $\Gamma$, that is the set of complete subgraphs of $\Gamma$ (including the empty graph).  Identifying vertices of $\Gamma$ with $S$, we consider the map $C \mapsto \prod_{s \in \rV(C)} s$.  By Remark~\ref{rem:finite-order-element-conjugacy} every finite order element is conjugate to an element in the image of this map.  Further, if $\prod_{s \in \rV(C)} s$ and $\prod_{s \in \rV(D)} s$ are conjugate, they must be equal by Tits solution to the word problem.  It follows that $C \mapsto \prod_{s \in \rV(C)} s$ defines a bijection between $\mathrm{Cliq}(\Gamma)$ and the set of conjugacy classes of finite order elements of $W$.
\end{remark}

\begin{theorem}
  \label{thm:right-angled-cocentre-basis}
  Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$ over an algebraically closed field $K$ of characteristic different from $2$. Let $(a_s)_{s\in S}$, $(b_s)_{s\in S}$ be the deformation parameters of $\cH$ and assume that $b_s \neq -\frac{a_s^2}{4}$ for each $s \in S$.  Then $(T_{\cO})_{\cO}$ where $\cO$ runs through the conjugacy classes of $W$ is a basis of the cocentre of $\cH$.
\end{theorem}

\label{sec:introduction}

The study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \cite{lott1999}, who investigated for example delocalised $\Ltwo$-Betti numbers and delocalised $\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.

While this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \cite{opdamsolleveld2010-discrete-series} and generalised principal series \cite{delormeopdam2011} of affine Hecke algebras.

Hecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \cite{caspersklisselarsen2021}, simplicity results \cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\ \cite[Corollary B]{raumskalski2023}.

The cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \cite{baumnistor2002} in the purely algebraic setup and by Solleveld \cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.

In this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.

\begin{theorem}
  \label{thm:delocalised-trace-identification}
  Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\cO \subseteq W$, the map $\vphi_\cO$ is a trace on $\cH$.  If $(T_{\cU})_{\cU}$ denotes the basis of the cocentre of $\cH$ satisfying $T_{\cU} = T_w + [\cH, \cH]$ for every element of minimal length $w \in \cU$, then $\vphi_\cO(T_\cU) = \delta_{\cO, \cU}$.
\end{theorem}

The next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.
\begin{introtheorem}
 \label{introthm:Schwartz-algebra-existence}
 Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$. There is a subalgebra $\cS(W, S, q) \subseteq \Cstarred(W, S, q)$, which
 \begin{itemize}
 \item is dense and closed under holomorphic functional calculus,
 \item and such that for every conjugacy class $\cO \subseteq W$, the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
 \end{itemize}
\end{introtheorem}

The proof of \cite[Theorem 5.3]{henie2014} applies verbatim to show the following result: the collection $(T_\cO)_{\cO}$ linearly generates the cocentre of any generic Hecke algebra. We will use the following description of the coefficients appearing in linear combinations of these elements.
\begin{lemma}
 \label{lem:non-minimal-element-cocentre}
 Let $\cH$ be a generic Hecke algebra of type $(W,S)$. Let $\cO \subseteq W$ be a conjugacy class and let $w \in \cO$. Then
 \begin{gather*}
 T_w + [\cH, \cH]
 =
 \sum_\cK c_{\cK, w} T_\cK
 \end{gather*}
 for certain coefficients $c_{\cK, w}$, where $\cK$ runs over all conjugacy classes of elements that are Bruhat subordinate to $w$.
\end{lemma}
\begin{proof}
 Let $\ell$ be the minimal length of an element in $\cO$. We prove the claim by induction on $|w| - \ell = k \in \NN_0$. For $k = 0$ there is nothing to prove.

\begin{notation}
 \label{not:delocalised-trace}
 Let $(W, S)$ be a Coxeter system of right-angled type and let $\cH$ be a generic Hecke algebra of type $(W, S)$ defined over $R$. For a conjugacy class $\cO \subseteq W$, we denote by $\Sigma_\cO\colon \cH \lra R$ the sum over all coefficients of minimal length elements in $\cO$, that is
 \begin{gather*}
 \Sigma_\cO(T_w)
 =
 \begin{cases}
 1 & \text{ if $w \in \cO$ and $w$ is of minimal length in $\cO$} \\
 0 & \text{ otherwise}\eqstop
 \end{cases}
 \end{gather*}
 Combining this with Notation~\ref{not:cyclic-reduction-map}, we will write $\vphi_\cO = \Sigma_\cO \circ \Phi$.
\end{notation}

\begin{theorem}
 \label{thm:delocalised-trace-identification}
 Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\cO \subseteq W$, the map $\vphi_\cO$ is a trace on $\cH$. If $(T_{\cU})_{\cU}$ denotes the basis of the cocentre of $\cH$ satisfying $T_{\cU} = T_w + [\cH, \cH]$ for every element of minimal length $w \in \cU$, then $\vphi_\cO(T_\cU) = \delta_{\cO, \cU}$.
\end{theorem}
\begin{proof}
 In order to show that $\vphi_\cO$ is a trace, it suffices to show that for all $w \in W$ and all $s \in S$ the equality $\vphi_\cO(T_s T_w) = \vphi_\cO(T_w T_s)$ holds. We prove this statement by induction on $|w| \in \NN_0$ and observe that the case $|w| = 0$ is clear. We distinguish several cases.

\begin{theorem}
 \label{thm:delocalised-trace-existence}
 Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and let $\cH$ be the Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$. Let $\cS'(W, S, q), \cS(W, S, q) \subseteq \Cstarred(W, S, q)$ be smooth subalgebras, such that $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$ and the natural quasi-derivation
 \begin{gather*}
 \Delta\colon T_w \mapsto \sum_{\substack{w = w_1w_2 \\ \text{ reduced decompositon}}} T_{w_1} \ot T_{w_2}
 \end{gather*}
 continuously extends to a map into the unconditional tensor product $\cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$. Then for every conjugacy class $\cO \subseteq W$ the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
\end{theorem}
\begin{proof}
 Let us first observe that the map $\Sigma_\cO$ summing over the finitely many coefficients of elements of minimal length in $\cO$ extends to a continuous map on $\cS'(W, S, q)$, as $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$. Consider maps $\gamma, \rho\colon W \lra W$ defined as in Lemma~\ref{lem:cyclic-reduction-hecke-algebra}. Since $\Delta$ extends continuously to a map $\cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$ and since $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$, the map $\Delta'\colon \cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$ satisfying $\Delta'(T_w) = T_{\gamma(w) \rho(w)} \ot T_{\gamma(w)^{-1}}$ is well-defined and continuous. Let $\Phi\colon \CC(W, S, q) \lra \CC(W, S, q)$ be the unique linear map satisfying $\Phi(T_w) = T_{\gamma(w)^{-1}} T_{\gamma(w)} T_{\rho(w)}$. Then we have $\vphi_\cO = \Sigma_\cO \circ \Phi$ as shown in Theorem~\ref{thm:delocalised-trace-identification}. So a continuous extension of $\vphi_\cO$ to $\cS(W, S, q)$ is obtained by the composition
 \begin{multline*}
 \cS(W, S, q) \stackrel{\Delta'}{\lra}
 \cS'(W, S, q) \ot \cS'(W, S, q) \stackrel{\text{tensor flip}}{\lra} \\
 \cS'(W, S, q) \ot \cS'(W, S, q) \stackrel{\text{multiplication}}{\lra}
 \cS'(W, S, q) \stackrel{\Sigma_\cO}{\lra}
 \CC
 \eqcomma
 \end{multline*}
This finishes the proof.
\end{proof}

Before proving Theorem~\ref{thmintro:pairing}, let us clarify its context and notation beyond what we have done in the introduction already. Recall from \cite[Theorem A.2.1]{bost1990-principe-oka} that the inclusion of a smooth subalgebra into a unital C*-algebra induces an isomorphism in K-theory. We also recall the K-theory calculations for right-angled Hecke C*-algebras from \cite[Section 4]{raumskalski2022}. To this end we use the description of conjugacy classes of finite order elements from Remark~\ref{rem:finite-order-element-cliques}. Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques in the commutation graph $\Gamma$ of a right-angled Coxeter system $(W, S)$, that is subsets $C \subseteq S$ such that $st = ts$ for all $s,t \in C$. For a clique $C \in \mathrm{Cliq}(\Gamma)$, we write $p_C = \prod_{s \in C} \chi_1^s$. Here $\chi_1^s = \pi_s(\chi_1)$ denotes the image of the projection $\chi_1 \in \CC[\ZZ/2]$ associated with the trivial representation under the linear map extending the group homomorphism $\pi_s\colon \ZZ/2 \lra \CC(W, S, q)^\times$ which satisfies $\pi_s(1) = \frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. Then for every deformation parameter $q \in \RR_{> 0}^S$ the map $\mathrm{Cliq}(\Gamma) \lra \rK_0(\Cstarred(W, S, q))\colon C \mapsto [p_C]$ induces an isomorphism $\ZZ^{\mathrm{Cliq}(\Gamma)} \lra \rK_0(\Cstarred(W, S, q))$.
\begin{proof}[Proof of Theorem~\ref{thmintro:pairing}]
 As described above, for $D \in \mathrm{Cliq}(\Gamma)$ we have $p_D = \prod_{s \in D} \chi_1^s$ where $\chi_1^s$ is the projection associated with the trivial representation of the unitary representation $\ZZ/2 \mapsto \CC(W, S, q)$ mapping $1$ to $\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. We have $\chi_1^s = \mathbb{1}_{(0,\infty)}(T_s) = \frac{1}{q_s^{1/2} + q_s^{-1/2}} (T_s + q_s^{-1/2}) =\frac{1}{q_s^{1/2} + q_s^{-1/2}} T_s + \frac{1}{1 + q_s}$. Writing $T_D = \prod_{s \in D} T_s$, we find that
 \begin{gather*}
 p_D = \sum_{C \subseteq D} \bigl ( \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \bigr ) \bigl ( \prod_{s \in D \setminus C} \frac{1}{1 + q_s} \bigr ) T_C
 \eqstop
 \end{gather*}
 Since all elements in the support of $p_D$ have minimal length and are pairwise non-conjugate, we find that
 \begin{gather*}
 \vphi_C(p_D)
 =
 \Sigma_C(p_D)
 =
 \begin{cases}
 0 & C \nsubseteq D \eqcomma \\
 \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \prod_{s \in D \setminus C} \frac{1}{1 + q_s} & C \subseteq D \eqstop
 \end{cases}
 \end{gather*}
 Now the claimed formula for $\vphi_C(p_D)$ follows.

We mention that in order to prove this result, we establish in Theorem~\ref{thm:right-angled-cocentre-basis} that the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.  The problem of describing the cocentre of Hecke algebras is well-known in the context of character theory.  See e.g. \cite{geckpfeiffer2000-characters} for spherical Coxeter types and \cite{henie2014} for affine Coxeter types.  The recent work \cite{chen2025-centralizers} explicitly states folklore conjectures about the cocentre for indefinite Coxeter types.

The next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.
\begin{introtheorem}
  \label{introthm:Schwartz-algebra-existence}
  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$.  There is a  subalgebra $\cS(W, S, q) \subseteq \Cstarred(W, S, q)$, which
  \begin{itemize}
  \item is dense and closed under holomorphic functional calculus,
  \item and such that for every conjugacy class $\cO \subseteq W$, the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
  \end{itemize}
\end{introtheorem}

From the point of view of operator algebras, one of the main purposes of delocalised traces is to determine K-theory classes via their pairing.  The K-theory of right-angled Hecke C*-algebras was calculated in \cite{raumskalski2022}.  We briefly recall that for every clique $C$ in the commutation graph of a right-angled Coxeter system there is a projection $p_C$, such that the classes $([p_C])_C$ freely generate K-theory.  In Remark~\ref{rem:finite-order-element-cliques}, we observe that these cliques are in natural bijection with finite order elements of a right-angled Coxeter system.  We obtain the following result, which determines the pairing of K-theory and delocalised traces.
\begin{introtheorem}
  \label{thmintro:pairing}
  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $(q_s)_{s \in S} \in \RR_{> 0}^S$ a deformation parameter.  Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques of the commutation graph $\Gamma$ of $(W, S)$, identified with the set of conjugacy classes of finite order elements in $W$.  For a clique $C \in \mathrm{Cliq}(\Gamma)$ let $p_C$ be the associated projection and let $\vphi_C$ be the delocalised trace associated with $C$.  Then the pairing between K-theory of $\Cstarred(W, S, q)$ and delocalised traces of finite order elements is determined by the formula
  \begin{gather*}
    \vphi_C(p_D) =
    \begin{cases}
      0 & C \nsubseteq D\eqcomma \\
      \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \prod_{s \in D \setminus C} \frac{1}{1 + q_s} & C \subseteq D \eqstop
    \end{cases}
  \end{gather*}
  In particular, the pairing $\rK_0(\Cstarred(W,S,q)) \times \RR^{\mathrm{Cliq(\Gamma)}} \lra \RR$ is non-degenerate.
\end{introtheorem}

As mentioned above, in the last section of \cite{puschnigg2010} Puschnigg used his main results to obtain certain decompositions of cyclic cohomology for his smooth algebras and the group C*-algebras of hyperbolic groups, close in spirit to the computation of cyclic homology of group rings due to Burghelea \cite{burghelea1985, burghelea2023-erratum-for-1985}. This line of research was later continued, using the Jolissaint's rapid decay subalgebras, for example in \cite{jiogleramsey14} and in \cite{engel20}. Our results in principle open the way to obtaining similar decompositions of cyclic cohomology in the context of Hecke algebras, such as the Fr{\'e}chet algebra $\cS(W, S, q)$ appearing in Theorem \ref{introthm:Schwartz-algebra-existence}. However, the setting of right-angled hyperbolic Coxeter types remains too narrow to make such an endeavour worthwhile at present, since higher cyclic cohomology is expected to vanish.  Further work in this direction, would chime with the recent progress in understanding noncommutative geometric properties of the Iwahori-Hecke algebras such as in the work \cite{klisseperovic25} already mentioned.

This paper is organised as follows: after this introduction, in the preliminaries we describe background material and fix notation, focusing on Coxeter groups and Hecke algebras.  Then in the third section, we investigate the combinatorics of the multiplication in right-angled Hecke algebras, which plays a major role in this work.  In the fourth section, we adapt work of Blackadar-Cuntz and Puschnigg on differential seminorms and quasi-derivations to the needed generality.  In the fifth section, we use this framework in order to exhibit specific smooth subalgebras of the Hecke algebras under consideration, to which the natural quasi-derivation extends.  In the sixth section, we describe traces on right-angled Hecke algebras, and in that last section combine this with the previous results on smooth subalgebras, to obtain delocalised traces on Hecke-Schwartz algebras and calculate their pairing with K-theory.

\subsection*{Acknowledgements}

SP was partially supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 677120-INDEX) and the Knut and Alice Wallenberg Foundation (grant number KAW 2020.0252).

SR was partially supported by the Swedish Research Council (grant number 2018-04243) and the German Research Foundation (DFG project no. 550184791).

PN was supported by the National Science Center Grant Maestro-13 UMO-2021/42/A/ST1/00306.

The authors would like to thank IMPAN and its B{\c e}dlewo Conference Center as well as the University to Potsdam for their hospitality during respective visits.

To prove Theorem~\ref{thmintro:delocalised-trace-identification-simplified}, the introduction says that the authors first establish in Theorem~\ref{thm:right-angled-cocentre-basis} that “the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.”

Question: Let \(\mathcal H\) be the Iwahori Hecke algebra associated with a right-angled Coxeter system \((W,S)\), and let \(\{T_w\}_{w\in W}\) denote its standard basis indexed by \(W\). Let \(\mathcal O\subseteq W\) be a conjugacy class. A trace on \(\mathcal H\) means a linear functional \(\varphi:\mathcal H\to\mathbb C\) such that \(\varphi(xy)=\varphi(yx)\) for all \(x,y\in\mathcal H\). For \(w\in W\), say that \(w\) is of minimal length in its conjugacy class if its Coxeter length is minimal among all elements conjugate to \(w\). Which existence statement holds under these assumptions?

There exists a unique trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.

Choices:
B. There exists a unique trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\), \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.
C. There exists a trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.
D. For each conjugacy class \(\mathcal O\subseteq W\), there exists a unique linear functional \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\).
E. There exists a trace \(\varphi:\mathcal H\to\mathbb C\) such that for every conjugacy class \(\mathcal O\subseteq W\) and every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.

trace property replaced by mere linearity

dependence on the conjugacy class parameter \(\mathcal O\)


\label{main1}
Let $X$ be a 1-nodal quadric threefold  and $\pi \colon \widetilde{X} \longrightarrow X$ be its blow-up.

Then there exist a stability condition $\sigma_{\mathcal{K}u (X)}=(Z_\mathcal{A}, \mathcal{A})$ on $\mathcal{K}u (X)$,
and a weak stability condition $\sigma_{\widetilde{\mathcal{D}}'} = (Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})$ on the categorical resolution $\widetilde{\mathcal{D}}'$ of $\mathcal{K}u(X)$,
such that they are related as follows.
\begin{enumerate}
    \item[(1)] $Z_{\widetilde{\mathcal{A}}} = Z_\mathcal{A} \circ \pi_\ast$;
    \item[(2)] $\mathbf{R}\pi_\ast \colon \widetilde{\mathcal{A}}\longrightarrow \mathcal{A}$ is an exact functor.
\end{enumerate}

Bridgeland stability conditions on triangulated categories, introduced in \cite{Bri07},
have become a central tool in the study of derived categories.
They are now known to exist on curves \cite{Mac07} and on
surfaces \cite{AB13}.
In dimension three, the tilt-stability approach of Bayer--Macr\`i--Toda
\cite{BMT14} and its refinements provides a general framework for constructing
Bridgeland stability conditions,
and has led to existence results in a number of cases (see for instance \cite{BMS16}).

On the other hand, Kuznetsov studied the semiorthogonal decompositions of Fano threefolds and introduced Kuznetsov components
\cite{Kuz04,Kuz05,Kuz09},
which have played a central role in the study of their derived categories and geometry.
It is therefore natural to investigate stability conditions on Kuznetsov components
of Fano threefolds.
In the smooth case this has been carried out in
\cite{BLMS} for all smooth Fano threefolds of Picard rank one.
This leads to the question of what happens in the presence of mild singularities.

In this article, we extend our previous result in \cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \cite{Kuz08b},
and we first address the simplest such case, namely a quadric threefold with a single ordinary double point.

Our main result is the following:

\begin{abstract}
Let $X \subset \mathbb{P}^4$ be a quadric threefold with a single ordinary double point, and let $\mathcal{K}u(X)$ be its Kuznetsov component.
In this paper, we construct a weak stability condition $\sigma_{\widetilde{\mathcal{D}}'}$ on its categorical resolution $\widetilde{\mathcal{D}}' \subset \mathrm{D^b}(\widetilde{X})$,
which is compatible with the Verdier localization $\mathbf{R}\pi_\ast$ and descends to a Bridgeland stability condition on $\mathcal{K}u(X)$.
This can be viewed as a three-dimensional analogue of our previous result in \cite{Cho25}.

We describe the geometry of the blow-up $\pi \colon \widetilde{X} \longrightarrow X$ and obtain two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$,
arising from the projective bundle structure of $\widetilde{X}$ and from Kuznetsov's categorical resolution.
Comparing them, we isolate an admissible subcategory $\widetilde{\mathcal{D}}' \subseteq \mathrm{D^b}(\widetilde{X})$ resolving $\mathcal{K}u(X)$ and show that it admits a full Ext-exceptional collection,
from which we construct $\sigma_{\widetilde{\mathcal{D}}'}$.
\end{abstract}

In this article, we extend our previous result in \cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \cite{Kuz08b},
and we first address the simplest such case, namely a quadric threefold with a single ordinary double point.

In order to construct stability conditions on the derived category of a singular threefold,
it is natural to first work on an appropriate ``resolution''.
For a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \cite{Kuz08b}
together with the construction of Kuznetsov--Shinder \cite{KS24} produce an admissible subcategory
$\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$,
equipped with a functor
$\mathbf{R}\pi_\ast \colon \widetilde{\mathcal{D}} \longrightarrow \mathrm{D^b}(X)$ which is a Verdier localization,
with kernel generated by a single spherical object.

\begin{definition}\label{stability}
A weak stability condition (resp. Bridgeland stability condition) on $\mathcal{C}$ is a pair $\sigma = (Z,\mathcal{H})$ consisting of a group homomorphism (called the central charge of $\sigma$) $Z \colon \Lambda \longrightarrow \mathbb{C}$ and a heart $\mathcal{H}$ of a bounded t-structure on $\mathcal{C}$, such that the following conditions hold:
\begin{enumerate}
 \item[(a)] The composition $Z \circ v \colon {\rm K}(\mathcal{H})={\rm K}(\mathcal{C}) \longrightarrow \Lambda \longrightarrow \mathbb{C}$ is a weak stability function (resp. stability function) on $\mathcal{H}$. This gives a notion of slope: for any $E \in \mathcal{H}$, we set
 \begin{equation*}
 \mu_\sigma(E)=\mu_Z(E):=
 \begin{cases}
 \frac{-\Re Z(E)}{\Im Z(E)}, & \text{if } \Im Z(E)>0, \\
 +\infty, & \text{if } \Im Z(E)=0.
 \end{cases}
 \end{equation*}
 We say that an object $0\neq E \in \mathcal{H}$ is $\sigma$-semistable (resp. $\sigma$-stable) if for every nonzero proper subobject $F \subset E$, we have $\mu_\sigma (F) \leq \mu_\sigma (E)$ (resp. $\mu_\sigma (F) < \mu_\sigma (E)$).

\begin{theorem}[{\normalfont\cite[Theorem 5.8]{KS24}}]\label{KS24}
Let $X$ be a variety of dimension $n \geq 2$ over an algebraically closed
field $k$ of characteristic not equal to $2$ with an ordinary double point
$x_0$ and no other singularities. Let
\[
 \pi : \widetilde{X} = \operatorname{Bl}_{x_0}(X) \longrightarrow X
\]
be the blowup of the singular point, let
$\epsilon \colon E \hookrightarrow \widetilde{X}$ be the embedding of the
exceptional divisor over $x_0$, and let $S$ be a spinor bundle on $E$.
Then the subcategory
$\widetilde{\mathcal{D}}:= \{F \in \mathrm{D^b}(\widetilde{X})|\epsilon^\ast F \in \langle S,\mathcal{O}_{E}\rangle\}$
is admissible in $\mathrm{D^b}(\widetilde{X})$. Moreover,
\begin{enumerate}
 \item the induced functor
 $\mathbf{R}\pi_{*} : \widetilde{\mathcal{D}} \to \mathrm{D^b}(X)$
 is a crepant Verdier localization;
 \item the kernel $\ker(\pi_\ast|_{\widetilde{{\mathcal{D}}}})$ is generated by a single spherical object $K \in \widetilde{\mathcal{D}}$;
 \item when $\dim(X)$ is even, $K=\epsilon_\ast S$ is $2$-spherical,
 and when $\dim(X)$ is odd, $K$ is $3$-spherical and fits into the distinguished triangle $K \longrightarrow \epsilon_\ast S \longrightarrow \epsilon_\ast S'[2]$, where $S'$ is another spinor bundle on $E$.
\end{enumerate}
\end{theorem}

Assume that $\mathcal{N}$ is generated by a single object $A \in\mathcal{C}$ and that $\mathcal{Q}\cap\mathcal{N}$ is a Serre subcategory of $\mathcal{Q}$ (i.e. closed under subobjects, quotients, and extensions).
Then the following are equivalent:
\begin{enumerate}
\item $\mathcal{Q}$ descends to the heart $\mathcal{Q}/(\mathcal{Q}\cap\mathcal{N})$ of a bounded $t$-structure on $\mathcal{C}/\mathcal{N}$.
\item There exists an object $B\in\mathcal{Q}$ such that
$\mathcal{N} = \langle B\rangle$.
\end{enumerate}
In this case, $q\colon \mathcal{Q}\longrightarrow \mathcal{Q}/(\mathcal{Q}\cap\mathcal{N})$ is exact.
\end{lemma}

\begin{proposition}
There exist a weak stability condition $\sigma_{\widetilde{\mathcal{D}}'} = (Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})$ on $\widetilde{\mathcal{D}}'$, satisfying the support property with respect to the quotient lattice $\mathrm{K_{num}}(\widetilde{\mathcal{D}}')/\ker (\pi_\ast|_{\widetilde{\mathcal{D}}'})$,
and a Bridgeland stability condition $\sigma_{\mathcal{K}u(X)} = (Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$,
such that
\begin{enumerate}
 \item $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{D}}' \longrightarrow \mathcal{K}u(X)$ is compatible with the central charges, i.e. $Z_{\widetilde{\mathcal{A}}} = Z_{\mathcal{A}} \circ \pi_\ast$.
 \item $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{A}} \longrightarrow \mathcal{A}$ is an exact functor.

In order to construct stability conditions on the derived category of a singular threefold,
it is natural to first work on an appropriate ``resolution''.
For a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \cite{Kuz08b}
together with the construction of Kuznetsov--Shinder \cite{KS24} produce an admissible subcategory
$\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$,
equipped with a functor
$\mathbf{R}\pi_\ast \colon \widetilde{\mathcal{D}} \longrightarrow \mathrm{D^b}(X)$ which is a Verdier localization,
with kernel generated by a single spherical object.

The category $\widetilde{\mathcal{D}}$ admits an explicit description thanks to the projective bundle structure of $\widetilde{X}$,
and it will serve as the starting point of our construction.

First, we exploit two natural semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$.
The first arises from Orlov’s projective bundle formula for $\tilde{X}\cong \mathbb{P}_E(\mathcal{O}_E \oplus \mathcal{O}_E(-1,-1))$,
which yields a semiorthogonal decomposition of $\mathrm{D^b}(\widetilde{X})$ given by eight exceptional line bundles.
The second decomposition comes from Kuznetsov’s categorical resolution of the nodal quadric.

The key point is that, by performing a sequence of mutations relating these two decompositions,
we exhibit a full Ext-exceptional collection of length $3$ in the categorical resolution
$\widetilde{\mathcal{D}}' \subseteq \widetilde{\mathcal{D}}$ of $\mathcal{K}u(X)$,
which greatly simplifies the construction of stability conditions.

More precisely, we have:
\begin{theorem}
The categorical resolution  $\widetilde{\mathcal{D}}'$ of $\mathcal{K}u(X)$ admits a full Ext-exceptional collection of length $3$.
\end{theorem}

Their extension closure immediately gives a heart $\mathcal{B}$ of bounded $t$-structure.
Nevertheless, it can be seen that no weak stability condition with heart $\mathcal{B}$ descends to $\mathcal{K}u(X)$,
and we therefore find a suitable tilt to construct a new heart $\widetilde{\mathcal{A}}\subset \widetilde{\mathcal{D}}'$,
so that its pushforward $\mathcal{A}:=\mathbf{R}\pi_\ast(\widetilde{\mathcal{A}})$ is a heart on $\mathcal{K}u(X)$ and we can then define a localization-compatible central charge $Z_{\widetilde{\mathcal{A}}}$
and induce a stability condition  $\sigma_{\mathcal{A}}=(Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$.
\subsection{Related work}
Our result fits into recent developments on Bridgeland stability conditions and moduli on Kuznetsov components of Fano varieties.

In the smooth case, stability conditions on Fano threefolds of Picard rank one and their Kuznetsov components are constructed in \cite{Li19} and \cite{BLMS}, respectively.

On the singular side, Kuznetsov and Shinder introduced the notion of categorical absorption of singularities and constructed, under natural hypotheses,
categorical absorptions for projective varieties with isolated ordinary double points \cite{KS24}.
They also studied the derived categories of 1-nodal prime Fano threefolds in \cite{KS25}.

Independently, in \cite{CGL+} the authors studied kernels of categorical resolutions of nodal singularities and proved that for nodal varieties the kernel is generated by a single spherical object, with applications to nodal cubic fourfolds.

Our weak stability condition on the categorical resolution also fits into the framework of partial compactifications of stability spaces by massless objects developed in \cite{BPPW22}. In their terminology it can be viewed as a lax stability condition lying on a boundary stratum,
with massless subcategory corresponding to the  kernel of the Verdier localization.

Finally, in our previous work \cite{Cho25} we constructed Bridgeland stability conditions on a singular surface and its resolution,
and the present paper can be viewed as a threefold counterpart.

\subsection{Organization of the paper}
In Section 2 we review background on stability conditions, hearts of $t$-structures, and tilting.

Section 3 is devoted to the geometry of the nodal quadric threefold $X$ and its resolution $\widetilde{X}$; we describe the projective bundle structure of $\widetilde{X}$ and derive a semiorthogonal decomposition of $\mathrm{D^b}(\widetilde{X})$ by Orlov's formula.

In Section 4, we introduce Kuznetsov’s categorical resolution $\widetilde{\mathcal{D}}$ of $\mathrm{D^b}(X)$ and isolate the admissible subcategory $\widetilde{\mathcal{D}}'$ which resolves the Kuznetsov component $\mathcal{K}u(X)$. We then perform a sequence of mutations, using the projective bundle decomposition from Section 3, to exhibit a full exceptional triple generating $\widetilde{\mathcal{D}}'$.

In Section 5, we use this exceptional collection to construct a new heart $\widetilde{\mathcal{A}}$ on $\widetilde{\mathcal{D}}'$ and prove that it descends to a heart $\mathcal{A}$ on $\mathcal{K}u(X)$.
We also carry out the construction of the stability condition on $\mathcal{K}u(X)$ and prove the main result.

Throughout the paper, we work over the complex numbers $\mathbb{C}$.

In order to prove Theorem~\ref{main1}, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$ equipped with a functor $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{D}}\to\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.

The category $\widetilde{\mathcal{D}}$ is described explicitly using the projective bundle structure of $\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\widetilde{\mathcal{D}}'\subseteq\widetilde{\mathcal{D}}$ of $\mathcal{K}u(X)$, which simplifies the stability construction.

The extension closure of this exceptional collection gives a heart $\mathcal{B}$, but “no weak stability condition with heart $\mathcal{B}$ descends to $\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\widetilde{\mathcal{A}}\subset \widetilde{\mathcal{D}}'$. This is arranged so that its pushforward $\mathcal{A}:=\mathbf{R}\pi_\ast(\widetilde{\mathcal{A}})$ is a heart on $\mathcal{K}u(X)$ and $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{A}}\to\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\widetilde{\mathcal{A}}}$ and induces a stability condition $\sigma_{\mathcal{A}}=(Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$ with $Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ\pi_\ast$.

In order to prove the main theorem, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$ equipped with a functor $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{D}}\to\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.

The category $\widetilde{\mathcal{D}}$ is described explicitly using the projective bundle structure of $\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\widetilde{\mathcal{D}}'\subseteq\widetilde{\mathcal{D}}$ of $\mathcal{K}u(X)$, which simplifies the stability construction.

The extension closure of this exceptional collection gives a heart $\mathcal{B}$, but “no weak stability condition with heart $\mathcal{B}$ descends to $\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\widetilde{\mathcal{A}}\subset \widetilde{\mathcal{D}}'$. This is arranged so that its pushforward $\mathcal{A}:=\mathbf{R}\pi_\ast(\widetilde{\mathcal{A}})$ is a heart on $\mathcal{K}u(X)$ and $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{A}}\to\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\widetilde{\mathcal{A}}}$ and induces a stability condition $\sigma_{\mathcal{A}}=(Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$ with $Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ\pi_\ast$.

Question: Let \(X\) be a 1-nodal quadric threefold, i.e. a quadric threefold with exactly one ordinary double point, and let \(\pi\colon \widetilde{X}\to X\) be the blow-up of the node. Let \(\mathcal{K}u(X)\) be the Kuznetsov component of \(X\), and let \(\widetilde{\mathcal{D}}'\) be a categorical resolution of \(\mathcal{K}u(X)\) inside \(\mathrm{D}^b(\widetilde{X})\). A Bridgeland stability condition or weak stability condition on a triangulated category is a pair \((Z,\mathcal{H})\) consisting of a central charge \(Z\) and the heart \(\mathcal{H}\) of a bounded \(t\)-structure. Which existence statement holds under these assumptions?

There exist a Bridgeland stability condition \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and a weak stability condition \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\) and \(\mathbf{R}\pi_*\colon \widetilde{\mathcal{A}}\to \mathcal{A}\) is an exact functor.

Choices:
B. There exist Bridgeland stability conditions \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\) and \(\mathbf{R}\pi_*\colon \widetilde{\mathcal{A}}\to \mathcal{A}\) is an exact functor.
C. There exist a Bridgeland stability condition \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and a weak stability condition \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\).
D. There exist a Bridgeland stability condition \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and a weak stability condition \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\) and \(\mathbf{R}\pi_*\colon \widetilde{\mathcal{A}}\to \mathcal{A}\) is a faithful functor.
E. There exist a weak stability condition \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and a Bridgeland stability condition \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\) and \(\mathbf{R}\pi_*\colon \widetilde{\mathcal{A}}\to \mathcal{A}\) is an exact functor.

weak-vs-Bridgeland status on the resolution

exactness of \(\mathbf{R}\pi_*\) replaced by faithfulness

assignment of Bridgeland vs weak stability to the two categories


\begin{thm}\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.
\end{thm}

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\subsection{Connections to literature}\label{discuss}

\par Work of Penner-Papadopoulos \cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.
\par In the setting of \cite[Theorem 4.1]{PP},
the support map $\sigma$ produced for the $f$-invariant train track $\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\tau$ could e.g. be mapped by $\sigma \circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\sigma \circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\sigma \circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\tau) \prec \tau$ has irreducible transition matrix while $f(\tau) \prec \tau$ does not.

\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \textit{\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \Cref{mainthm} necessarily produces train tracks which are not generic.
This motivates the following question:

\begin{qn}
    For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par \textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.

\section{Background}\label{background}

\par A \textit{\textbf{strongly connected component}} of a directed graph $G$ is a maximal subset $C$ of the vertices of $G$ such that for any two vertices in $C$, there is a path in $G$ from the first vertex to the second.
If $G$ has only one strongly connected component, it is called \textit{\textbf{strongly connected}}.
A nonnegative square matrix $A$ is called \textit{\textbf{irreducible}} if for every pair of indices $i$, $j$ there is an $n>0$ such that the $(i,j)$-th entry of $A^n$ is positive. $A$ is irreducible if and only if it is the adjacency matrix of a strongly connected graph.
\par
    Given an orientable finite-type surface $S$ without boundary and negative Euler characteristic, a homeomorphism $f\colon S \to S$ is called \textit{\textbf{pseudo-Anosov}} if there is a pair of transverse measured singular foliations $(\Lambda^u,\mu_u)$ and $(\Lambda^s,\mu_s)$, called the \textit{\textbf{unstable}} and \textit{\textbf{stable}} foliations, and a constant $\lambda > 1$, called the \textit{\textbf{stretch factor}}, such that $f(\Lambda^u)=\Lambda^u$, $f(\Lambda^s)=\Lambda^s$ and $f(\mu^u) = \lambda \mu^u$, $f(\mu^s) = \lambda^{-1} \mu^s$. No nontrivial power of a pseudo-Anosov map fixes the homotopy class of any essential closed curve in $S$. A pseudo-Anosov map is a diffeomorphism except at the singular points of $\Lambda^u$.

    \par The \textit{\textbf{mapping torus of $f$}} is the 3-manifold
    $$ M_f = \faktor{(S \times [0,1])}{ (f(p),0) \sim (p,1)} $$ which fibers over the circle with fiber $S$ and monodromy $f$.

    \par If all of the singularities of the foliations are at the punctures of $S$, we say $f$ is \textit{\textbf{fully punctured}}. We will study general pseudo-Anosovs by deleting the singularities of the foliations to obtain a new surface $S^\circ$ on which the restriction of $f$ is fully punctured.
For full background on pseudo-Anosov homeomorphisms, see Fathi--Laudenbach--Poénaru \cite{FLP}.

\subsection{Train tracks}\label{tts}
\par A \textit{\textbf{train track}} $\tau$ is a closed 1-complex embedded in a surface $S$ with a ``smoothing" at each vertex so that $\tau$ has a well-defined tangent space at each vertex, and $S \setminus \tau$ contains no nullgons, unpunctured monogons or unpunctured bigons. See Penner-Harer \cite{PH}. The vertices of $\tau$ are called \textit{\textbf{switches}} and the edges are called \textit{\textbf{branches}}. If the switches of $\tau$ all have degree three, $\tau$ is called \textit{\textbf{generic}}.

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
    \begin{enumerate}
        \item $\sigma$ is homotopic to the identity,
        \item $\sigma(\tau_1) \subseteq \tau_2$,
        \item for all points $x$ in $\tau_1$,
    $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
    \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$  is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
    \par
    Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
    the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
    $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
    A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
    A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
    except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
    If there is a $\tau_1 \prec_\phi \tau_2$
    we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
    \begin{figure}[h]
    \centering
    \includegraphics[width=0.66\textwidth]{foldgood.pdf}
    \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
    \label{fig:fold}
\end{figure}
    \par
    A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
    If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\begin{thm}[\protect{\cite[Theorem 3.5]{A}}]\label{ttfs} Given $f\colon S \to S$ pseudo-Anosov, there exists a generic invariant train track $\tau$ for $f$ such that $f(\tau)$ folds to $\tau$, i.e. there exist a finite sequence of train tracks
$$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau,$$
where $\tau_i$ is carried by $\tau_{i+1}$ with support map given by a single fold.
\end{thm}

\label{intro}
Given a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.
The existence of invariant train tracks with irreducible transition matrix
allows the application of Perron-Frobenius theory to the study of the properties of these maps.
For example, when an invariant train track is irreducible it can be used to build a Markov partition for the map.
The existence of irreducible invariant train tracks is often taken for granted
(See e.g. \cite{FR}, \cite{F}, \cite{J}, \cite{EF}, \cite{M}, \cite{Pe})
but to our knowledge no complete proof is currently in the literature. See \Cref{discuss} for more details.

\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.
One important datum attached to a veering triangulation is its \textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \cite{LMT1}.
When the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with
irreducible transition matrices.
\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component" from which there are paths to every vertex, and some number of ``infinitesimal cycles" which do not have any paths back to the rest of the graph \cite{AT}.
By studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\begin{qn}
 For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
 \begin{enumerate}
 \item $\sigma$ is homotopic to the identity,
 \item $\sigma(\tau_1) \subseteq \tau_2$,
 \item for all points $x$ in $\tau_1$,
 $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
 \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$ is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
 \par
 Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
 the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
 $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
 A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
 A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
 except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
 If there is a $\tau_1 \prec_\phi \tau_2$
 we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
 \begin{figure}[h]
 \centering
 \includegraphics[width=0.66\textwidth]{foldgood.pdf}
 \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
 \label{fig:fold}
\end{figure}
 \par
 A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
 If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\par Note that the referenced theorem is stated in terms of \textit{splits} rather than folds, which are the combinatorial inverse of folds.
 Additionally, the referenced theorem is stronger than what we state here because the splitting sequence is shown to be canonical up to ``commuting maximal splits": at each step in the sequence, the train track is split at a branch carrying maximal measure of the invariant lamination, and the splits of maximal branches commute if there is not a unique branch carrying maximal measure.
 This theorem motivates the following definition:
\begin{defn}
 An invariant train track $\tau$ for a pseudo-Anosov map $f$ is called \textit{\textbf{veering}} if it
 is generic and $\tau$ can be obtained from $f(\tau)$ by a sequence of folds, in which case there is a sequence of train tracks
 $$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau$$
 where $\tau_i$ is carried by $\tau_{i+1}$ with support map consisting of a single fold.
\end{defn}
\par In \Cref{clvts} we will see that a veering train track as defined here is sufficient to produce the veering triangulation of the fully punctured mapping torus of $f$. While veering train tracks are not necessarily irreducible, they will be the beginning of our construction of irreducible tracks.
\par
 Given two train tracks $\tau_1$, $\tau_2$ on $S$, a (surface) \textit{\textbf{train track map}} is a map $t: S \to S$ so that $t(\tau_1)$ is contained in $\tau_2$, and the restriction of $t$ to $\tau_1$ is
 such that for any train path $p\colon [0,1] \to \tau_1$, $t \circ p\colon [0,1] \to \tau_2$ is also a train path. In other words, a train track map is like a combinatorial support map except that it does not have to be homotopic to the identity map.
 \par The following lemma will be useful for producing support maps:
\begin{lemma}\label{supplemma}
 Let $t\colon S \to S$ be a train track map taking $\tau_1$ to $\tau_2$, and $g\colon S \to S$ be a diffeomorphism.
 If $t$ is homotopic to $g$, then $g(\tau_1) \prec \tau_2$ with support map given by $t \circ g^{-1}$.
\end{lemma}
\begin{proof}
 Since $t$ and $g$ are homotopic, $t \circ g^{-1}$ is homotopic to the identity. Since $t(\tau_1) \subseteq \tau_2$, $(t \circ g^{-1})(g(\tau_1)) = t(\tau_1) \subseteq \tau_2$. Let $p \in g(\tau_1)$. Since $g$ is a diffeomorphism, the restriction of the differential
 $$d_p g^{-1}_{|g(\tau_1)}\colon T_p g(\tau_1) \to T_{g^{-1}(p)} \tau_1$$
 is an isomorphism. Since $t$ is a train track map, $d_{g^{-1}(p)} t_{| \tau_1}$ is an isomorphism, so the composition
 $$d_p (t \circ g^{-1})_{|g(\tau_1)} = (d_{g^{-1}(p)} t_{| \tau_1}) \circ (d_p g^{-1}_{|g(\tau_1)})\colon T_p g(\tau_1) \to T_{(t \circ g^{-1})(p)} \tau_2$$
 is also an isomorphism.
\end{proof}

Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\colon S\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.

\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.,

Question: Let \(f\colon S\to S\) be a pseudo-Anosov homeomorphism. A train track \(\tau\) on \(S\) is called invariant for \(f\) if \(f(\tau)\) is carried by \(\tau\) via a combinatorial support map. If the branches of \(f(\tau)\) are \(\{b^1_j\}\) and the branches of \(\tau\) are \(\{b^2_i\}\), the associated transition matrix is the nonnegative matrix \(T=[t_{i,j}]\), where \(t_{i,j}\) is the number of times the image of \(b^1_j\) under the support map passes over \(b^2_i\). A nonnegative matrix is irreducible if for every \(i,j\) there exists \(n\ge 1\) such that \((T^n)_{i,j}>0\). Under these assumptions, which existence statement holds?

For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.

Choices:
B. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), every invariant train track \(\tau\) for \(f\) has the property that the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.
C. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that the transition matrix associated to \(f(\tau)\prec \tau\) is nonzero.
D. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that for every pair of branches \(b^2_i,b^2_j\) of \(\tau\), one has \(t_{i,j}>0\) in the transition matrix associated to \(f(\tau)\prec \tau\).
E. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that each strongly connected component of the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.

existential choice of modified track after contracting obstructing branches

irreducibility requirement dropped to mere existence of a nontrivial transition matrix

irreducible replaced by strictly positive one-step transition matrix

infinitesimal-cycle obstruction bypassed only after contracting branches, not for the original component decomposition