\begin{example}\label{simple-ex}
Let $A$ be a finite simple group. The Schur multiplier $H_2(A;\mathbb{Z})$ has odd order precisely in the following cases:
\begin{itemize}
\item[(1).] Cyclic groups: $\mathbb{Z}/p$ with $p$ a prime number;
\item[(2).] Groups of Lie type (with $q=p^k$ for a prime number $p$ and positive $k$):
  \begin{itemize}
  \item Chevalley groups:
     \begin{itemize}
     \item[(Ch1).] linear groups $A_n(q)$ with $(n+1,q-1)$ odd and $(n, q)\neq (1, 4)$, $(2, 2)$, $(2, 4)$, $(3, 2)$;
     \item[(Ch2).] orthogonal groups $B_n(q)$ with $n>1$, $q$ even and $(n, q)\neq (2, 2)$, $(3, 2)$;
     \item[(Ch3).] symplectic groups $C_n(q)$ with $n>2$, $q$ even and $(n, q)\neq (3, 2)$;
     \item[(Ch4).] orthogonal groups $D_n(q)$ with $n>3$, $(4,q^n-1)=1$ and $(n, q)\neq (4, 2)$;
     \item[(Ch5).] $E_6(q)$;
     \item[(Ch6).] $E_7(q)$ with $q$ even;
     \item[(Ch7).] $E_8(q)$;
     \item[(Ch8).] $F_4(q)$ with $q\neq 2$;
     \item[(Ch9).] $G_2(q)$ with $q\neq 4$;
     \end{itemize}
 \item Steinberg groups:
   \begin{itemize}
     \item[(S1).] unitary groups ${}^{2}A_n(q^2)$ with $n>1$, $(n+1,q+1)$ odd and $(n, q)\neq (3, 2)$, $(5,2)$;
     \item[(S2).] orthogonal groups ${}^{2}D_n(q^2)$ with $n>3$, $(4,q^n+1)=1$;
     \item[(S3).] ${}^{2}E_6(q^2)$ with $q\neq 2$;
     \item[(S4).] ${}^{3}D_4(q^3)$;
     \end{itemize}
 \item Suzuki groups: ${}^{2}B_2(2^{2n+1})$ with $n\geq 2$;
  \item Ree groups and Tits group:
    \begin{itemize}
     \item[(R1)] Ree groups ${}^{2}F_4(2^{2n+1})$ with $n\geq 1$;
     \item[(R2)] Tits group ${}^{2}F_4(2)'$;
     \item[(R3)] Ree groups ${}^{2}G_2(3^{2n+1})$ with $n\geq 1$;
     \item[(R4)] the derived group ${}^{2}G_2(3)'$;
     \end{itemize}
     \end{itemize}
\item[(3).] Sporadic groups:
  \begin{itemize}
  \item Mathieu groups:
     \begin{itemize}
     \item[(M1)] $M_{11}$;
     \item[(M2)] $M_{23}$;
     \item[(M3)] $M_{24}$;
     \end{itemize}
  \item Janko groups:
     \begin{itemize}
     \item[(J1)] $J_{1}$;
     \item[(J2)] $J_{3}$;
     \item[(J3)] $J_{4}$;
     \end{itemize}
  \item Conway groups:
     \begin{itemize}
     \item[(C1)] $Co_{2}$;
     \item[(C2)] $Co_{3}$;
     \end{itemize}
  \item Fischer groups:
    \begin{itemize}
     \item[(F1)] $Fi_{23}$;
     \item[(F2)] $Fi_{24}'$;
     \end{itemize}
  \item McLaughlin group: $McL$;
  \item Held group: $He$;
  \item O'Nan group: $O'N$;
  \item Harada-Norton group: $HN$;
  \item Lyons group: $Ly$;
  \item Thompson group: $Th$;
  \item Fischer-Griess Monster group, $M$.
   \end{itemize}
\end{itemize}
\end{example}

\begin{example}\label{Xio-ex}
Theorem \ref{spinc-thm-intro} also recovers \cite[Theorem 1.2]{Xio16}, where $L$ is assumed to be a simply connected $2$-step nilpotent Lie group with ${\rm dim}[L,L] = 1$ and $G$ acts trivially on the center of $L$.
Indeed, as shown in the proof of \cite[Theorem 1.2]{Xio16} there, the classifying map $\varphi: M\larrow BSO(n)$ factors as
\[
\varphi: M\stackrel{\eta}{\larrow} BG\stackrel{}{\larrow} BU(m)\stackrel{}{\larrow} BSO(n)
\]
for a suitable unitary group $U(m)$. Consequently, $M$ is spin$^c$ and Theorem \ref{spinc-thm-intro} applies to show that $M$ bounds.
\end{example}

A closed manifold \( M \) is said to be \emph{almost flat} if, for every \( \epsilon > 0 \), there exists a Riemannian metric \( g_\epsilon \) on \( M \) such that
\[
\operatorname{diam}(M, g_\epsilon) \le 1, \qquad |K_{g_\epsilon}| < \epsilon ,
\]
where \( K_{g_\epsilon} \) denotes the sectional curvature of \( g_\epsilon \).
Thus, \( M \) admits metrics of arbitrarily small curvature and uniformly bounded diameter. For each dimension \( n \), there exists a constant \( \epsilon_n > 0 \) with the property that if an \( n \)-dimensional manifold admits an \( \epsilon_n \)-flat metric of diameter at most one, then it is almost flat \cite{CG86}.

A classical theorem of Gromov~\cite{Gro78} asserts that every almost flat manifold is finitely covered by a nilmanifold. Conversely, Farrell and Hsiang~\cite{FH83} proved that any manifold finitely covered by a nilmanifold is homeomorphic to an almost flat manifold. Ruh~\cite{Ruh82} subsequently strengthened Gromov’s result, showing that every almost flat manifold is in fact diffeomorphic to an infranilmanifold.

A long-standing conjecture, proposed independently by Farrell--Zdravkovska~\cite{FZ83} and Yau~\cite{Yau93}, predicts that every almost flat manifold bounds a compact manifold.
In the special case of flat manifolds, this was established by Hamrick and Royster~\cite{HR82} in a striking theorem.
More recently, Davis and Fang~\cite{DF16} confirmed the conjecture under the additional assumption that the \(2\)-Sylow subgroup of the holonomy group is cyclic or generalized quaternionic; their result was later rederived by Xiong~\cite{Xio16} through a different argument.
In particular, the theorem of Davis and Fang encompasses earlier advances due to Farrell--Zdravkovska~\cite{FZ83} and Upadhyay~\cite{Upa01}.

The general case of the conjecture remains open. Moreover, Davis and Fang~\cite{DF16} pointed out that determining whether every almost flat \emph{spin} manifold, possibly after changing the spin structure, bounds a spin manifold is a subtle and difficult problem. It is therefore natural, and of considerable interest, to ask the corresponding question for almost flat \(\mathrm{spin}^c\) manifolds.

The main result of this paper provides a confirmation of the Farrell--Zdravkovska--Yau conjecture when the almost flat manifold is spin\(^c\) as follows.

The main result of this paper provides a confirmation of the Farrell--Zdravkovska--Yau conjecture when the almost flat manifold is spin\(^c\) as follows.

The existence of a spin\(^c\) structure on an almost flat manifold is intimately connected with the nature of its holonomy group (which is finite).
We now consider a particular instance of this relationship and obtain the following theorem as a consequence of Theorem~\ref{spinc-thm-intro}.
Recall that the classical \emph{Schur multiplier} of a group \(G\) is its second group homology \(H_2(G;\mathbb{Z})\), introduced by Issai Schur in 1904. It is clear that the Schur multiplier of a finite group is finite.

\begin{theorem}\label{G-thm-intro}
Let \(M\) be an almost flat oriented manifold with holonomy group \(G\).
If the Schur multiplier of \(G\) is of odd order, then \(M\) bounds a compact manifold.

In dimension 4, we can give a complete verification of the Farrell--Zdravkovska--Yau conjecture, both in the orientable and the spin cases.
\begin{theorem}\label{4-thm-intro}
Let \(M\) be an almost flat orientable \(4\)-manifold. Then \(M\) bounds a spin\(^c\) manifold. Furthermore, if \(M\) is spin, then \(M\) with any spin structure bounds a spin manifold.
\end{theorem}

\begin{lemma}\label{sw-lemma}
Let $(M, c)$ be an $n$-dimensional almost flat spin$^c$ manifold. Then its Stiefel-Whitney numbers all vanish.
\end{lemma}
\begin{proof}
Since the rational Pontryagin classes of the almost flat manifold $M$ all vanish by Lemma \ref{chern-lemma}, each integral spin$^c$ Wu classes $\mu _{2i}^{Spin^c}(M)=a_i c^i$, where $a_i$ is the coefficient of $c^i$ in the expression of the universal integral spin$^c$ Wu class $\mu _{2i}^{Spin^c}$. Hence, any integral spin$^c$ Wu number vanishes if $n$ is odd, or is of the form $a_I c^{m}$ with $I=\{j_1,\ldots, j_r\}$ a partition of $m$ if $n=2m$ is even. In the latter case, recall the classical Legendre formula
$$m!=2^{m-s_2(m)}\cdot (\text{an odd number})$$
 and the fact $s_2(m)\leq s_2(j_1)+\cdots+s_2(j_r)$. Then by Lemmas \ref{denom-lemma} and \ref{index-lemma},
\[
\langle a_I c^{m}, [M]\rangle \in 2^m m! a_I \mathbb{Z}\subseteq 2^{s_2(j_1)+\cdots+s_2(j_r)}m! \mathbb{Z} \subseteq
2^{m}\mathbb{Z}.
\]
In particular, the mod-$2$ reduction of any integral spin$^c$ Wu number of $M$ vanishes, or equivalently, the mod-$2$ Wu numbers of $M$ all vanish. Since Stiefel-Whitney numbers are equivalent to mod-$2$ Wu numbers, the Stiefel-Whitney numbers of $M$ all vanish as well.
\end{proof}

The following proposition provides a simple criterion when almost flat spin$^c$ manifold bounds.
\begin{proposition}\label{cn=0-prop}
Let $(M, c)$ be an $n$-dimensional almost flat spin$^c$ manifold. Then $(M, c)$ bounds a spin$^c$ manifold if and only if $\langle c^{\lfloor \frac{n}{2} \rfloor }, [M]\rangle =0$.
\end{proposition}
\begin{proof}
By Lemma \ref{spinc-bord-lemma}, $(M, c)$ bounds a spin$^c$ manifold if and only if the Stiefel-Whitney numbers and $c$-Pontryagin numbers all vanish. By Lemma \ref{sw-lemma}, its Stiefel-Whitney numbers all vanish. By Lemma \ref{chern-lemma}, only the number $\langle c^{\lfloor \frac{n}{2} \rfloor }, [M]\rangle$ can be nontrivial among all the $c$-Pontryagin numbers. Hence, the proposition follows.
\end{proof}

\begin{remark}\label{nasen}
 The proof in fact yields a more general result than Theorem \ref{spinc-thm-intro}. Indeed, we show that if a spin$^c$ manifold $M$ has all rational Pontryagin classes vanishing, then it bounds an orientable manifold. Moreover, if $n$ is odd, then $M$, equipped with any spin$^c$ structure, bounds a spin$^c$ manifold.
 \end{remark}

The following proposition recovers the main theorem of \cite{DF16} for the {\em oriented case}.
\begin{proposition}\label{DFprop}
Let $M$ be an $n$-dimensional almost flat oriented manifold. If the $2$-Sylow subgroup ${\rm Syl}_2(G)$ of the holonomy group $G$ of $M$ is cyclic or generalized quaternionic, then $M$ bounds an orientable manifold.
\end{proposition}
\begin{proof}
The assumption that ${\rm Syl}_2(G)$ is cyclic or generalized quaternionic implies $H^3({\rm Syl}_2(G);\mathbb{Z})=0$ by Lemma \ref{DFlemma}, and then implies that $M$ is spin$^c$ by Lemma \ref{H3lemma}. Hence, $M$ bounds by Theorem \ref{spinc-thm-intro}.
\end{proof}

\begin{lemma}\label{denom-lemma}
In the expression of the universal integral spin$^c$ Wu class $\mu_{2n}^{Spin^c}$, the denominator of the coefficient $a_n$ of $c^{n}$ satisfies
\[
{\rm denom}~(a_n)~|~2^{n-s_2(n)},
\]
where $s_2(n)$ denotes the sum of the binary digits of $n$.
\end{lemma}

\begin{theorem}\label{spinc-thm-intro}
Let $M$ be an \(n\)-dimensional almost flat spin\(^c\) manifold. Then $M$ bounds an orientable manifold. Moreover, if \(n\) is odd, then $M$ with any spin$^c$ structure bounds a spin\(^c\) manifold.
\end{theorem}

The existence of a spin\(^c\) structure on an almost flat manifold is intimately connected with the nature of its holonomy group (which is finite).
We now consider a particular instance of this relationship and obtain the following theorem as a consequence of Theorem~\ref{spinc-thm-intro}.
Recall that the classical \emph{Schur multiplier} of a group \(G\) is its second group homology \(H_2(G;\mathbb{Z})\), introduced by Issai Schur in 1904. It is clear that the Schur multiplier of a finite group is finite.

\begin{theorem}\label{G-thm-intro}
Let \(M\) be an almost flat oriented manifold with holonomy group \(G\).
If the Schur multiplier of \(G\) is of odd order, then \(M\) bounds a compact manifold.

In particular, if the \(2\)-Sylow subgroup of \(G\) is cyclic, generalized quaternionic, or coincides with the \(2\)-Sylow subgroup of any finite simple group listed in Example~\ref{simple-ex}, then \(M\) is the boundary of a compact manifold.
\end{theorem}

Theorem~\ref{G-thm-intro} combined with Theorem \ref{spinc-thm-intro} extends the result of Davis and Fang~\cite{DF16} for almost flat {\em oriented} manifolds. Additionally, as noted in Example~\ref{Xio-ex}, the result of Xiong~\cite{Xio16} appears as a special case of Theorem~\ref{spinc-thm-intro}.

\medskip

In dimension 4, we can give a complete verification of the Farrell--Zdravkovska--Yau conjecture, both in the orientable and the spin cases.
\begin{theorem}\label{4-thm-intro}
Let \(M\) be an almost flat orientable \(4\)-manifold. Then \(M\) bounds a spin\(^c\) manifold. Furthermore, if \(M\) is spin, then \(M\) with any spin structure bounds a spin manifold.
\end{theorem}

Ontaneda~\cite{Ont20} established an important converse to the classical Cheeger--Fukaya--Gromov theorem~\cite{CFG92}:
if an almost flat manifold \(M\) bounds, then there exists a compact manifold \(W\) with boundary \(M\) such that \(W \setminus M\) carries a complete, finite-volume Riemannian metric with negatively pinched sectional curvature.

Farrell and Zdravkovska~\cite{FZ83} went further to conjecture that an almost flat (respectively, flat) manifold bounds a compact manifold whose interior admits a complete, finite-volume metric with negative (respectively, constant negative) sectional curvature.
In the flat case this conjecture was disproved by Long and Reid~\cite{LR20}, while in the almost flat case it was verified by Davis and Fang~\cite{DF16} when the holonomy group is cyclic or quaternionic.

In the present work, combining Theorem~\ref{spinc-thm-intro} with Ontaneda’s theorem~\cite{Ont20}, we confirm the conjecture for almost flat spin\(^c\) manifolds.

\begin{theorem}
An almost flat spin\(^c\) manifold bounds a compact manifold whose interior admits a complete, finite-volume Riemannian metric with negatively pinched sectional curvature.
\end{theorem}

$\, $

Our approach to the proofs of the foregoing theorems departs in a substantial way from earlier work. Whereas infranilmanifolds have traditionally been studied largely through group-theoretic methods, we adopt a more topological point of view. The central tool in our proof is the integral Wu class for
spin\(^c\)
 manifolds, developed in Section~\ref{sec: spincWu}. This construction is inspired by the integral spin Wu class introduced by Hopkins and Singer~\cite{HS05}. The guiding idea is that, while the Stiefel–Whitney classes themselves are only indirectly related to geometry, the integral lift of Wu classes admits a more transparent geometric interpretation. As a result, questions concerning the vanishing of Stiefel–Whitney numbers may be reduced to parity properties of these integral Wu classes.

The essential point in the proof of the main theorem for
spin\(^c\) almost flat manifolds is that the Atiyah–Singer index theorem yields strong divisibility properties for the top power of the
spin\(^c\) characteristic class $c$. This divisibility, in turn, guarantees the evenness of the integral Wu numbers, thereby forcing the vanishing of the relevant Stiefel–Whitney numbers.

$\, $

The paper is organized as follows.
In Section~\ref{sec: spincWu}, we recall the integral spin Wu class of Hopkins and Singer and introduce the corresponding integral Wu class for spin\(^c\) vector bundles, establishing along the way an estimate (Lemma~\ref{denom-lemma}) that will be used later.
Section~\ref{sec: spincbound} contains the proof of our main theorem, Theorem~\ref{spinc-thm-intro}, which confirms the Farrell--Zdravkovska--Yau conjecture for almost flat spin\(^c\) manifolds.
In Section~\ref{sec: holo}, we derive Theorem~\ref{G-thm-intro} as an application of Theorem~\ref{spinc-thm-intro}.
Finally, Section~\ref{sec: 4} is devoted to the 4-dimensional case, where Theorem~\ref{4-thm-intro} is proved.

\bigskip

\noindent{\bf Acknowledgements.}
Fei Han was partially supported by the grant AcRF A-8000451-00-00 from National University of Singapore. Ruizhi Huang was supported in part by the National Natural Science Foundation of China (Grant nos. 12331003 and 12288201), the National Key R\&D Program of China (No. 2021YFA1002300) and the Youth Innovation Promotion Association of Chinese Academy Sciences. Weiping Zhang was partially supported by NSFC Grant No. 11931007, National Key R\&D Program of China (2024YFA1013202) and Nankai Zhide Foundation.

The authors would like to thank Haibao Duan for helpful discussions on integral Wu classes. Ruizhi Huang also thanks Xiaolei Wu for valuable conversations on group homology, and Nansen Petrosyan for pointing out Remark \ref{nasen}.

The post-theorem introduction indicates that the proof of Theorem~\ref{spinc-thm-intro} uses a topological approach whose "central tool" is the integral Wu class for spin$^c$ manifolds (introduced in Section~\ref{sec: spincWu}), motivated by Hopkins--Singer’s integral spin Wu class. The "guiding idea" is that the integral lift of Wu classes has a clearer geometric interpretation, so that "questions concerning the vanishing of Stiefel--Whitney numbers may be reduced to parity properties of these integral Wu classes." The "essential point" is that the Atiyah--Singer index theorem gives "strong divisibility properties" for the top power of the spin$^c$ characteristic class $c$; this divisibility "guarantees the evenness of the integral Wu numbers," which in turn "forc[es] the vanishing of the relevant Stiefel--Whitney numbers." The paper roadmap adds that Section~\ref{sec: spincbound} contains the proof of Theorem~\ref{spinc-thm-intro}.

The post-theorem introduction indicates that, in establishing the main theorem, the proof uses a topological approach whose "central tool" is the integral Wu class for spin$^c$ manifolds (introduced next), motivated by Hopkins--Singer’s integral spin Wu class. The "guiding idea" is that the integral lift of Wu classes has a clearer geometric interpretation, so that "questions concerning the vanishing of Stiefel--Whitney numbers may be reduced to parity properties of these integral Wu classes." The "essential point" is that the Atiyah--Singer index theorem gives "strong divisibility properties" for the top power of the spin$^c$ characteristic class $c$; this divisibility "guarantees the evenness of the integral Wu numbers," which in turn "forc[es] the vanishing of the relevant Stiefel--Whitney numbers." The paper roadmap adds that later we give the proof of the main theorem.

Question:
Let \(M\) be an \(n\)-dimensional almost flat spin\(^c\) manifold, meaning that \(M\) is a closed manifold admitting a spin\(^c\) structure and, for every \(\epsilon>0\), there exists a Riemannian metric \(g_\epsilon\) on \(M\) such that
\[
\operatorname{diam}(M,g_\epsilon)\le 1,\qquad |K_{g_\epsilon}|<\epsilon,
\]
where \(K_{g_\epsilon}\) denotes the sectional curvature. Which of the following conclusions holds for \(M\)?

\(M\) is the boundary of a compact orientable manifold. Moreover, if \(n\) is odd, then for any chosen spin\(^c\) structure on \(M\), there exists a compact spin\(^c\) manifold whose induced boundary spin\(^c\) structure is that given one on \(M\).

Choices:
B. \(M\) is the boundary of a compact orientable manifold. Moreover, for any \(n\) and for any chosen spin\(^c\) structure on \(M\), there exists a compact spin\(^c\) manifold whose induced boundary spin\(^c\) structure is that given one on \(M\).
C. If \(n\) is odd, then for any chosen spin\(^c\) structure on \(M\), there exists a compact spin\(^c\) manifold whose induced boundary spin\(^c\) structure is that given one on \(M\).
D. \(M\) is the boundary of a compact orientable manifold if and only if, when \(n\) is odd, for any chosen spin\(^c\) structure on \(M\), there exists a compact spin\(^c\) manifold whose induced boundary spin\(^c\) structure is that given one on \(M\).
E. \(M\) is the boundary of a compact spin\(^c\) manifold. Moreover, if \(n\) is odd, then there exists a choice of spin\(^c\) structure on \(M\) for which some compact spin\(^c\) manifold induces that boundary spin\(^c\) structure.

odd-dimensional restriction for spin^c bounding with any chosen structure

dropped the unconditional orientable-bounding conclusion

logical linkage between unconditional orientable bounding and the odd-dimensional spin^c enhancement

replaced orientable bounding by unconditional spin^c bounding and weakened quantifier from any structure to some structure

The stem gives only the definition of an almost flat spin^c manifold and does not explicitly or implicitly reveal the odd-dimensional restriction or the precise bounding conclusion.

The item is close to a direct theorem-recall question: it essentially asks for the correct conclusion under the theorem's hypotheses. The nearby variants add some comparison, but it is still largely a reformulation of the result.

Some reasoning is needed to distinguish the exact scope of the conclusion, especially the odd-dimensional spin^c-bounding clause versus stronger or weaker alternatives. However, it mainly tests precise recall of the statement rather than deeper generative reasoning.

Most distractors are plausible overgeneralizations or weakened variants, which is good. However, choice C is a weaker statement that is still true, so it functions poorly as a distractor and creates ambiguity in a single-answer format.

Good at avoiding answer leakage, but the question is mostly theorem-statement recognition rather than substantive reasoning. Its main flaw is the inclusion of a weaker true option, which weakens distractor quality and makes the item somewhat ambiguous.


\label{thm:twistor}
The twistor bundle $q:\mathbb{C} P^5\to \mathbb{H} P^2$ has
\[
2= \mathsf{secat}(q)=\mathrm{Swgt}(q;\F_2)>\mathrm{Mwgt}(q;\F_2)=1>\mathrm{wgt}(q;\F_2)=0.
\]

\begin{thm}\label{thm:twistor}
The twistor bundle $q:\C P^5\to \Ha P^2$ has
\[
2= \secat(q)=\Swgt(q;\F_2)>\Mwgt(q;\F_2)=1>\wgt(q;\F_2)=0.
\]
\end{thm}

\label{sec:intro}

The {\em sectional category} of a fibration $q:E\to B$, denoted $\mathsf{secat}(q)$, is defined to be the minimal integer $k$ such that $B$ admits a cover by open sets $U_0,\ldots , U_k$, on each of which $q$ admits a continuous local section $s_i:U_i\to E$ \cite{Schwarz,James}. Prominent special cases include the Lusternik--Schnirelmann category $\mathsf{cat}(X)$ \cite{CLOT}, which is the sectional category of the based path fibration $P_0 X\to X$, and Farber's topological complexity ${\sf TC}(X)$ \cite{Far03,Far04}, which is the sectional category of the free path fibration $PX\to X\times X$, as well as its many variants \cite{FG07,Rudyak,CFW}. Note that here we use the normalized version, for which $\mathsf{secat}(q)=0$ if and only if $q$ admits a (global) section.

An important foundational result due to Schwarz \cite{Schwarz} (see also \cite{James}) states that if $B$ is paracompact then $\mathsf{secat}(q)$ agrees with the minimal $k$ such that the $(k+1)$-fold fibred join $q(k):E(k)\to B$ admits a section. If $F$ is the fibre of $q$, then the fibre of $q(k)$ is $F(k)=F\ast \cdots \ast F$, the iterated join of $(k+1)$ copies of $F$. When $F$ is $(r-1)$-connected, $F(k)$ is $(r+1)(k+1)-2$-connected. Then (assuming $B$ is a CW complex) cellular obstruction theory gives an upper bound
\[
\mathsf{secat}(q)< \frac{\mathrm{dim}(B)+1}{r+1}.
\]

A standard cohomological argument using the open sets definition gives that $\mathsf{secat}(q)$ is bounded below by $\operatorname{nil}\operatorname{ker} (q^*:H^*(B;R)\to H^*(E;R))$, the nilpotency of the kernel of the induced map in cohomology. Here we take coefficients in an arbitrary commutative ring $R$ (local coefficients may also be used). This bound can sometimes be improved upon using the fibred join interpretation. In particular, we define following Farber--Grant \cite{FG07,FG08} the \emph{sectional category weight} of $q$ to be
\[
\mathrm{wgt}(q;R)=\min \{k \mid q(k)^*:H^*(B;R)\to H^*(E(k);R)\mbox{ is injective}\}.
\]
Then (since a section $s:B\to E(k)$ of $q(k)$ clearly implies injectivity of $q(k)^*$) one has $\mathsf{secat}(q)\geq\mathrm{wgt}(q;R)$. It can be shown that $\mathrm{wgt}(q;R)\geq \operatorname{nil}\operatorname{ker} (q^*:H^*(B;R)\to H^*(E;R))$, and examples of strict inequality are known.

A further improvement to this lower bound was found by Iwase and Kono \cite{IwaseKono}. They observed that a section $s$ of $q(k)$, in addition to implying injectivity of the induced map $q(k)^*$, would entail a retraction $s^*: H^*(E(k);R)\to H^*(B;R)$ which commutes with all cohomology operations. They therefore defined the \emph{module weight} of $q$ at the prime $p\ge2$ to be
\begin{multline*}
\mathrm{Mwgt}(q;\F_p)=\min \{k \mid q(k)^*:H^*(B;\F_p)\to H^*(E(k);\F_p)\mbox{ admits}\\
\mbox{a retraction in the category of $\mathcal{A}_p$-modules}\}.
\end{multline*}
Here $\mathcal{A}_p$ denotes the mod $p$ Steenrod algebra. Clearly, one has inequalities
\[
\mathsf{secat}(q)\ge \mathrm{Mwgt}(q;\F_p) \ge \mathrm{wgt}(q;\F_p)\ge \operatorname{nil}\operatorname{ker}(q^*: H^*(B;\F_p)\to H^*(E;\F_p))
\]
for each prime $p$. Examples of strict inequality $\mathrm{Mwgt}(q;\F_p) > \mathrm{wgt}(q;\F_p)$ can be found in \cite{IwaseKono, Choi09, Choi15}.

The ideas in the previous two paragraphs originated in the work of Fadell and Husseini \cite{FH} on category weight, and were developed in the context of $q$ being the based path fibration $P_0 X\to X$ by Rudyak \cite{Rudyak99} and Strom \cite{Strom}. We have given here the straightforward generalisation to arbitrary fibrations $q: E\to B$.

In this paper we will improve on the module weight using secondary cohomology operations. Recall that whenever stable cohomology operations $\theta$ and $\varphi$ satisfy a relation $\varphi\circ\theta\equiv 0$, this gives rise to a \emph{secondary cohomology operation}
\[
\Phi=\Phi(\varphi,\theta) : \operatorname{ker} \,\theta \to \mathrm{coker} \,\varphi
\]
with \emph{indeterminacy} $\operatorname{im} \varphi$. Such secondary operations are natural, in the following sense: if $f: X\to Y$ is a map of spaces, then the diagram
 \[
 \xymatrix{
 \operatorname{ker}\, \theta_Y \ar[r]^-{\Phi} \ar[d]_{f^*} & \mathrm{coker}\, \varphi_Y \ar[d]^{f^*} \\
 \operatorname{ker}\, \theta_X \ar[r]^-{\Phi} & \mathrm{coker}\, \varphi_X
 }
 \]
 commutes. (This makes sense because $f^* (\operatorname{ker}\,\theta_Y)\subseteq \operatorname{ker}\,\theta_X$ and $f^*(\operatorname{im} \varphi_Y)\subseteq \operatorname{im} \varphi_X$.) If we consider only stable cohomology operations with $\F_p$-coefficients, we can say that $f^*:H^*(Y;\F_p)\to H^*(X;\F_p)$ is a map of $\A_p$-modules which commutes with all secondary operations.  This leads us to our main definition.

\begin{defn}\label{def:Swgt}
The \emph{secondary module weight} of the fibration $q:E\to B$ at the prime $p$ is
 \begin{multline*}
 \mathrm{Swgt}(q;\F_p) := \min \{k \mid q(k)^*:H^*(B;\F_p)\to H^*(E(k);\F_p)\mbox{ admits a retraction } \\
 \mbox{in the category of $\mathcal{A}_p$-modules that commutes with all secondary operations}\}.
\end{multline*}
\end{defn}

We illustrate the utility of this definition by giving examples where $\mathsf{secat}(q)=\mathrm{Swgt}(q;\F_p)>\mathrm{Mwgt}(q;\F_p)$. Our first example is the so-called \emph{twistor bundle}
\[
\xymatrix{
S^2 \ar[r] & \mathbb{C} P^5 \ar[r]^q & \mathbb{H} P^2
}
\]
under which $q$ sends a complex line in $\mathbb{C}^6\cong\mathbb{H}^3$ to the quaternionic line in $\mathbb{H}^3$ containing it. Standard techniques give $1\le \mathsf{secat}(q)\le 2$, but are inconclusive in determining the exact value of $\mathsf{secat}(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \ref{def:Swgt}.

The {\em sectional category} of a fibration $q:E\to B$, denoted $\mathsf{secat}(q)$, is defined to be the minimal integer $k$ such that $B$ admits a cover by open sets $U_0,\ldots , U_k$, on each of which $q$ admits a continuous local section $s_i:U_i\to E$ \cite{Schwarz,James}. Prominent special cases include the Lusternik--Schnirelmann category $\mathsf{cat}(X)$ \cite{CLOT}, which is the sectional category of the based path fibration $P_0 X\to X$, and Farber's topological complexity ${\sf TC}(X)$ \cite{Far03,Far04}, which is the sectional category of the free path fibration $PX\to X\times X$, as well as its many variants \cite{FG07,Rudyak,CFW}. Note that here we use the normalized version, for which $\mathsf{secat}(q)=0$ if and only if $q$ admits a (global) section.

An important foundational result due to Schwarz \cite{Schwarz} (see also \cite{James}) states that if $B$ is paracompact then $\mathsf{secat}(q)$ agrees with the minimal $k$ such that the $(k+1)$-fold fibred join $q(k):E(k)\to B$ admits a section. If $F$ is the fibre of $q$, then the fibre of $q(k)$ is $F(k)=F\ast \cdots \ast F$, the iterated join of $(k+1)$ copies of $F$. When $F$ is $(r-1)$-connected, $F(k)$ is $(r+1)(k+1)-2$-connected. Then (assuming $B$ is a CW complex) cellular obstruction theory gives an upper bound
\[
\mathsf{secat}(q)< \frac{\mathrm{dim}(B)+1}{r+1}.
\]

A standard cohomological argument using the open sets definition gives that $\mathsf{secat}(q)$ is bounded below by $\operatorname{nil}\operatorname{ker} (q^*:H^*(B;R)\to H^*(E;R))$, the nilpotency of the kernel of the induced map in cohomology. Here we take coefficients in an arbitrary commutative ring $R$ (local coefficients may also be used). This bound can sometimes be improved upon using the fibred join interpretation. In particular, we define following Farber--Grant \cite{FG07,FG08} the \emph{sectional category weight} of $q$ to be
\[
\mathrm{wgt}(q;R)=\min \{k \mid q(k)^*:H^*(B;R)\to H^*(E(k);R)\mbox{ is injective}\}.
\]
Then (since a section $s:B\to E(k)$ of $q(k)$ clearly implies injectivity of $q(k)^*$) one has $\mathsf{secat}(q)\geq\mathrm{wgt}(q;R)$. It can be shown that $\mathrm{wgt}(q;R)\geq \operatorname{nil}\operatorname{ker} (q^*:H^*(B;R)\to H^*(E;R))$, and examples of strict inequality are known.

A further improvement to this lower bound was found by Iwase and Kono \cite{IwaseKono}. They observed that a section $s$ of $q(k)$, in addition to implying injectivity of the induced map $q(k)^*$, would entail a retraction $s^*: H^*(E(k);R)\to H^*(B;R)$ which commutes with all cohomology operations. They therefore defined the \emph{module weight} of $q$ at the prime $p\ge2$ to be
\begin{multline*}
\mathrm{Mwgt}(q;\F_p)=\min \{k \mid q(k)^*:H^*(B;\F_p)\to H^*(E(k);\F_p)\mbox{ admits}\\
\mbox{a retraction in the category of $\mathcal{A}_p$-modules}\}.
\end{multline*}
Here $\mathcal{A}_p$ denotes the mod $p$ Steenrod algebra. Clearly, one has inequalities
\[
\mathsf{secat}(q)\ge \mathrm{Mwgt}(q;\F_p) \ge \mathrm{wgt}(q;\F_p)\ge \operatorname{nil}\operatorname{ker}(q^*: H^*(B;\F_p)\to H^*(E;\F_p))
\]
for each prime $p$. Examples of strict inequality $\mathrm{Mwgt}(q;\F_p) > \mathrm{wgt}(q;\F_p)$ can be found in \cite{IwaseKono, Choi09, Choi15}.

\begin{defn}\label{def:Swgt}
The \emph{secondary module weight} of the fibration $q:E\to B$ at the prime $p$ is
 \begin{multline*}
 \mathrm{Swgt}(q;\F_p) := \min \{k \mid q(k)^*:H^*(B;\F_p)\to H^*(E(k);\F_p)\mbox{ admits a retraction } \\
 \mbox{in the category of $\mathcal{A}_p$-modules that commutes with all secondary operations}\}.
\end{multline*}
\end{defn}

We illustrate the utility of this definition by giving examples where $\mathsf{secat}(q)=\mathrm{Swgt}(q;\F_p)>\mathrm{Mwgt}(q;\F_p)$. Our first example is the so-called \emph{twistor bundle}
\[
\xymatrix{
S^2 \ar[r] & \mathbb{C} P^5 \ar[r]^q & \mathbb{H} P^2
}
\]
under which $q$ sends a complex line in $\mathbb{C}^6\cong\mathbb{H}^3$ to the quaternionic line in $\mathbb{H}^3$ containing it. Standard techniques give $1\le \mathsf{secat}(q)\le 2$, but are inconclusive in determining the exact value of $\mathsf{secat}(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \ref{def:Swgt}.

\begin{defn}\label{def:Swgt}
The \emph{secondary module weight} of the fibration $q:E\to B$ at the prime $p$ is
 \begin{multline*}
 \Swgt(q;\F_p) := \min \{k \mid q(k)^*:H^*(B;\F_p)\to H^*(E(k);\F_p)\mbox{ admits a retraction } \\
 \mbox{in the category of $\mathcal{A}_p$-modules that commutes with all secondary operations}\}.
\end{multline*}
\end{defn}

\def\Ab{\mathrm{Ab}}
\def\Rel{\mathrm{Rel}}
\def\cat{\mathsf{cat}}
\def\secat{\mathsf{secat}}
\def\wgt{\mathrm{wgt}}
\def\rank{\mathrm{rank}}
\def\ind{\mathrm{ind}}
\def\dim{\mathrm{dim}}
\def\ker{\operatorname{ker}}
\def\nil{\operatorname{nil}}
\def\coker{\mathrm{coker}}
\def\im{\mathrm{im}}
\def\aut{\mathrm{aut}_1\,}
\def\ev{\mathrm{ev}}
\def\cupQ{\mathrm{cup}_0}
\def\rk{\mathrm{rank}}
\def\eQ{\mathrm{e}_0}
\def\res{\mathrm{res}}
\def\tto{\twoheadrightarrow}

\newcommand{\co}{\colon\thinspace}
\newcommand{\nsub}{\mathrel{\unlhd}}
\newcommand{\PB}{\mathcal{P}}
\newcommand{\TC}{{\sf TC}}
\newcommand{\MTC}{{\sf MTC}}
\newcommand{\TCS}{\mathbf{TC}^S}
\newcommand{\swgt}{\mathrm{swgt}}
\def\Mwgt{\mathrm{Mwgt}}
\def\Swgt{\mathrm{Swgt}}
\newcommand{\cld}{{\sf cd}}
\newcommand{\gld}{{\sf gd}}
\newcommand{\catG}{{\sf cat}_G}
\newcommand{\secatG}{{\sf secat}_G}
\newcommand{\cldG}{{\sf cd}_\mathcal{G}}
\newcommand{\gldG}{{\sf gd}_\mathcal{G}}
\newcommand{\OG}{\mathcal{O}G}
\newcommand{\OF}{\mathcal{O}_\mathcal{F}}
\newcommand{\G}{\Gamma}
\newcommand{\GG}{\pi\rtimes G}
\newcommand{\gen}{\mathfrak{genus}}
\newcommand{\stab}{\stackrel{\simeq}{\longrightarrow}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\OFG}{\mathcal{O}_\F\Gamma}
\newcommand{\OFGG}{\mathcal{O}_\FG (\pi\rtimes G)}
\newcommand{\pspan}{\operatorname{pspan}}
\newcommand{\spn}{\operatorname{span}}
\newcommand{\paths}[1]{P#1}
\newcommand{\eval}[1]{\pi_{#1}}
\newcommand{\evalG}[1]{p_{#1}}
\newcommand{\evaln}[1]{\pi_{n,#1}}

\begin{defn}\label{def:Swgt}
The \emph{secondary module weight} of the fibration $q:E\to B$ at the prime $p$ is
 \begin{multline*}
 \Swgt(q;\F_p) := \min \{k \mid q(k)^*:H^*(B;\F_p)\to H^*(E(k);\F_p)\mbox{ admits a retraction } \\
 \mbox{in the category of $\mathcal{A}_p$-modules that commutes with all secondary operations}\}.
\end{multline*}
\end{defn}

We illustrate the utility of this definition by giving examples where $\secat(q)=\Swgt(q;\F_p)>\Mwgt(q;\F_p)$. Our first example is the so-called \emph{twistor bundle}
\[
\xymatrix{
S^2 \ar[r] & \C P^5 \ar[r]^q & \Ha P^2
}
\]
under which $q$ sends a complex line in $\C^6\cong\Ha^3$ to the quaternionic line in $\Ha^3$ containing it. Standard techniques give $1\le \secat(q)\le 2$, but are inconclusive in determining the exact value of $\secat(q)$. In fact, it was our attempts to determine whether $q(1)$ admits a section using obstruction theory that led us to Definition \ref{def:Swgt}.

Our second example concerns Lusternik--Schnirelmann category. As is standard, when $q:P_0 X\to X$ is the based path fibration on a space $X$ we denote $\wgt(q;\F_p)$, $\Mwgt(q;\F_p)$ and $\Swgt(q;\F_p)$ by $\wgt(X;\F_p)$, $\Mwgt(X;\F_p)$ and $\Swgt(X;\F_p)$, respectively.

\begin{thm}\label{thm:2cell}
Let $\eta\in \pi_7(S^6)$ and $\omega\in \pi_6(S^3)$ be generators, and let $\alpha\in\pi_7(S^3)$ be the homotopy class of the composition $\omega\circ\eta$. If $X=S^3\cup_\alpha e^8$ is the homotopy cofibre of $\alpha$, then
\[
2=\cat(X)=\Swgt(X;\F_2)>\Mwgt(X;\F_2)=\wgt(X;\F_2)=1.
\]
\end{thm}

\begin{lem}\label{lem:Mwgttwistor}
For $q$ the twistor bundle, $\Mwgt(q;\F_2)=1$.
\end{lem}

\begin{lem}\label{lem:Swgttwistor}
 For $q$ the twistor bundle, $\Swgt(q;\F_2)=2$.
 \end{lem}

Theorem \ref{thm:twistor} answers a question on MathOverflow asked by Prateep Chakraborty \cite{Chakra}. Note that the rational sectional category $\secat_0(q)$ in this example is $1$, as follows from results of Stanley \cite{Stanley}.

Our second example concerns Lusternik--Schnirelmann category. As is standard, when $q:P_0 X\to X$ is the based path fibration on a space $X$ we denote $\mathrm{wgt}(q;\F_p)$, $\mathrm{Mwgt}(q;\F_p)$ and $\mathrm{Swgt}(q;\F_p)$ by $\mathrm{wgt}(X;\F_p)$, $\mathrm{Mwgt}(X;\F_p)$ and $\mathrm{Swgt}(X;\F_p)$, respectively.

\begin{thm}\label{thm:2cell}
Let $\eta\in \pi_7(S^6)$ and $\omega\in \pi_6(S^3)$ be generators, and let $\alpha\in\pi_7(S^3)$ be the homotopy class of the composition $\omega\circ\eta$. If $X=S^3\cup_\alpha e^8$ is the homotopy cofibre of $\alpha$, then
\[
2=\mathsf{cat}(X)=\mathrm{Swgt}(X;\F_2)>\mathrm{Mwgt}(X;\F_2)=\mathrm{wgt}(X;\F_2)=1.
\]
\end{thm}

In the proofs of both Theorems \ref{thm:twistor} and \ref{thm:2cell}, the secondary operation $\Phi$ based on the Adem relation $Sq^3Sq^1+Sq^2Sq^2\equiv 0$ is shown to be nonzero with zero indeterminacy in the total space of the fibred join. In each case, this operation detects the Hopf invariant of the attaching map of the top cell \cite{BH,CLOT,GGV}, which turns out to be represented by the class of the composition $\eta\circ\eta\in \pi_7(S^5)$. The main technique to prove non-triviality of $\Phi$ is what Harper calls ``compatibility with exact sequences" in his book \cite{Harper}. This method sometimes allows to compute secondary operations in a space $Y$ in terms of primary operations in the other spaces in a cofiber sequence involving $Y$. We review the method in Section 2 below. Thereafter the proofs diverge: Theorem \ref{thm:twistor} uses the Thom isomorphism relating Steenrod squares in the total space of a sphere bundle with Stiefel-Whitney classes, while Theorem \ref{thm:2cell} uses results of Kono--Kozima \cite{KonoKozima} on Steenrod squares in the loop space of $\operatorname{Spin}(n)$.

In the proofs of both Theorems \ref{thm:twistor} and \ref{thm:2cell}, the secondary operation $\Phi$ based on the Adem relation $Sq^3Sq^1+Sq^2Sq^2\equiv 0$ is shown to be nonzero with zero indeterminacy in the total space of the fibred join. In each case, this operation detects the Hopf invariant of the attaching map of the top cell \cite{BH,CLOT,GGV}, which turns out to be represented by the class of the composition $\eta\circ\eta\in \pi_7(S^5)$. The main technique to prove non-triviality of $\Phi$ is what Harper calls ``compatibility with exact sequences" \cite{Harper}, allowing one to compute secondary operations in a space $Y$ in terms of primary operations in the other spaces in a cofiber sequence involving $Y$ (reviewed in Section 2). For Theorem \ref{thm:twistor} specifically, the proof then uses the Thom isomorphism relating Steenrod squares in the total space of a sphere bundle with Stiefel--Whitney classes.

In establishing the main theorem, the secondary operation $\Phi$ based on the Adem relation $Sq^3Sq^1+Sq^2Sq^2\equiv 0$ is shown to be nonzero with zero indeterminacy in the total space of the fibred join. We also use the following theorem.
\begin{thm}\label{thm:2cell}
Let $\eta\in \pi_7(S^6)$ and $\omega\in \pi_6(S^3)$ be generators, and let $\alpha\in\pi_7(S^3)$ be the homotopy class of the composition $\omega\circ\eta$. If $X=S^3\cup_\alpha e^8$ is the homotopy cofibre of $\alpha$, then
\[
2=\cat(X)=\Swgt(X;\F_2)>\Mwgt(X;\F_2)=\wgt(X;\F_2)=1.
\]
\end{thm}
In each case, this operation detects the Hopf invariant of the attaching map of the top cell Berrick--Hess, Cohen--Lada--May (Title not available, year not available), and Gugenheim--May--Vigu\'{e}-Poirrier (Title not available, year not available), which turns out to be represented by the class of the composition $\eta\circ\eta\in \pi_7(S^5)$. The main technique to prove non-triviality of $\Phi$ is what Harper calls ``compatibility with exact sequences'' Harper (Title not available, year not available), allowing one to compute secondary operations in a space $Y$ in terms of primary operations in the other spaces in a cofiber sequence involving $Y$ (reviewed later). For the main theorem specifically, the proof then uses the Thom isomorphism relating Steenrod squares in the total space of a sphere bundle with Stiefel--Whitney classes.

Question: Let q:\mathbb{C}P^5\to\mathbb{H}P^2 be the twistor bundle, where q sends a complex line in \mathbb{C}^6\cong\mathbb{H}^3 to the quaternionic line in \mathbb{H}^3 containing it. Write \mathsf{secat}(q) for the normalized sectional category of q, i.e. the least k such that \mathbb{H}P^2 can be covered by open sets U_0,\dots,U_k on each of which q admits a continuous local section. For the (k+1)-fold fibred join q(k):E(k)\to \mathbb{H}P^2, define \mathrm{wgt}(q;\mathbb{F}_2)=\min\{k\mid q(k)^*:H^*(\mathbb{H}P^2;\mathbb{F}_2)\to H^*(E(k);\mathbb{F}_2)\text{ is injective}\}, \mathrm{Mwgt}(q;\mathbb{F}_2) as the least k for which q(k)^* admits a retraction in the category of \mathcal{A}_2-modules, and \mathrm{Swgt}(q;\mathbb{F}_2) as the least k for which q(k)^* admits such a retraction commuting with all secondary operations. Which quantitative statement holds for this bundle?

2=\mathsf{secat}(q)=\mathrm{Swgt}(q;\mathbb{F}_2)>\mathrm{Mwgt}(q;\mathbb{F}_2)=1>\mathrm{wgt}(q;\mathbb{F}_2)=0.

Choices:
B. 2=\mathsf{secat}(q)=\mathrm{Swgt}(q;\mathbb{F}_2)=\mathrm{Mwgt}(q;\mathbb{F}_2)>\mathrm{wgt}(q;\mathbb{F}_2)=0.
C. 2=\mathsf{secat}(q)=\mathrm{Swgt}(q;\mathbb{F}_2)>\mathrm{Mwgt}(q;\mathbb{F}_2)\ge \mathrm{wgt}(q;\mathbb{F}_2).
D. 1=\mathsf{secat}(q)=\mathrm{Swgt}(q;\mathbb{F}_2)>\mathrm{Mwgt}(q;\mathbb{F}_2)=\mathrm{wgt}(q;\mathbb{F}_2)=0.
E. 2=\mathsf{secat}(q)=\mathrm{Mwgt}(q;\mathbb{F}_2)>\mathrm{Swgt}(q;\mathbb{F}_2)=1>\mathrm{wgt}(q;\mathbb{F}_2)=0.

nontrivial secondary operation obstructs only secondary-compatible retraction, not ordinary module retraction

exact numerical values of \mathrm{Mwgt} and \mathrm{wgt}

nonzero secondary obstruction at the first fibred join rules out a section of q(1)

ordering between \mathrm{Swgt} and \mathrm{Mwgt} forced by secondary-operation obstruction

The stem defines the invariants but does not reveal the numerical values or the correct ordering. There is no explicit or near-explicit statement of choice A in the prompt.

The item is not tautological in the strict sense, since the student must choose among several competing quantitative statements. However, it is very close to asking for recall of a specific known theorem/result about this bundle rather than deriving a conclusion from supplied data.

Some reasoning is needed to compare the invariants and reject impossible orderings, but the problem mainly tests knowledge/recall of the exact computed values rather than substantial generative mathematical reasoning from first principles.

The distractors are mathematically plausible and target realistic confusions: collapsing Swgt and Mwgt, giving only a weaker true inequality, misstating the sectional category, or reversing the secondary-operation obstruction. They are distinct and well aligned with common failure modes.

A solid high-level recall question with strong distractors and no answer leakage, but it leans more toward theorem recollection than genuine generative reasoning.


\label{t:main}
  Let $q\ge4$ be a prime power and $G$ be almost simple with socle
  $S=\PSL_2(q)$, that is $S\le G\le\aut(S)$. Suppose that $G$ acts
  primitively on $\Omega$. Then for every distinct
  $\alpha, \beta\in\Omega$ there exists $s\in S$ such that
  $\alpha^s=\beta$ and $s$ fixes no point of $\Omega$.

It is an easy exercise using the orbit-counting theorem to show that
if $G$ is a transitive permutation group on a finite set $\Omega$,
then the set of derangements of $G$ generates a transitive subgroup.

It is also easy to see that in general the set of derangements,
together with the identity, need not be transitive on $\Omega$. A
small example is the alternating group $A_4$ acting on the $2$-subsets
of $\{1,2,3,4\}$: An element $g\in A_4$ moving $\{1,2\}$ to
$\{3,4\}$ moves every element in its natural degree $4$ action, so
it is a double transposition, and therefore fixes some $2$-subset.

While there are many more such examples, it seemed hard to find one
under the additional assumption that $G$ acts primitively. In fact the
following Problem 8.75 of the Kourovka Notebook
\cite{kourovka_notebook}, attributed to John G.~Thompson and
designated ``A known problem'', was open for more than 40 years.
\begin{question}\label{q:8.75}
 Suppose $G$ is a finite primitive permutation group on $\Omega$, and
 $\alpha$, $\beta$ are distinct points of $\Omega$. Does there exist
 an element $g\in G$ such that $\alpha^g=\beta$ and $g$ fixes no
 point of $\Omega$?
\end{question}
In \cite{mueller___8.75} we give a negative answer. The group there is
the Steinberg triality group ${}^{3}D_{4}(2)$ of order
$211{,}341{,}312 = 2^{12}\cdot 3^{4}\cdot 7^{2}\cdot 13$ in a
primitive permutation action on a set of size $4{,}064{,}256$.

Using Magma \cite{magma}, we checked that this is the only example if
we assume that $G$ is almost simple with socle of size less than or
equal $20{,}158{,}709{,}760$. It would have been computationally too
expensive to check all the groups with socle $\PSL_2(q)$ in this
range. The purpose of this note is to show that these groups do not
produce negative examples. In fact a somewhat stronger result holds.

In some cases there are lengthy calculations which, in principle,
could be carried out by hand. Instead, we provide SageMath code at
\cite{dera_psl2_verify} that confirms the claims.
\section{Quartic hyperelliptic curves}
If $D$ is a plane algebraic curve over a field $F$ given by
$P(X, Y)\in F[X, Y]$, then $D(F)=\defset{(x, y)\in F^2}{P(x,y)=0}$
denotes the set of affine $F$-points on $D$.

Finally assume that $f(X)=(aX^2-2X+d)(aX^2+2X+d)$ is
 inseparable. The two factors are relatively prime (because the
 difference is $4X$). Furthermore, the discriminant of both factors
 is $4(1-ad)$, hence $ad=1$ and we are done.
\end{proof}
\begin{remark}
 One can check that $q=13$, $17$, and $19$ do not give
 exceptions. However, $q=3$, $5$, $7$, $9$, and $11$ are indeed
 exceptions. This makes it unlikely that there is a ``cleaner'' proof
 of the lemma for odd $q$.
\end{remark}
\begin{corollary}\label{c:split}
 Let $q\ge4$ be a prime power and $a, b, c, d\in\FF_q$ with
 $ad-bc=1$. Set
 $T=\defset{r+1/r}{r\in\Fqs}\cup\set{0}$. Suppose that
 $ar+d/r\in T$ and $-cr+b/r\in T$ for all $r\in\Fqs$. Then one
 of the following holds:
 \begin{itemize}
 \item[(a)] $a=d=0$ or $b=c=0$;
 \item[(b)] $q=9$ and $ar+d/r=2$ for some $r\in\Fqs$.
 \end{itemize}
\end{corollary}
\begin{proof}
 Using the previous lemma, one only need to check the cases where
 $q\le19$ is an odd prime power. This is most conveniently done with
 the SageMath code at \cite{dera_psl2_verify}.
\end{proof}
\begin{lemma}
 Let $q$ be a prime power with $q\ge23$ if $q$ is odd. For
 $F=\FF_{q^2}$ set $N=\defset{r\in F}{r^{q+1}=1}$. Suppose that for
 $0\ne a\in F$ the following holds: For all $r\in N$ with
 $ar+a^q/r\ne0$ there is $s\in N$ such that $ar+a^q/r = s+1/s$. Then
 $a\in N$.
\end{lemma}
\begin{proof} We first handle the easier case that $q$ is even.
As the square map is an automorphism of $F$, there is $r_0\in F$ with
$a^{q-1}=r_0^2$. Note that $a^{q-1}\in N$, hence $r_0\in N$.

We see that the curve $Y^2=\gamma P(X)Q(X)$ has at least $2(q-6)$
 affine points over $\FF_q$. On the other hand, by Lemma
 \ref{l:hasse}, this number is at most $q+1+2\sqrt{q}$. From this one
 quickly obtains $q\le19$.
\end{proof}
\begin{remark}
 One can check that $q=11$, $17$, and $19$ do not give exceptions.
 However, the cases $q=3$, $5$, $7$, $9$, $13$ are indeed
 exceptions. This is most easily seen for $q=3$, because then
 $\defset{s+1/s}{s\in N}=\FF_q$, so the condition on $a$ is vacuous.
\end{remark}
\begin{corollary}\label{c:nonsplit}
 Let $q$ be a prime power. Set $N=\defset{r\in\FF_{q^2}}{r^{q+1}=1}$
 and $T=\defset{r+1/r}{r\in N}$. For $a, b\in\FF_{q^2}$ with
 $a^{q+1} + b^{q+1} = 1$ suppose that $ar+a^q/r\in T$ and
 $br+b^q/r\in T$ for all $r\in\Fqs$. Then $a=0$ or $b=0$.
\end{corollary}
\begin{proof}
 If $q\ge23$, then the previous lemma shows that $a^{q+1}=1$, hence
 $b=0$, or $b^{q+1}=1$, hence $a=0$. The cases $q\le19$ are
 checked in \cite{dera_psl2_verify}.
\end{proof}
\section{The groups of quaternions $2A_4$, $2S_4$, and $2A_5$}
\begin{lemma}
 Let $F$ be a field and $\bH$ be a subgroup of $\PSL_2(F)$ isomorphic
 to $A_4$, $S_4$, or $A_5$. Let $H$ be the preimage of $\bH$ in
 $\SL_2(F)$ and $x, y\in F^{2\times 2}$. If the characteristic of $F$
 is $0$ or relatively prime to $\abs{\bH}$, then
 \[
 \sum_{h\in H}\trace(hy)\cdot\trace(hx)^{2m+1} =\abs{\bH}\cdot c_m\cdot (\trace(x)\cdot\trace(y)-\trace(x\cdot y))\cdot\det(x)^m,
 \]
 where $m\in\set{0, 1}$ if $\bH\cong A_4$, $m\in\set{0, 1, 2}$ if
 $\bH\cong S_4$, $m\in\set{0, 1, 2, 3, 4}$ if $\bH\cong A_5$, and
 $c_0, c_1, c_2, c_3, c_4$ = $1, 2, 5, 14, 42$ are the first five
 Catalan numbers (with index shifted by $1$).
\end{lemma}
\begin{proof}
 We assume that the characteristic of $F$ is $0$ or relatively prime
 to $\abs{\bH}$. Let $\bar F$ be an algebraic closure of $F$. It is
 well known that $\PSL_2(\bar F)$ contains exactly one conjugacy
 class of subgroups isomorphic to $\bH$, see e.g.~\cite[Proposition
 4.1]{beauville___pgl2}. (In fact more is true. By \cite[Proposition
 4.2]{beauville___pgl2} $\PGL_2(F)$ contains at most one conjugacy
 class of subgroups isomorphic to $\bH$.)

Note that for $h\in H$ (recall that $H\le\SL_2(\bar F)$), we
 have $f(h'x, h'y)=f(x, y)$, so $f$ is an invariant of
 $H$. Working out the low degree invariants of $H$ could
 provide a more conceptual explanation. (At a first glance, it
 might not be obvious that the right hand side is an invariant
 under simultaneous left multiplication of $x$ and $y$ by
 $h$. But this follows from the identity
 $\trace(x)\cdot\trace(y)-\trace(x\cdot y)=\trace(\adj(y)\cdot
 x)$, where $\adj(y)$ denotes the adjugate of $y$.)
 \end{itemize}
\end{remark}
\begin{lemma}\label{l:polyhedral}
 Let $F$ be a field and $\bH$ be a subgroup of $\PSL_2(F)$ isomorphic
 to $A_4$, $S_4$, or $A_5$. Assume that the characteristic of $F$ is
 $0$ or relatively prime to $\abs{\bH}$. Let $H$ be the preimage of
 $\bH$ in $\SL_2(F)$. Set
 $T=\defset{\trace(h)}{h\in H}\setminus\set{-2, 2}$, and for
 $x = \begin{pmatrix}a & b\\c & d\end{pmatrix}\in\SL_2(F)$ set
 $y = \begin{pmatrix}0 & 0\\c & d\end{pmatrix}$. Then
 \[
 \sum_{h\in H}\left(\trace(hy)\prod_{t\in T}(\trace(hx)-t)\right) =\abs{\bH}.
 \]
\end{lemma}
\begin{proof}
 With this choice of $x$ and $y$ we have
 \[\trace(x)\cdot\trace(y)-\trace(x\cdot y)=ad-bc=\det(x)=1,\]
 so the previous lemma yields
 \[
 \sum_{h\in H}\trace(hy)\cdot\trace(hx)^{2m+1} =\abs{\bH}\cdot c_m.
 \]
 Let $Z$ be a variable over $F$. Then \cite{dera_psl2_verify} verifies
 \[
 \prod_{t\in T}(Z-t) =
 \begin{cases}
 Z^3 - Z,&\text{ if }\bH\cong A_4\\
 Z^5 - 3Z^3 + 2Z,&\text{ if }\bH\cong S_4\\
 Z^7 - 4Z^5 + 4Z^3 - Z,&\text{ if }\bH\cong A_5.\\
 \end{cases}
 \]
This yields
 \[
 \sum_{h\in H}\left(\trace(hy)\prod_{t\in
 T}(\trace(hx)-t)\right) =
 \begin{cases}
 \abs{\bH}(c_1-c_0)=\abs{\bH},&\text{ if }\bH\cong A_4\\
 \abs{\bH}(c_2-3c_1+2c_0)=\abs{\bH},&\text{ if }\bH\cong S_4\\
 \abs{\bH}(c_3-4c_2+4c_1-c_0)=\abs{\bH},&\text{ if }\bH\cong A_5.\\
 \end{cases}
 \]
\end{proof}
\section{Proof of Theorem \ref{t:main}}
The maximal subgroups of almost simple groups with socle $\PSL_2(q)$
are known; see \cite[Table~8.1, Table~8.2]{bray_etal___maximal} or
\cite{giudici___psl2}. The results in \cite{bray_etal___maximal} and
\cite{giudici___psl2} are more detailed than those in the following
lemma, which lists only the restrictions on $q$ that we shall use.
\begin{lemma}\label{l:giudici}
 Let $q\ge4$ be a prime power and $S=\PSL_2(q)\le
 G\le\PGammaL_2(q)$. Let $M$ be a maximal subgroup of $G$ with
 $S\not\le M$. Set $\bH=M\cap S$. Then one of the following holds.
 \begin{itemize}
 \item[(a)] $[S:\bH] = q+1$,
 \item[(b)] $[S:\bH] = q(q+1)/2$,
 \item[(c)] $[S:\bH] = q(q-1)/2$,
 \item[(d)] $\bH\cong A_4$, and $q$ is and odd prime,
 \item[(e)] $\bH\cong S_4$, and $q$ is a prime with $q\equiv\pm1\mod{8}$,
 \item[(f)] $\bH\cong A_5$, and $q\equiv\pm1\mod{10}$ and $q=9$ if $q$ is
 power of $3$,
 \item[(g)] $\bH\cong\PSL_2(r)$ or $\bH\cong\PGL_2(r)$, where $q=r^e$
 for $e\ge2$.
 \end{itemize}
 Furthermore, in each case $\bH$ is unique up to conjugacy in
 $\aut(S)$ (with two classes in case (g) if $r$ is odd).
\end{lemma}
Let $S$ and $G$ be as in Theorem \ref{t:main}, $M$ be a stabilizer of
a point in $\Omega$, and $\bH=M\cap S$. Then $M$ is a maximal subgroup
of $G$ by primitivity. We handle the cases (a) to (g) separately.

The remaining cases (b) to (g) require some work. Suppose that there
is a counterexample. Then there exist distinct
$\alpha, \beta\in\Omega$ such that no derangement in $S$ moves
$\alpha$ to $\beta$. Let $\bH$ be the stabilizer in $S$ of $\alpha$
and $\bx\in S$ with $\alpha^{\bx}=\beta$. Then $\bH\bx$ is the set of
elements which moves $\alpha$ to $\beta$. As this set does not contain
a derangement, every element in the coset $\bH\bx$ is conjugate to an
element from $\bH$.

If $G$ acts primitively on $\Omega$, then the possibilities for a point
stabilizer $S_\omega < S$ are known. While some cases are relatively
easy to handle, the more difficult ones arise when $S_\omega$ is
dihedral, or when $S_\omega$ is one of the polyhedral groups $A_4$,
$S_4$, or $A_5$. In the former case we require Hasse’s bound on the
number of points on elliptic curves over finite fields, and to rule out
the latter cases we derive trace identities for the binary polyhedral
groups $2A_4$, $2S_4$, and $2A_5$.

In some cases there are lengthy calculations which, in principle,
could be carried out by hand. Instead, we provide SageMath code at
\cite{dera_psl2_verify} that confirms the claims.

To prove Theorem~\ref{t:main}, one uses that when $G$ acts primitively on $\Omega$, “the possibilities for a point stabilizer $S_\omega < S$ are known.” The argument then splits by the type of $S_\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\omega$ is dihedral, or when $S_\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \cite{dera_psl2_verify} that confirms the claims.”

To prove the main theorem, one uses that when $G$ acts primitively on $\Omega$, “the possibilities for a point stabilizer $S_\omega < S$ are known.” The argument then splits by the type of $S_\omega$: “some cases are relatively easy to handle,” while the “more difficult ones arise when $S_\omega$ is dihedral, or when $S_\omega$ is one of the polyhedral groups $A_4$, $S_4$, or $A_5$.” In the dihedral case, the proof “require[s] Hasse’s bound on the number of points on elliptic curves over finite fields,” and to exclude the polyhedral cases one “derive[s] trace identities for the binary polyhedral groups $2A_4$, $2S_4$, and $2A_5$.” The authors note that “in some cases there are lengthy calculations” and they “provide SageMath code at \cite{dera_psl2_verify} that confirms the claims.”

Question: Let $q\ge 4$ be a prime power, let $S=\PSL_2(q)$, and let $G$ be an almost simple group with socle $S$, meaning $S\le G\le \operatorname{Aut}(S)$. Assume that $G$ acts primitively on a set $\Omega$ (that is, $G$ preserves no nontrivial partition of $\Omega$). For distinct points $\alpha,\beta\in\Omega$, which conclusion holds?

For every pair of distinct points $\alpha,\beta\in\Omega$, there exists an element $s\in S$ such that $\alpha^s=\beta$ and $s$ fixes no point of $\Omega$; equivalently, $s$ is a derangement in the action on $\Omega$.

Choices:
B. For every pair of distinct points $\alpha,\beta\in\Omega$, there exists an element $g\in G$ such that $\alpha^g=\beta$ and $g$ fixes no point of $\Omega$; equivalently, one may always choose a derangement from the whole almost simple group $G$ sending $\alpha$ to $\beta$.
C. For every pair of distinct points $\alpha,\beta\in\Omega$, there exists an element $s\in S$ such that $\alpha^s=\beta$.
D. For every point $\alpha\in\Omega$, there exists an element $s\in S$ that fixes no point of $\Omega$ and moves $\alpha$; in particular, $S$ contains derangements in this primitive action.
E. For every pair of distinct points $\alpha,\beta\in\Omega$, every element $s\in S$ with $\alpha^s=\beta$ fixes no point of $\Omega$; equivalently, the entire transporter from $\alpha$ to $\beta$ inside $S$ consists of derangements.

socle-membership of the witnessing element

pairwise transporter conclusion weakened to mere existence of some derangements

existential statement on a coset replaced by universal statement on the whole transporter coset

The stem sets up the group-action context but does not explicitly state the conclusion about derangements in S. The correct answer is not leaked directly, though the focus on elements of S narrows the target domain.

The correct option is very close to a theorem-style conclusion stated almost verbatim from the setup. However, the presence of nearby variants involving quantifiers, weakening, and ambient-group confusion makes it slightly more than a pure restatement.

Some reasoning is needed to distinguish the exact claim from stronger, weaker, or mis-scoped alternatives, especially between A, B, and C. Still, the item mainly tests recall/recognition of the precise theorem statement rather than substantial generative reasoning.

Several distractors are meaningful and target common errors: confusing S with G, dropping the derangement condition, or altering quantifiers. But one option (E) is fairly implausible, so overall distractor quality is mixed.

A decent theorem-recognition MCQ with limited answer leakage and some plausible distractors, but it leans heavily on recall of the exact statement rather than deeper mathematical generation or derivation.


Question: Let $q\ge 4$ be a prime power, let $S=\PSL_2(q)$, and let $G$ be an almost simple group with socle $S$, meaning $S\le G\le \operatorname{Aut}(S)$. Assume that $G$ acts primitively on a set $\Omega$ (that is, $G$ preserves no nontrivial partition of $\Omega$). For distinct points $\alpha,\beta\in\Omega$, which conclusion holds?

Choices:
B. For every pair of distinct points $\alpha,\beta\in\Omega$, there exists an element $g\in G$ such that $\alpha^g=\beta$ and $g$ fixes no point of $\Omega$; equivalently, one may always choose a derangement from the whole almost simple group $G$ sending $\alpha$ to $\beta$.
C. For every pair of distinct points $\alpha,\beta\in\Omega$, there exists an element $s\in S$ such that $\alpha^s=\beta$.
D. For every point $\alpha\in\Omega$, there exists an element $s\in S$ that fixes no point of $\Omega$ and moves $\alpha$; in particular, $S$ contains derangements in this primitive action.
E. For every pair of distinct points $\alpha,\beta\in\Omega$, every element $s\in S$ with $\alpha^s=\beta$ fixes no point of $\Omega$; equivalently, the entire transporter from $\alpha$ to $\beta$ inside $S$ consists of derangements.

The stem gives only the group-action setup and asks which conclusion follows; it does not explicitly state or strongly hint at the derangement-transporter conclusion in choice A.

The correct option is essentially a theorem-style conclusion stated in full, so the item is close to a direct restatement. However, the presence of nearby quantifier/strength variants means the task is not purely tautological.

The student must distinguish subtle logical variations: witness in S versus G, existence versus universality, and pairwise transport versus mere existence of derangements. This creates moderate reasoning demand, though it is still mainly recognition of the precise theorem statement.

The distractors are mathematically plausible and target common errors: overstrengthening to G, weakening by dropping the derangement condition, replacing pairwise transport with mere existence, and switching existential to universal quantification.

A solid MCQ with no real answer leakage and strong distractors, but it remains fairly close to theorem recognition rather than deep generative reasoning.


\begin{rem}\label{rem:DMexs}
Some nonarithmetic Deligne--Mostow lattices \cite{DeligneMostow} also admit homomorphisms onto cocompact Fuchsian groups from \emph{forgetful maps} associated with contracting weights. See for example \cite[Thm.\ 3.1(iv)]{Deraux}.
\end{rem}

\subsection{Automorphisms of Fuchsian groups}\label{ssec:FuchsianAut}

The purpose of this subsection is to prove \Cref{thm:Fixpsubs}.

\begin{pf}[Proof of \Cref{thm:Fixpsubs}]
Let $p$ be a prime, $T$ be a surface group of genus $g \ge2$, and $\phi \in \Aut(T)$ have the property that $\phi(N) = N$ for every finite index normal subgroup with index a power of $p$. Consider the pro-$p$ completion $T_{\wh{p}}$ of $T$. Then $\varphi$ extends to a continuous automorphism of $T_{\wh{p}}$ that is \emph{normal}, i.e., so that $\varphi(\wh{N})$ for every closed normal subgroup of $T_{\wh{p}}$. Indeed, every such normal subgroup is an intersection of finite index normal subgroups of $T_{\wh{p}}$, and finite index normal subgroups of $T_{\wh{p}}$ are in natural one-to-one correspondence with the normal $p$-power index subgroups of $T$.

Then $T_{\wh{p}}$ is topologically finitely presented, since $T$ is finitely presented, hence $T_{\wh{p}}$ is \emph{psuedo $p$-free} in the sense of Jarden--Ritter (see \cite[\S 1]{JardenRitter} for the definition) by \cite[Thm. 4]{JardenRitter}. Thus the extension of $\varphi$ to $T_{\wh{p}}$ is inner by \cite[Thm.\ 1]{JardenRitter}. Moreover, Paris proved that $T$ is $p$-conjugacy separable \cite[Thm.\ 1.7]{Paris}, which implies that $\varphi$ must preserve conjugacy classes of $T$ (i.e., it is \emph{pointwise inner}). Therefore $\varphi$ is a normal automorphism of $T$, meaning that it preserves all normal subgroups of $T$. Finally, it is a theorem of Bogopolski, Kudryavtseva, and Zieschang that normal automorphisms of higher-genus surface groups are inner \cite[Thm.\ 1.3]{BKZ}, which completes the proof of the theorem.
\end{pf}

\section{The main result}\label{sec:Main}

Suppose that $\Lam < \PU(2,1)$ is a cocompact nonarithmetic lattice with nontrivial universal homomorphism $\rho$. Following \Cref{ssec:PU21}, let $\Gam < \PU(2,1)$ be the unique maximal lattice containing $\Lam$. By \Cref{lem:TFFuchsian} and by passing to a finite index subgroup if necessary, we can assume that the universal homomorphism for $\Lam$ has a torsion-free factor and that $\Lam$ is a characteristic subgroup of $\Gam$. If $M_{\mathrm{tf}} \coloneqq \ker(\rho_{\mathrm{tf}})$ denotes the kernel of the torsion-free universal homomorphism for $\Lam$, then $M_{\mathrm{tf}}$ is an infinite index characteristic subgroup of $\Gam$ by \Cref{lem:UniversalChar} and because a characteristic subgroup of a characteristic subgroup is characteristic in the larger group. Set $T \coloneqq \Lam / M_{\mathrm{tf}}$, so $T = \rho_{\mathrm{tf}}(\Lam)$.

As described in \Cref{ssec:PU21}, away from finitely many prime ideals $\frakp$ of the ring $\Tr\Ad \Gam$, there is a well-defined congruence quotient $\wh{Q} \coloneqq \Gam / \Gam(\frakp)$ with trivial center and commutator subgroup $Q$ that is a nonabelian finite simple group. Moreover, $\Lam$ and $M_{\mathrm{tf}}$ are Zariski dense in $\PU(2,1)$, hence after possibly passing to a finite index subgroup of $\Lam$, we can assume that they both map onto $Q$ for infinitely many $\frakp$. Fix one such $\frakp$ for the remainder of this section. Then define $\Lam(\frakp) \coloneqq \Gam(\frakp) \cap \Lam$ and $M_{\mathrm{tf}}(\frakp) \coloneqq \Gam(\frakp) \cap M_{\mathrm{tf}}$, noting that both $\Lam(\frakp)$ and $M_{\mathrm{tf}}(\frakp)$ are normal subgroups of $\Gam$, since each is the intersection of normal subgroups of $\Gam$.

\begin{lem}\label{lem:QuoSetup}
With the notation established in this section,
\begin{align*}
\rho_{\mathrm{tf}}(\Lam(\frakp)) &= T \\
\Lam / M_{\mathrm{tf}}(\frakp) &\cong Q \times T \\
\Gam / M_{\mathrm{tf}}(\frakp) &\cong \wh{Q} \times \wh{T}
\end{align*}
where $\wh{T} \coloneqq \Gam / M_{\mathrm{tf}}$ contains $T$ as a finite index subgroup.
\end{lem}
\begin{pf}
The first equality holds since $M_{\mathrm{tf}}$ contains coset representatives for $\Lam(\frakp)$ in $\Lam$. The second holds since $M_{\mathrm{tf}} / M_{\mathrm{tf}}(\frakp)$ and $\Lam(\frakp) / M_{\mathrm{tf}}(\frakp)$ are disjoint normal subgroups of $\Lam / M_{\mathrm{tf}}(\frakp)$, and the third is similar.
\end{pf}

\begin{rem}
Note that $T$ is then a product of factors of the torsion-free universal homomorphism of $\Lam(\frakp)$, but there may be additional factors.
\end{rem}

\begin{equation}\label{eq:ReductionModp}
\pi_\frakp : \wt{\bfG}(\calO_k) \lra \wt{\bfG}(\bbF_\frakp)
\end{equation}
to any $\wt{\Del} \le \wt{\bfG}(\calO_k)$ has kernel the \emph{congruence subgroup} $\wt{\Del}(\frakp) \trianglelefteq \wt{\Del}$. Strong approximation for Zariski dense subgroups of simply connected algebraic groups says that if $\wt{\Del}$ is Zariski dense in $\wt{\bfG}(k)$, then $\pi_\frakp(\wt{\Del}) = \wt{\bfG}(\bbF_\frakp)$ for all but finitely many $\frakp$ \cite[Thm.\ 1.1]{Weisfeiler}; also see Cor.\ 3 in Window 9 of \cite{LubotzkySegal} along with discussion on p.\ 392 implying the more general statement for number fields other than $\bbQ$. If $\wt{\Del}$ is the preimage of $\Del$ in $\SU(2,1)$, then congruence subgroups of $\Del$ are those whose preimage in $\wt{\Del}$ are congruence subgroups.

Since $\PU(2,1)$ has type ${}^2 \mathrm{A}_2$, there is a quadratic extension $\ell$ of $k$ canonically associated with $\bfG$. If $\Del \le \Gam$ is Zariski dense in $\PU(2,1)$, then away from a finite set of exceptional primes, the group $\Del / \Del(\frakp)$ has commutator subgroup $\PSL_3(\bbF_\frakp)$ if $\frakp$ splits in $\ell$ or $\PU(3; \bbF_\frakp)$ if $\frakp$ is inert in $\ell$. In particular, the congruence quotient of $\Del$ has commutator subgroup a nonabelian finite simple group for all but finitely many primes $\frakp$. For example in the split case, observe that $\Del / \Del(\frakp)$ is then naturally a subgroup of $\PGL_3(\bbF_\frakp)$ containing $\PSL_3(\bbF_\frakp)$, so it has trivial center.

\subsection{K\"ahler groups}\label{ssec:KahlerGroups}

A \emph{K\"ahler group} is the fundamental group of a compact K\"ahler manifold. See \cite{KahlerBook, Py} for basics on K\"ahler groups. Note that finite index subgroups of K\"ahler groups are naturally K\"ahler groups.

Let $\Gam < \PU(2,1)$ be a cocompact lattice, possibly with torsion. Since $\PU(2,1)$ is a linear Lie group and $\Gam$ is finitely generated, $\Gam$ is residually finite and thus there is a finite index subgroup $\Lam < \Gam$ that is torsion-free. Since smooth projective varieties are compact K\"ahler manifolds, by pullback of the Fubini--Study metric on $\bbP^N$ under a projective embedding, it follows that any such $\Lam$ is a K\"ahler group. It then follows from \cite[Lem.\ 1.1.15]{KahlerBook} that all cocompact lattices in $\PU(2,1)$ are K\"ahler groups.

\medskip

Fundamental to this paper is the \emph{universal homomorphism} attached to any K\"ahler group. See \cite[\S 2.C]{HLPSV} for discussion and references for the following result defining the universal homomorphism.

\begin{thm}\label{thm:Universal}
Suppose $\Lam$ is a K\"ahler group. Then there is a (possibly empty) finite collection of cocompact Fuchsian groups $\{T_\al\}_{\al \in \frakU}$ and a homomorphism
\begin{equation}\label{eq:UniversalHom}
\rho : \Lam \lra T \coloneqq \prod_{\al \in \frakU} T_\al
\end{equation}

\label{sec:Intro}

Throughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \Cref{sec:Main}, is the following.

\begin{abstract}
In response to a question raised by Belolipetsky and the first author in \cite{BelolipetskyLubotzky}, we prove that for every finite group $G$ there are infinitely many isomorphism classes of compact complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.
\end{abstract}

Throughout this paper, an automorphism of a complex manifold will always mean a biholomorphism. The main result of this paper, proved in \Cref{sec:Main}, is the following.

Greenberg \cite{Greenberg} proved \Cref{thm:Main} for Riemann surfaces, i.e., complex hyperbolic $1$-manifolds. Belolipetsky and the first author \cite{BelolipetskyLubotzky} proved the analogue of \Cref{thm:Main} for isometry groups of hyperbolic $n$-manifolds, $n \ge 3$, and this paper answers a question raised in \cite[\S 6.5]{BelolipetskyLubotzky}. Our argument follows the broad outline of that paper, but differences at key steps create many new challenges we must overcome.

\begin{rem}\label{rem:Highern}
Whether or not there are cocompact nonarithmetic lattices in $\PU(n,1)$ for all $n \ge 3$ is one of the major open problems in the study of lattices in Lie groups. For any $n \ge 3$, if there is a cocompact nonarithmetic lattice in $\PU(n,1)$ with nontrivial universal homomorphism, then this paper produces infinitely many isomorphism classes of closed complex hyperbolic $n$-manifolds with isometry group any finite group $G$ \emph{verbatim}.
\end{rem}

\begin{thm}\label{thm:ProjectiveVersion}
For every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.
\end{thm}

In what follows, $\pi : T \to T_r$ will denote projection onto the $r^{th}$ surface group factor of $T$ for an arbitrary $r$ that will be fixed for the remainder of this section. Let $\sig$ denote projection of $\Gam$ onto $\Gam / M_{\mathrm{tf}}(\frakp)$ and also the restriction of $\sig$ to any finite index subgroup of $\Gam$. This leads to the diagram of groups and surjective homomorphisms given in \Cref{fig:BigDiagram}.
\begin{figure}[h]
\centering
\begin{tikzcd}
& \Gam \arrow[ddr, -] \arrow[dl, -] \arrow[rr, ->, "\sig"] \arrow[rrrr, bend left = 35, ->, "\textrm{mod}\, M_{\mathrm{tf}}"] & & \wh{Q} \times \wh{T} \arrow[d, -] \arrow[rr, ->, "\textrm{mod}\, \wh{Q}" below] & & \wh{T} \arrow[d, -] \\
\Gam(\frakp) \arrow[dd, -] & & & Q \times T \arrow[rr, ->, "\textrm{mod}\, Q"] && T \arrow[rr, ->, "\pi"] & & T_r \\
& & \Lam \arrow[dll, -] \arrow[ur, ->, "\sig"] \arrow[urrr, bend right=20, ->, "\rho_{\mathrm{tf}}" {xshift=0.75em, yshift=-1.25em}] \\
\Lam(\frakp) \arrow[ddd, -] \arrow[uurrrrr, ->, bend right=35, "\sig\, =\, \rho_{\mathrm{tf}}|_{\Lam(\frakp)}" {xshift=6em, yshift=-0.5em}] & & \\
& & \\
& & M_{\mathrm{tf}} \arrow[dll, -] \arrow[uuu, -, crossing over] & \\
M_{\mathrm{tf}}(\frakp) & &
\end{tikzcd}
\caption{The relationships between various groups}\label{fig:BigDiagram}
\end{figure}
Define
\begin{equation}\label{eq:TauDef}
\tau \coloneqq \pi \circ \rho_{\mathrm{tf}} : \Lam \lra T_r
\end{equation}
and $M \coloneqq \ker(\tau|_\Del)$. For a finite index subgroup $\Del \le \Gam$, let $N_\Gam(\Del)$ denote its normalizer in $\Gam$. If $\Del$ contains $M$, then any $\gam \in N_\Gam(\Del)$ normalizing $M$ induces an automorphism of $\Del / M$ through $\pi \circ \rho_{\mathrm{tf}}$. The following proposition/construction is essential for proving \Cref{thm:Main}.

\begin{pf}[Proof of \Cref{thm:Main}]
The only input to begin this section of the paper was a cocompact nonarithmetic lattice $\Lam < \PU(2,1)$ with nontrivial universal homomorphism. Such examples exist by \Cref{thm:Examples}. If $G$ is a nontrivial group (resp.\ the trivial group), then \Cref{prop:FinalNontrivialG} (resp.\ \Cref{prop:TrivialCase}) produces a torsion-free lattice $B \le \Lam$ whose normalizer $N(B) < \PU(2,1)$, which equals $N_\Gam(B)$ for $\Gam$ the unique maximal lattice in $\PU(2,1)$ containing $B$ by the Margulis dichotomy, has $N(B) / B \cong G$. Mostow rigidity now implies that $G$ is the automorphism group (equivalently, holomorphic isometry group) of the smooth compact complex hyperbolic $2$-manifold $B \bs \bbB^n$. Varying the primes $\{p_j\}$ in our construction gives infinitely many lattices with covolume tending to infinity for any fixed finite group $G$, hence another application of Mostow rigidity implies that there are infinitely many isomorphism classes of lattices. This proves the theorem.
\end{pf}

Complex hyperbolic manifolds are of general type \cite[\S V.20]{BPV}, hence \Cref{thm:Main} immediately implies \Cref{thm:ProjectiveVersion}. Surfaces of general type all have finite automorphism group, which is directly analogous to the classical Hurwitz bound $|\Aut(C)| \le 84(g-1)$ for a smooth projective curve of genus $g$. In fact, $|\Aut(X)|$ can be bounded from above in terms of the self-intersection of the first Chern class of the canonical bundle; see for example \cite{Xiao}. Thus it is natural that the volume (or Euler characteristic, by Gauss--Bonnet) goes to infinity with $|G|$ for our examples.

\begin{thm}\label{thm:Examples}
There are cocompact nonarithmetic lattices $\Gam$ in $\PU(2,1)$ with nontrivial universal homomorphism.
\end{thm}

\begin{thm}\label{thm:Main}
For every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds with automorphism group isomorphic to $G$.
\end{thm}

Greenberg \cite{Greenberg} proved \Cref{thm:Main} for Riemann surfaces, i.e., complex hyperbolic $1$-manifolds. Belolipetsky and the first author \cite{BelolipetskyLubotzky} proved the analogue of \Cref{thm:Main} for isometry groups of hyperbolic $n$-manifolds, $n \ge 3$, and this paper answers a question raised in \cite[\S 6.5]{BelolipetskyLubotzky}. Our argument follows the broad outline of that paper, but differences at key steps create many new challenges we must overcome.

Recall that a compact complex hyperbolic $n$-manifold is a quotient $\Gamma \backslash \mathbb{B}^n$ of the unit ball in $\mathbb{C}^n$ by a torsion-free cocompact lattice $\Gamma$ in the Lie group $\PU(n,1)$. Note that $\mathbb{B}^1$ is simply the Poincar\'e disk with the hyperbolic metric. For $n \ge 2$, Mostow rigidity then implies that the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group, and the proof of \Cref{thm:Main} is through this interpretation.

One strong similarity between the proofs is in the input: a cocompact nonarithmetic lattice in the isometry group of the symmetric space. Moreover, \cite{BelolipetskyLubotzky} uses a nonabelian free quotient to begin the construction, whereas we use a quotient that is a cocompact Fuchsian group. The subgroup counting methods in both papers are nearly identical. Note that it is quite easy, given a free or surface group quotient of a finite index subgroup $\Lambda$ of a lattice $\Gamma$, to produce a finite index subgroup $B$ of $\Gamma$ so that $N_\Gamma(B) / B$ \emph{contains} $G$, where $N_\Gamma(B)$ is the normalizer of $B$ in $\Gamma$, but ensuring that $B$ is torsion-free and that $N_\Gamma(B) / B$ is \emph{exactly} $G$ is quite subtle.

The nonarithmetic lattices used in \cite{BelolipetskyLubotzky} are the famous examples of Gromov and Piatetski-Shapiro \cite{GPS}. These lattices naturally have the structure of a free product with amalgamation over the fundamental group of a codimension one totally geodesic subspace, and the nonabelian free quotient of a finite index subgroup is derived from this (see \cite{LubotzkyFreeQuo}). In the complex hyperbolic setting, there are no real codimension one totally geodesic subspaces \cite[\S 3.1]{Goldman}, hence no such splittings.

Our replacement, from the theory of K\"ahler groups, is the \emph{universal homomorphism} to a direct product of cocompact Fuchsian groups; see \cite{HLPSV} for an introduction. Cocompact nonarithmetic lattices in $\PU(2,1)$ with nontrivial universal homomorphism are found among the famous examples of Livn\'e \cite{Livne} and Deligne--Mostow \cite{DeligneMostow}; see \Cref{thm:Examples}. With this different input, a key lattice $\Delta$ constructed early in the outline of \cite{BelolipetskyLubotzky} now fails to be normal in the maximal lattice $\Gamma$ in $\PU(2,1)$. In a certain sense, much of the work in \Cref{sec:Main} is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \cite{BelolipetskyLubotzky}.

\begin{rem}\label{rem:Highern}
Whether or not there are cocompact nonarithmetic lattices in $\PU(n,1)$ for all $n \ge 3$ is one of the major open problems in the study of lattices in Lie groups. For any $n \ge 3$, if there is a cocompact nonarithmetic lattice in $\PU(n,1)$ with nontrivial universal homomorphism, then this paper produces infinitely many isomorphism classes of closed complex hyperbolic $n$-manifolds with isometry group any finite group $G$ \emph{verbatim}.
\end{rem}

In this paper, a genus $g$ surface group will always mean the fundamental group of a closed Riemann surface of genus $g$, despite the slight clash of terminology with fundamental groups of complex surfaces also appearing. In \Cref{ssec:FuchsianAut} we will prove the following result on automorphisms of surface groups, which is an analogue of a theorem of the first author in the free group case \cite[Thm.\ 1]{LubotzkyFree} that may be of independent interest.

\begin{thm}\label{thm:Fixpsubs}
Let $T$ be a surface group of genus $g \ge 2$, $p$ be a prime, and suppose that $\varphi \in \Aut(T)$ has the property that $\varphi(N) = N$ for all finite index normal subgroups of $T$ with index a power of $p$. Then $\varphi$ is inner.
\end{thm}

\noindent\textbf{Interpretation as a complex projective surface}

\medskip

Compact complex hyperbolic $2$-manifolds are also minimal smooth complex projective surfaces of general type \cite[\S V.20]{BPV}. The appendix to this paper recasts \Cref{thm:Main} in that context to prove the following result.

\begin{thm}\label{thm:ProjectiveVersion}
For every finite group $G$, there are infinitely many isomorphism classes of irreducible, minimal smooth complex projective surfaces of general type with automorphism group $G$.
\end{thm}

A surface is irreducible if it is not finitely (\'etale) covered by a product of complex curves. Greenberg's theorem easily produces products of complex curves with any finite automorphism group, so only the irreducible case is of interest. As described in the appendix, one may be able to construct irreducible examples using other methods. For example, there may exist a nonsingular fibration $X \to C$ over a curve $C$ (e.g., a ruled surface) where $\Aut(C) \cong G$ by Greenberg and there is a natural isomorphism $\Aut(X) \cong \Aut(C)$. However, we were not able to find a reference for \Cref{thm:ProjectiveVersion} in the literature.

\begin{rem}\label{rem:HardPart}
Our construction is also related to a holomorphic map to a smooth projective curve (see \Cref{ssec:KahlerGroups}). However, the bulk of the proof is precisely in showing that our surface has no extra automorphisms, which is highly unclear from the nature of the construction.
\end{rem}

\medskip

\noindent\textbf{Organization of the paper}

\medskip

\noindent
This paper is organized as follows. \Cref{sec:Prelim} contains preliminaries about lattices in $\PU(2,1)$, universal homomorphisms for K\"ahler groups, and the proof of \Cref{thm:Fixpsubs}. \Cref{sec:Main} proves \Cref{thm:Main}. The appendix discussions automorphism groups of smooth complex projective surfaces.

\medskip

\noindent \textbf{Acknowledgments}

\medskip

\noindent
Much of the work on this paper was done while the authors were at the conference `Discrete and profinite groups' at the Isaac Newton Institute for Mathematical Sciences, and thank the institute for its hospitality. The first author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 (N.\ 882751). The second author was supported by Grants DMS-2203555 and DMS-2506896 from the National Science Foundation and the Simons Foundation [SFI-MPS-TSM-00014184, MS].

The proof of \Cref{thm:Main} is “through” the fact that for $n\ge 2$, Mostow rigidity implies “the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group.” The argument “follows the broad outline” of \cite{BelolipetskyLubotzky}: start from “a cocompact nonarithmetic lattice” in the relevant isometry group, use a quotient of a finite index subgroup $\Lambda<\Gamma$ (here “a quotient that is a cocompact Fuchsian group,” rather than a free quotient), and then apply “subgroup counting methods” (said to be “nearly identical” to \cite{BelolipetskyLubotzky}) to build finite-index subgroups $B<\Gamma$.

A key step is to arrange, for such $B$, that the normalizer quotient realizes the desired finite group: it is “quite easy…to produce a finite index subgroup $B$ of $\Gamma$ so that $N_\Gamma(B) / B$ \emph{contains} $G$,” but “ensuring that $B$ is torsion-free and that $N_\Gamma(B) / B$ is \emph{exactly} $G$ is quite subtle,” and the “bulk of the proof is precisely in showing that our surface has no extra automorphisms.”

Because in the complex hyperbolic setting “there are no real codimension one totally geodesic subspaces…hence no such splittings,” the paper replaces the free-quotient input by the “universal homomorphism to a direct product of cocompact Fuchsian groups.” With this different input, “a key lattice $\Delta$…now fails to be normal in the maximal lattice $\Gamma$,” and “much of the work in \Cref{sec:Main} is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \cite{BelolipetskyLubotzky}.”

To prove the main theorem, the argument is “through” the fact that for $n\ge 2$, Mostow rigidity implies “the automorphism group of a complex hyperbolic manifold is precisely the holomorphic isometry group.” The argument “follows the broad outline” of \cite{BelolipetskyLubotzky}: start from “a cocompact nonarithmetic lattice” in the relevant isometry group, use a quotient of a finite index subgroup $\Lambda<\Gamma$ (here “a quotient that is a cocompact Fuchsian group,” rather than a free quotient), and then apply “subgroup counting methods” (said to be “nearly identical” to \cite{BelolipetskyLubotzky}) to build finite-index subgroups $B<\Gamma$.

A key step is to arrange, for such $B$, that the normalizer quotient realizes the desired finite group: it is “quite easy…to produce a finite index subgroup $B$ of $\Gamma$ so that $N_\Gamma(B) / B$ \emph{contains} $G$,” but “ensuring that $B$ is torsion-free and that $N_\Gamma(B) / B$ is \emph{exactly} $G$ is quite subtle,” and the “bulk of the proof is precisely in showing that our surface has no extra automorphisms.”

Because in the complex hyperbolic setting “there are no real codimension one totally geodesic subspaces…hence no such splittings,” the paper replaces the free-quotient input by the “universal homomorphism to a direct product of cocompact Fuchsian groups.” With this different input, “a key lattice $\Delta$…now fails to be normal in the maximal lattice $\Gamma$,” and “much of the work later is using properties of the universal homomorphism to circumvent this significant obstruction to executing several steps in the strategy from \cite{BelolipetskyLubotzky}.”

Question: Let an automorphism of a complex manifold mean a biholomorphism. For an arbitrary finite group $G$, which statement holds about closed complex hyperbolic $2$-manifolds (that is, compact complex manifolds of complex dimension $2$ carrying a complex hyperbolic structure)?

There exist infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose automorphism group is isomorphic to $G$.

Choices:
B. There exist infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose automorphism group contains a subgroup isomorphic to $G$.
C. There exists at least one closed complex hyperbolic $2$-manifold whose automorphism group is isomorphic to $G$.
D. There exists a closed complex hyperbolic $2$-manifold $X$ such that every finite group $G$ occurs as a subgroup of $\operatorname{Aut}(X)$.
E. For every finite group $G$, there are infinitely many isomorphism classes of closed complex hyperbolic $2$-manifolds whose full holomorphic isometry group is isomorphic to $G$.

The stem does not explicitly reveal the correct choice and gives only definitions plus a broad existence question. There are no direct textual cues favoring the strongest statement.

The item is close to theorem recognition: it asks which existence statement holds among nearby formulations. It is not a pure verbatim restatement because the options vary in quantifier strength and exactness, but it still mainly tests recall of the precise theorem.

Some reasoning is needed to compare 'isomorphic to G' versus 'contains G,' and 'exists one' versus 'infinitely many,' but the question mostly rewards knowing the exact statement rather than generating a conclusion from underlying mathematical arguments.

The distractors are plausible and target common failure modes: weakening the theorem (C), confusing exact automorphism group with containing a subgroup (B, D), and introducing an overreaching uniform-family claim (E). They are distinct and mathematically meaningful.

A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward precise statement recall than genuine generative mathematical reasoning.