\begin{lemma}\label{lem:comparison}
Let $C:=\sup_{\Omega_\epsilon}|K_M|$. Then
$$
|\M_r(\Gamma^\epsilon)-\M_r(\Gamma)|\leq \big((r+1)\M_{r+1}(\Gamma^\epsilon) + C\M_{r-1}(\Gamma^\epsilon)\big)\epsilon.
$$
\end{lemma}

\begin{theorem}\label{thm:main}
Let $\Gamma$ be a closed hypersurface with positive reach embedded in a Riemannian manifold $M$. Suppose there exists a sequence of closed embedded hypersurfaces $\Gamma_i\subset M$ with uniformly positive reach such  that $\Gamma_i\to\Gamma$ with respect to Hausdorff distance. Then $\mathcal{M}_r(\Gamma_i)\to \mathcal{M}_r(\Gamma)$.
\end{theorem}

\label{sec:intro}
Let $\Gamma$ be an oriented $\mathcal{C}^{1,1}$ hypersurface in a Riemannian $n$-manifold $M$. Then the principal curvatures
$\kappa:=(\kappa_1,\dots,\kappa_{n-1})$ of $\Gamma$ are well-defined almost everywhere, by Rademacher's theorem, and the \emph{total $r^{th}$ mean curvature} of $\Gamma$ is given by
$$
\M_{r}(\Gamma):=\int_\Gamma \sigma_r(\kappa),
$$
where
$\sigma_r(\kappa):=\sum_{1\leq i_1<\dots<i_r\leq n-1}\kappa_{i_1}\dots \kappa_{i_r}$ are the elementary symmetric polynomials, for $1\leq r\leq n-1$.
By convention, $\sigma_0:=1$, and $\sigma_r:=0$ for $r\geq n$.
 Up to multiplicative constants, depending only on $n$, $\mathcal{M}_r(\Gamma)$ form the coefficients of the generalized Steiner's polynomial and Weyl's tube formula \cite{gray2004}. They are also known as quermassintegrals when $\Gamma$ is a convex hypersurface in Euclidean space $\mathbf{R}^n$ \cite{schneider2014}. Here we study the continuity of these fundamental objects. In particular, we show:

\section{Introduction}\label{sec:intro}
Let $\Gamma$ be an oriented $\C^{1,1}$ hypersurface in a Riemannian $n$-manifold $M$. Then the principal curvatures
$\kappa:=(\kappa_1,\dots,\kappa_{n-1})$ of $\Gamma$ are well-defined almost everywhere, by Rademacher's theorem, and the \emph{total $r^{th}$ mean curvature} of $\Gamma$ is given by
$$
\M_{r}(\G):=\int_\G \sigma_r(\kappa),
$$
where
$\sigma_r(\kappa):=\sum_{1\leq i_1<\dots<i_r\leq n-1}\kappa_{i_1}\dots \kappa_{i_r}$ are the elementary symmetric polynomials, for $1\leq r\leq n-1$.
By convention, $\sigma_0:=1$, and $\sigma_r:=0$ for $r\geq n$.
 Up to multiplicative constants, depending only on $n$, $\mathcal{M}_r(\Gamma)$ form the coefficients of the generalized Steiner's polynomial and Weyl's tube formula \cite{gray2004}. They are also known as quermassintegrals when $\Gamma$ is a convex hypersurface in Euclidean space $\R^n$ \cite{schneider2014}. Here we study the continuity of these fundamental objects. In particular, we show:

The \emph{reach} of $\Gamma$, denoted by $\text{reach}(\Gamma)$, is the supremum of $\epsilon\geq 0$ such that through each point of $\Gamma$ there pass a pair of (geodesic) balls of radius $\epsilon$ whose interiors are disjoint from $\Gamma$. If $\text{reach}(\Gamma)>0$, then $\Gamma$ is $\C^{1,1}$ \cite[Lem. 2.6]{ghomi-spruck2022}. Thus $\M_r(\Gamma)$ is well-defined. Theorem \ref{thm:main} was established in $\R^n$ by Federer \cite[Thm. 5.9]{federer1959}. The general version should follow from the theory of smooth valuations \cite{alesker2007} assuming convergence of normal cycles \cite{fu1994,zahle1986}; however, the latter has not been explicitly developed in Riemannian manifolds. We give a more direct and fairly self-contained argument via Stokes theorem and universal differential forms introduced by Chern \cite{chern1945}.

The prime motivation for this work is the next result, which follows from Theorem \ref{thm:main} via recent results
for total curvatures \cite{ghomi-spruck2023total}. A \emph{Cartan-Hadamard manifold} $M$ is a complete, simply connected manifold with nonpositive curvature. A subset of $M$ is \emph{convex} if it contains the geodesic connecting every pair of its points. A \emph{convex hypersurface} $\Gamma\subset M$ is the boundary of a compact convex set with interior points. We define
\begin{equation}\label{eq:def}
\mathcal{M}_r(\Gamma):=\lim_{\epsilon\searrow 0}\M_r(\Gamma^\epsilon),
\end{equation}
 where $\Gamma^\epsilon$ denotes the outer parallel hypersurface of $\Gamma$ at distance $\epsilon$. Note that $\mathcal{M}_r(\Gamma^\epsilon)$ is well-defined since $\text{reach}(\Gamma^\epsilon)\geq\epsilon$ and thus $\Gamma^\epsilon$ is $\C^{1,1}$ for $\epsilon>0$. Furthermore,
the limit exists since
\begin{equation}\label{eq:nondecreasing}
\epsilon\mapsto\mathcal{M}_r(\Gamma^\epsilon) \;\,\text{is nondecreasing},
\end{equation}
by \cite[Cor. 4.4]{ghomi-spruck2023total}, and $\mathcal{M}_r(\Gamma^\epsilon)\geq 0$ since $\Gamma^\epsilon$ is convex (assuming proper orientation, so that the principal curvatures are nonnegative). See \cite{ghomi-spruck2022} for basic facts about convex sets and their distance functions in Cartan-Hadamard manifolds.

\begin{theorem}\label{thm:main2}
The total curvature functionals $\mathcal{M}_r$ are continuous on the space of convex hypersurfaces in a Cartan-Hadamard manifold with respect to Hausdorff distance.
\end{theorem}

\section{Proof of Theorem \ref{thm:main}}
Let $\nu$ be a unit normal vector field along $\Gamma$, and $u\colon M\to\R$ be the signed distance function of $\Gamma$ with respect to $\nu$. For $0<\delta<\text{reach}(\Gamma)$, let $U:=u^{-1}([-\delta,\delta])$ be the \emph{tubular neighborhood} of $\Gamma$ of radius $\delta$. Then $u\in\C^{1,1}(U)$ \cite[Lem. 2.6]{ghomi-spruck2022}, which means that $u$ is $\C^{1,1}$ in a collection of local coordinate charts of $M$ covering $U$. Fix $\delta<\min\{\text{reach}(\Gamma)/2$, $\text{reach}(\Gamma_i)/2\}$. We assume $i$ is so large that $\Gamma_i\subset U$.

\begin{lemma}\label{lem:Gi}
$\Gamma_i\to \Gamma$ in $\C^{1,1}$ topology.
\end{lemma}
\begin{proof}
By \cite[Prop. 2.8]{ghomi-spruck2022}, the Hessians of $u$ and $u_i$ are uniformly bounded almost everywhere on $U$. Hence, on $U$, the gradients $\nabla u$ and $\nabla u_i$ are uniformly Lipschitz. It only remains to check that $\nabla u_i\to\nabla u$ pointwise on $U$.
For any point $p\in U\setminus \Gamma$ there exists a (geodesic) sphere $S\subset U$ of radius $|u(p)|>0$ centered at $p$ with $S\cap\Gamma=\{\ol p\}$. Similarly, assuming $i$ is so large that $p\not\in\Gamma_i$, there exists a sphere $S_i\subset U$ of radius $|u_i(p)|>0$ centered at $p$ with $S_i\cap\Gamma_i=\{\ol p_i\}$. Since $\Gamma_i\to\Gamma$ in Hausdorff distance, $u_i\to u$ in $\C^0(U)$. Thus
$$
\dist(\ol p_i,S)\leq\dist(S_i,S)\to 0,\quad\quad\text{and}\quad\quad
\dist(\ol p_i,\Gamma)\leq \dist(\Gamma_i,\Gamma)\to 0,
$$
where $\dist$ is the Riemannian distance in $M$.
Thus any limit point of $\ol p_i$ lies in $\Gamma\cap S=\{\ol p\}$, or $\ol p_i\to \ol p$. It follows that $\nabla u_i(p)\to \nabla u(p)$, since
these gradients are unit tangent vectors at $p$ to geodesic segments $p\ol p$ and $p \ol p_i$. Since the gradients are uniformly Lipschitz, $\nabla u_i(p)\to\nabla u(p)$ for all $p\in U$, which completes the proof.
\end{proof}

To prove the general case, let $\Gamma'\subset U$ be a hypersurface parallel to $\Gamma$, i.e., a level set of $u$ different from $\Gamma$. Let $\Omega'$ be the region between $\Gamma'$ and $\Gamma$. Then $\Gamma_i$ will be disjoint from $\Gamma'$ for $i$ sufficiently large. Let $\Omega_i'$ be the region between $\Gamma'$ and $\Gamma_i$. Since $\Gamma$, $\Gamma'$, and $\Gamma_i$ all have uniformly positive reach, the same argument for \eqref{eq:Mr} shows that
$$
|\M_r(\Gamma_i)-\M_r(\Gamma')|\leq C|\Omega_i'|, \quad\quad\text{and}\quad\quad |\M_r(\Gamma')-\M_r(\Gamma)|\leq C|\Omega'|,
$$
for some constant $C$. Thus, by the triangle inequality,
$$
\lim_{i\to\infty}|\M_r(\Gamma_i)-\M_r(\Gamma)|
\leq
\lim_{i\to\infty} C(|\Omega_i'|+|\Omega'|)
\leq
2C|\Omega'|.
$$
As $\Gamma'\to\Gamma$, we have $|\Omega'|\to 0$, which completes the proof.

Suppose there exists a sequence of convex hypersurfaces
$\Gamma_i\subset M$ such that $\Gamma_i\to\Gamma$ with respect to Hausdorff distance. By the triangle inequality,
\begin{multline*}
|\M_r(\Gamma_i)-\M_r(\Gamma)|
\leq \\
|\M_r(\Gamma_i)-\M_r(\Gamma_i^\epsilon)|
+
|\M_r(\Gamma_i^\epsilon)-\M_r(\Gamma^\epsilon)|
+
|\M_r(\Gamma^\epsilon)-\M_r(\Gamma)|.
\end{multline*}
As $i\to \infty$, the middle term on the right hand side vanishes by Theorem \ref{thm:main}. To bound the first term, let $B\subset M$ be a closed ball which contains $\Gamma$ in its interior, and set $C:=\sup_B|K_M|$. For $i$ sufficiently large and small $\epsilon$, we have $\Gamma_i$, $\Gamma_i^\epsilon\subset B$; therefore, Lemma \ref{lem:comparison} yields
$$
|\M_r(\Gamma_i)-\M_r(\Gamma^\epsilon_i)|
\leq
\big((r+1)\M_{r+1}(\Gamma^\epsilon_i) + C\M_{r-1}(\Gamma^\epsilon_i)\big)\epsilon.
$$
But $\M_{r+1}(\Gamma^\epsilon_i)\to \M_{r+1}(\Gamma^\epsilon)$ and $\M_{r-1}(\Gamma^\epsilon_i)\to \M_{r-1}(\Gamma^\epsilon)$ by Theorem \ref{thm:main}, since these hypersurfaces have uniformly positive reach. Thus
$$
\lim_{i\to\infty} |\M_r(\Gamma_i)-\M_r(\Gamma)|
\leq
\big((r+1)\M_{r+1}(\Gamma^\epsilon) + C\M_{r-1}(\Gamma^\epsilon)\big)\epsilon+
|\M_r(\Gamma^\epsilon)-\M_r(\Gamma)|.
$$
Letting $\epsilon\to 0$ and recalling \eqref{eq:def} completes the proof.

The \emph{reach} of $\Gamma$, denoted by $\text{reach}(\Gamma)$, is the supremum of $\varepsilon\geq 0$ such that through each point of $\Gamma$ there pass a pair of (geodesic) balls of radius $\varepsilon$ whose interiors are disjoint from $\Gamma$. If $\text{reach}(\Gamma)>0$, then $\Gamma$ is $\mathcal{C}^{1,1}$ \cite[Lem. 2.6]{ghomi-spruck2022}. Thus $\M_r(\Gamma)$ is well-defined.  Theorem \ref{thm:main} was established in $\mathbf{R}^n$ by Federer \cite[Thm. 5.9]{federer1959}. The general version should follow from the theory of smooth valuations \cite{alesker2007} assuming convergence of normal cycles \cite{fu1994,zahle1986}; however, the latter has not been explicitly developed  in Riemannian manifolds. We give a more direct and fairly self-contained argument via Stokes theorem and universal differential forms introduced by Chern \cite{chern1945}.

The prime motivation for this work is the next result, which follows from Theorem \ref{thm:main} via recent results
for total curvatures \cite{ghomi-spruck2023total}. A \emph{Cartan-Hadamard manifold} $M$ is a  complete, simply connected manifold with nonpositive curvature. A subset of $M$ is \emph{convex} if it contains the geodesic connecting every pair of its points. A \emph{convex hypersurface} $\Gamma\subset M$ is the boundary of a compact convex set with interior points. We define
\begin{equation}\label{eq:def}
\mathcal{M}_r(\Gamma):=\lim_{\varepsilon\searrow 0}\M_r(\Gamma^\varepsilon),
\end{equation}
 where $\Gamma^\varepsilon$ denotes the outer parallel hypersurface of $\Gamma$ at distance $\varepsilon$. Note that $\mathcal{M}_r(\Gamma^\varepsilon)$ is well-defined since $\text{reach}(\Gamma^\varepsilon)\geq\varepsilon$ and thus $\Gamma^\varepsilon$ is $\mathcal{C}^{1,1}$ for $\varepsilon>0$. Furthermore,
the limit exists since
\begin{equation}\label{eq:nondecreasing}
\varepsilon\mapsto\mathcal{M}_r(\Gamma^\varepsilon)  \;\,\text{is nondecreasing},
\end{equation}
by \cite[Cor. 4.4]{ghomi-spruck2023total}, and $\mathcal{M}_r(\Gamma^\varepsilon)\geq 0$ since $\Gamma^\varepsilon$ is convex (assuming proper orientation, so that the principal curvatures are nonnegative). See \cite{ghomi-spruck2022} for basic facts about convex sets and their distance functions in Cartan-Hadamard manifolds.

\begin{theorem}\label{thm:main2}
The total curvature functionals $\mathcal{M}_r$ are continuous on the space of convex hypersurfaces in a Cartan-Hadamard manifold with respect to Hausdorff distance.
\end{theorem}

This result simplifies a number of arguments, e.g., see \cite[Note 3.7]{ghomi-spruck2022} and \cite[Lem. 3.3]{ghomi-spruck2023minkowski},  related to the \emph{Cartan-Hadamard conjecture} on the isoperimetric inequality in spaces of nonpositive curvature. The conjecture follows  if the \emph{total Gauss-Kronecker curvature}
\begin{equation}\label{eq:conjecture}
\M_{n-1}(\Gamma)\geq|\mathbf{S}^{n-1}|
\end{equation}
for convex hypersurfaces $\Gamma$ in Cartan-Hadamard manifolds, where $|\mathbf{S}^{n-1}|$ is the volume of the unit sphere in $\mathbf{R}^n$  \cite{ghomi-spruck2022}. Our proof of Theorem \ref{thm:main2} employs an estimate for total curvatures of parallel hypersurfaces (Lemma \ref{lem:comparison}) which may be of further interest.

Theorem~\ref{thm:main} is said to admit a "more direct and fairly self-contained argument via Stokes theorem and universal differential forms introduced by Chern \cite{chern1945}." The introduction also notes an alternative route: "The general version should follow from the theory of smooth valuations \cite{alesker2007} assuming convergence of normal cycles \cite{fu1994,zahle1986}; however, the latter has not been explicitly developed in Riemannian manifolds."

To prove the main theorem, one may use a "more direct and fairly self-contained argument via Stokes theorem and universal differential forms introduced by Chern \cite{chern1945}." The introduction also notes an alternative route: "The general version should follow from the theory of smooth valuations \cite{alesker2007} assuming convergence of normal cycles \cite{fu1994,zahle1986}; however, the latter has not been explicitly developed in Riemannian manifolds."

Question:
Let $M$ be a Riemannian $n$-manifold, let $1\le r\le n-1$, and let $\Gamma\subset M$ be a closed embedded hypersurface with positive reach. For a $\mathcal C^{1,1}$ hypersurface, define its total $r^{\text{th}}$ mean curvature by
$$
\mathcal M_r(\Gamma):=\int_\Gamma \sigma_r(\kappa),
$$
where $\kappa=(\kappa_1,\dots,\kappa_{n-1})$ are the principal curvatures and
$$
\sigma_r(\kappa)=\sum_{1\le i_1<\cdots<i_r\le n-1}\kappa_{i_1}\cdots \kappa_{i_r}.
$$
Assume there is a sequence of closed embedded hypersurfaces $\Gamma_i\subset M$ with uniformly positive reach, meaning $\inf_i \operatorname{reach}(\Gamma_i)>0$, and suppose $\Gamma_i\to\Gamma$ in Hausdorff distance. Which statement holds?

The total $r^{\text{th}}$ mean curvatures converge: $$\mathcal M_r(\Gamma_i)\to \mathcal M_r(\Gamma).$$

Choices:
B. The total $r^{\text{th}}$ mean curvatures converge provided, in addition, that $\Gamma_i\to\Gamma$ in the $\mathcal C^{1,1}$ topology on a common tubular neighborhood; Hausdorff convergence with uniformly positive reach alone is not sufficient in general.
C. After passing to a subsequence, one has $$\mathcal M_r(\Gamma_{i_j})\to \mathcal M_r(\Gamma).$$
D. The total $r^{\text{th}}$ mean curvatures converge for every sequence of closed embedded hypersurfaces $\Gamma_i\subset M$ with $\Gamma_i\to\Gamma$ in Hausdorff distance, even without any uniform positive lower bound on $\operatorname{reach}(\Gamma_i)$.
E.
The total $r^{\text{th}}$ mean curvatures converge, and in fact there is a constant $C=C(M,n,r,\Gamma)>0$ such that for all sufficiently large $i$,
$$
\bigl|\mathcal M_r(\Gamma_i)-\mathcal M_r(\Gamma)\bigr|\le C\, d_H(\Gamma_i,\Gamma),
$$
where $d_H$ denotes the Hausdorff distance.

The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the valid conclusion, without giving away that full-sequence convergence must hold.

This is essentially a direct theorem-identification item: the hypotheses are stated in the stem and the correct choice is the theorem's conclusion almost verbatim. It mainly tests recall of the stated continuity result rather than selection among substantively different mathematical outcomes.

There is some reasoning pressure because the student must distinguish the exact conclusion from nearby variants: weaker subsequential convergence, unnecessary stronger regularity assumptions, removal of the reach hypothesis, and an unjustified Lipschitz estimate. However, once the theorem is recognized, the answer is straightforward.

The distractors are strong and mathematically plausible. B adds a tempting extra-regularity requirement, C offers a weaker true-looking statement, D removes a crucial hypothesis, and E overstates continuity to a Lipschitz rate. These reflect realistic failure modes.

A reasonably well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological because the correct answer is essentially the theorem's conclusion under the exact hypotheses given.


(\cite[conjecture~1.20]{KaKuMGSt})
\label{conj1}
$\xi_{g,1}$ is the unique closed, embedded minimal surface $M$ of genus $g\geq 2$ in $\mathbb{S}^3$ whose isometry group contains a Klein subgroup $\Z_2\times \Z_2=\langle \phi,\phi'\rangle$ such that $\phi,\phi'$ are reflections across orthogonal spheres $S,S'\subset \mathbb{S}^3$ (orientation-reversing on $M$), the closures $\Omega^+,\Omega^-=\phi(\Omega^+)$ of the components of $M\setminus S$
have genus zero, and all the boundary components of $\partial \Omega^+=\partial \Omega^-$ intersect the great circle $S\cap S'$.

\begin{theorem}\label{mainthm}
$\xi_{2,1}$ is the unique closed embedded minimal surface in $\esf^3$ with genus 2 and {\em $D_{4h}$-symmetric}.
\end{theorem}

\begin{conjecture}(\cite[conjecture~1.20]{KaKuMGSt})
\label{conj1}
$\xi_{g,1}$ is the unique closed, embedded minimal surface $M$ of genus $g\geq 2$ in $\esf^3$ whose isometry group contains a Klein subgroup $\Z_2\times \Z_2=\langle \phi,\phi'\rangle$ such that $\phi,\phi'$ are reflections across orthogonal spheres $S,S'\subset \esf^3$ (orientation-reversing on $M$), the closures $\Omega^+,\Omega^-=\phi(\Omega^+)$ of the components of $M\setminus S$
have genus zero, and all the boundary components of $\partial \Omega^+=\partial \Omega^-$ intersect the great circle $S\cap S'$.
\end{conjecture}

The existence and classification of closed embedded minimal surfaces in compact three-manifolds is a rich theory in differential geometry, with the classical and most important case of the closed embedded minimal surfaces in the three-sphere $\mathbb{S}^3$.

Classifying closed embedded minimal surfaces in $\mathbb{S}^3$
is a central and challenging problem. A classical result of 1966 by Almgren~\cite{alm3} asserts that the only closed immersed minimal surface in $\mathbb{S}^3$ of genus zero is the totally geodesic $\mathbb{S}^2$. In his pioneering 1970 paper~\cite{la3}, Lawson settled the foundations of the
theory of minimal surfaces in $\mathbb{S}^3$ and constructed infinitely many examples, both embedded and immersed. In particular, he proved the
existence of closed embedded (hence orientable) minimal surfaces of every genus in $\mathbb{S}^3$, by producing a family $\xi_{m,k}\subset \mathbb{S}^3$ of examples
of genus $mk$ for every $m,k\in \hbox{\bb N}$. In the simplest case $m=k=1$,
$\xi_{1,1}$ is the Clifford torus, whose uniqueness among embedded minimal tori in $\mathbb{S}^3$ gave rise to the famous Lawson conjecture, solved by Brendle~\cite{bren1} one decade ago. Another remarkable result concerning $\xi_{1,1}$ is the recent solution by Marques and Neves~\cite{mane1} of the well-known Willmore conjecture, where they showed that among all minimal surfaces in $\mathbb{S}^3$ of genus $g\geq 1$, the one with the smallest area is the Clifford torus.
In the second simplest case, the Lawson surface $\xi_{2,1}$ of genus two has a remarkably large symmetry group inside $O(4)$, isomorphic to the group $O_{48}=\Z_2\times S_4$ of isometries of a regular octahedron (Iso$(\xi_{2,1})$ is not contained in any direct inclusion of $O(3)$ into $O(4)$ by a hyperplane of $\mathbb{R}^4$). A fundamental problem arises:
\begin{quote}
Is $\xi_{2,1}$ the unique closed embedded minimal surface of genus two
in $\mathbb{S}^3$?
\end{quote}
This question remains open nowadays, and it is known to be false starting in genus three: In 1988, Karcher, Pinkall and Sterling~\cite{kps1} used tessellations of $\mathbb{S}^3$ with the symmetries of Platonic solids in $\mathbb{R}^3$
and conjugation to produce new examples of closed, embedded minimal surfaces in $\mathbb{S}^3$ of genera $3,5,6,7,11,17,19,73,601$. Subsequent
examples have been found by Choe and Soret~\cite{cs16} and Bai, Wang and Wang~\cite{BWW21} using similar methods.

There are other techniques available to produces examples of closed embedded minimal surfaces in $\mathbb{S}^3$, each with a different flavor. A PDE-based method is desingularization-doubling-gluing, that have given rise to a rich family of examples with the common feature of having high genus (Kapouleas and Yang~\cite{kapya}, Kapouleas and McGrath~\cite{KaMcGr1}, Wiygul~\cite{Wiy20}, Kapouleas and Wiygul~\cite{KapWiy22}). Equivariant min-max theory has been
used to produce closed embedded minimal surfaces that bear resemblance to both doublings and desingularizations of stationary varifolds within $\mathbb{S}^3$, as in Ketover \cite{Ket16}. Another fruitful technique is
via maximization of eigenvalues under symmetry constraints: if one seeks
for critical eigenvalues of the Laplace (or Steklov) eigenvalues for metrics
on compact surfaces which are invariant under discrete symmetry groups, we will find minimal immersions into spheres (or free boundary minimal immersions into Euclidean balls), and the problem amounts to control both the dimension of the ambient space (which is equivalent to controlling the dimension of the corresponding equivariant eigenspace) and embeddedness. This technique was started by Takahashi~\cite{taka1}, El Soufi and Ilias~\cite{sl1}, Montiel and Ros~\cite{mro2}, Nadirashvili~\cite{na2},
and then expanded by Jakobson, Levitin, Nadirashvili, Nigam and Polterovich~\cite{jako},
Jakobson, Nadirashvili and Polterovich~\cite{jako1}, Cianci, Karpukhin and Medvedev~\cite{ckm1}, Petrides~\cite{ptr2}, Karpukhin, Petrides and Stern~\cite{kapeSt1}, among others.
Very recently, Karpukhin, Kusner, McGrath and Stern~\cite{KaKuMGSt} have used this technique to produce embedded, free boundary minimal surfaces in the unit ball $\hbox{\bb B}^3$ of $\mathbb{R}^3$ of arbitrary topology, and many other examples in $\mathbb{S}^3$.

As for uniqueness, few results are known apart from those mentioned above~\cite{alm3,bren1,mane1}. Kapouleas and Wiygul~\cite{KaWiy0} have characterized the Lawson surfaces $\xi_{m,k}$ among closed embedded minimal surfaces in $\mathbb{S}^3$ with their genus and symmetry group.
Their approach is based on using the topology and symmetry assumptions to decompose every candidate surface into geodesic quadrilaterals with prescribed vertices, and then proving the uniqueness of the solution of
the Plateau problem for this contour.
In relation to this uniqueness result under symmetry, a relevant question consists of relaxing the isometry assumption on $\xi_{m,k}$ by only imposing a certain subgroup of the whole isometry group while keeping the topology, and still wonder if it is possible to characterize $\xi_{m,k}$. In this line, it is worth mentioning the following conjecture:

\item In Section~\ref{secConjugacy}, we will study the conjugate minimal disk $\Sigma_{l,\omega}^*\subset \esf^3$ to $\Sigma_{l,\omega}$ and related half $\mathcal{F}^*$ of $\Sigma_{l,\omega}^*$, for every $(l,\omega)\in \mathfrak{C}_1\cup \mathfrak{C}_2$, and similarly for the conjugate $\Sigma_{\sigma}^*$ of the unique solution $\Sigma_{\sigma}$ to the Plateau problem with contour $\mathcal{P}_{\sigma}$.
We will be interested in conditions under which successive Schwarz reflections of $\mathcal{F}^*$ across its boundary edges produce a closed embedded minimal surface in $\esf^3$ (in short, we will call this the {\it closing problem} for $(l,\omega)\in \mathfrak{C}_1\cup \mathfrak{C}_2$ or for $\sigma \in [0,\pi/2]$).
The boundary $\partial \mathcal{F}^*$ of $\mathcal{F}^*$ consists of four analytic arcs,
three of which, $\alpha^*_+,\be_+^*,\de_+^*$, are geodesics of reflective symmetry
(lying in totally geodesic two-spheres); the fourth boundary arc inside $\partial \mathcal{F}^*$ is a great circle arc $\g^*$ with $\mbox{Length}(\g^*)=\mbox{Length}(\g)$. Up to congruencies, we will normalize $\mathcal{F}^*$ in $\esf^3$ so that
\[
\gamma^*\subset \G_{{\bf k},{\bf v}_+}\quad \mbox{ and }\quad \de_+^*\subset \mathcal{S}_2 \cap \overline{\B ^+_3}.
\]
We will discard that $(l,\omega)$ solves the closing problem when
this pair lies in $\mathfrak{C}_2$ (Lemma~\ref{reduceC2}) or in the
closed triangle $\mathfrak{T}^-\cup \mathfrak{D}\subset \mathfrak{C}_1$ given by
the inequality $l+\omega \leq \pi/2$ (Lemma~\ref{reduceT-}), and also for every
choice of the parameter $\sigma\in [0,\pi/2]$ (Lemma~\ref{reducesigma}). These
lemmas will reduce our search of parameters $(l,\omega)$ that solve the
closing problem to those those lying in $\mathfrak{T}^+=\{ l+\omega > \pi/2\} \subset \mathfrak{C}_1$, which is
foliated by level curves of the function $\tau$ corresponding to values $\tau(l, \omega)\in (0,\pi/2)$.

\begin{lemma}
\label{deltaLawson}
In the above situation, we have:
\begin{enumerate}
\item $(l_{\mathcal{L}},\omega_{\mathcal{L}})$ is the unique solution to the closing problem along the curve given by~\eqref{eq:hexagon}.
\item ${\rm Length}(\de ^* _{\mathcal{L}}) = {\rm Length}(\de _+ ) < \pi /2$.
\end{enumerate}
\end{lemma}
\begin{proof}
If $(l,\omega)\in \mathfrak{C}_1$ satisfies $\cos l + \cos \omega = 1$, then the union of the compact minimal disk $\Sigma _{l ,\omega}$ with its
$\pi$-rotation about the great circle containing the edge $\a$ inherits the reflection symmetries across the totally geodesic two-spheres $\Pi _{\pm , \omega}$ given by \eqref{Piomega}, recall that $\bx \in \mathcal S _2 \cap \Pi _{+, \omega} \cap \Pi _{-, \omega}$. Hence, its conjugate minimal disk $\Sigma ^* _{l, \omega}$ contains three ambient geodesics passing through $\bx ^*$. This implies that if $\Sigma ^* _{l, \omega}$ produces a closed embedded minimal surface $\Sigma ^* \subset \esf ^3$ of genus $2$ after Schwarz reflection, then $\Sigma ^*$ has the symmetry group of the Lawson surface of genus $2$ and, hence, $\Sigma ^* = \xi _{2,1}$ by \cite{KaWiy0}.
So, item~1 is proved.

We will assume $\Theta \in (-\pi/2,0]$ in order to obtain a contradiction, which will be based on estimating by below the length of the curve $N^*\circ \de_+^*
\subset \mathcal{S}_2$.
Given $\varphi\in[-\pi/2,\pi/2]$, let $c_{\varphi}\subset\mathcal S_2$ be the half great circle with end points $\pm {\bf j}$ that contains the point $\cos\varphi \,{\bf k}+\sin\varphi \,{\bf e}$. Then, the family $\{ c_{\varphi}\setminus\set{\pm {\bf j}}\ | \ \varphi\in [-\pi/2,\pi/2]\}$ foliates the closed hemisphere $\mathcal S_2\cap \overline{\B_4^+}$ minus $\pm {\bf j}$. Decreasing $\varphi$ from $\pi/2$ to zero we find a first contact point between $c_{\varphi_1}$ and $\delta_+^*([0,l])$, for some $\varphi_1\in (0,\pi/2)$. Let $\delta_+^*(t_1)$ be this first contact point, where
$t_1\in (0,l)$. As $c_{\varphi_1}$ and $\delta_+^*$ are tangent at $\delta_+^*(t_1)$ and $\delta_+^*$ is a geodesic of reflective symmetry of $\Sigma^*$, we have
\begin{equation}\label{nt1}
N^* (\delta_+^*(t_1))=\cos\varphi_1\,{\bf e} - \sin\varphi_1\,{\bf k},
\end{equation}
hence the strictly convex curve $N^* \circ \de ^* _+$ satisfies
\begin{equation}\label{longitud1}
{\rm Length}(N^* \circ \de ^* _+)_0^{t_1}\geq \frac{\pi}{2}.
\end{equation}
Since $\langle {\bf z}_+^*,{\bf e}\rangle =\langle \de_+^*(l),{\bf e}\rangle <0$ and $\langle \de_+^*(t_1),{\bf e}\rangle >0$, by continuity there exists $t_2\in (t_1,l)$ such that $\langle \de_+^*(t_2),{\bf e}\rangle =0$.
\par\noindent
{\bf Claim:} $\varphi_1<\pi/4$.
\begin{proof}[Proof of the claim]
Arguing by contradiction, suppose $\varphi_1\geq\pi/4$. Then, both $d_{\esf ^3} (\delta_+^*(0),\delta_+^*(t_1))$ and $d_{\esf ^3} (\delta_+^*(t_1),\delta_+^*(t_2))$ are at least $\pi/4$. Therefore,
\[
\frac{\pi}{2}\leq d_{\esf ^3} (\delta_+^*(0),\delta_+^*(t_1)) +d_{\esf ^3} (\delta_+^*(t_1),\delta_+^*(t_2)) <{\rm Length}(\delta_+^*),
\]
which contradicts that ${\rm Length}(\delta_+^*)=l<\pi/2$. Hence the claim follows.
\end{proof}

Let $\Sigma^*\subset \esf^3$ be a $D_{4h}$-symmetric, embedded minimal surface of genus $2$. By Proposition~\ref{lem3.2}, $\Sigma^*\cap (\cap_{i=i}^4\overline{\B_i^+})$ is a minimal disk $\mathcal{F}^*$ whose
boundary consists of four regular arcs with consecutive end points,
\[
\partial \mathcal F ^* = \delta_+^* \cup \beta_+^* \cup \alpha_+^* \cup \gamma^*,
\]
where $\delta_+^*\subset \mathcal{S}_2$, $\beta_+^*\subset \mathcal{S}_1$,
$\alpha_+^*\subset \mathcal{S}_4$, and $\gamma ^*=[{\bf k},{\bf x}^*]$, being
${\bf x}^*={\bf v}_+$ the unique umbilic point of $\mathcal{F}^*$. $\Sigma^*$ is the union of 16 congruent copies of $\mathcal{F}^*$ (obtained by the action of $D_{4h}$ over $\mathcal{F}^*$) glued along their boundaries. The conjugate minimal
disk $\mathcal{F}$ to $\mathcal{F}^*$ can be normalized so that the boundary arc $\g\subset \partial \mathcal{F}$ corresponding to $\g^*$, satisfies that
$\g\subset \mathcal{S}_2\cap \overline{\B_4^+}$ and $\g$ joins the point ${\bf k}$ (corresponding by conjugation
to the point $\de_+^*\cap \g^*$) with ${\bf x}$ (corresponding to ${\bf x}^*$).
Hence, $\mathcal{F}$ can be reflected across $\mathcal{S}_2$ and $\mathcal{F}
\cup \mathcal{R}_2(\mathcal{F})$ is an $\mathcal{R}_2$-symmetric, (smooth) minimal disk bounded by great circle arcs
\[
\de_+,\de_-=\mathcal{R}_2(\de_+),\be_+,\be_-=\mathcal{R}_2(\be_+),\a=\a_+\cup \mathcal{R}_2(\a_+),
\]
meeting at right angles. In other words, $\partial
(\mathcal{F}\cup \mathcal{R}_2(\mathcal{F}))$ is a geodesic pentagon
either of the form $\mathcal{P}_{l,\omega}$ for some $(l,\omega)\in \mathfrak{C}\setminus \{ (\pi/2,0)\}$ (in fact, we can assume
$(l,\omega)\in \mathfrak{C}_1\cup \mathfrak{C}_2$ by the reduction of parameters explained in Section~\ref{sec4.6}), or of the form $\mathcal{P}_{\sigma}$,
$\sigma \in [0,\pi/2]$. By Lemmas~\ref{reduceC2}, \ref{reduceT-} and~\ref{reducesigma},
the only possible case for $\partial (\mathcal{F}\cup \mathcal{R}_2(\mathcal{F}))$ is $\mathcal{P}_{l,\omega}$ for some $(l,\omega)\in \mathfrak{T}^+\subset \mathfrak{C}_1$. By Theorem~\ref{thm3.30},
$\mathcal{P}_{l,\omega}$ is the boundary of a unique compact minimal
surface $\Sigma_{l,\omega}\subset \esf^3$, in particular
\[
\mathcal{F}\cup \mathcal{R}_2(\mathcal{F})=\Sigma_{l,\omega}.
\]
Since the length of $\g^*$ is $\pi/2$, the pair $(l,\omega)\in \mathfrak{T}_+$
that gives rise to $\mathcal{F}\cup \mathcal{R}_2(\mathcal{F})$ satisfies $L(l,\omega)=\pi/2$. As explained in Section~\ref{secL}, this implies
that $(l,\omega)$ is of the form $\Xi(\tau)$ for some $\tau\in (0,\pi/2)$. Since
the boundary arc $\be^*$ of $\mathcal{F}^*$ is contained in $\mathcal{S}_1$,
the function $\Theta$ introduced by Lemma~\ref{l63} vanishes at $\tau$. By Proposition~\ref{propos7.12}, there is a unique $\tau\in (0,\pi/2)$
with the properties $L(\Xi(\tau))=\pi/2$ and $\Theta(\tau)=0$.

It is worth mentioning a recent work by Kusner, L\"{u} and Wang~\cite{kuluwa1} where they characterize the Lawson surface $\xi_{m,k}$ among closed embedded surfaces of genus $mk$ in $\mathbb{S}^3$ that minimize the Willmore functional and are invariant under any group conjugate
to the maximal rotation-symmetry group within Iso$(\xi_{m,k})$. As a consequence, the same authors prove (Theorem~4.2 in~\cite{kuluwa1})
a similar characterization for $\xi_{m,k}$ as the one by Kapouleas and
Wiygul~\cite{KaWiy0} among closed embedded minimal surfaces of genus $mk$ in $\mathbb{S}^3$ which are invariant under certain index-two subroups of Iso$(\xi_{m,k})$.

We will prove a result closely related to Conjecture~\ref{conj1} in the case of genus $g=2$, strengthening the assumption on the Klein subgroup of $O_{48}$ by the product $D_{4h}=\Z_2\times D_4$ of $\Z_2$ (generated by a reflection $\phi$ across a sphere $S\subset \mathbb{S}^3$ so that $S$ disconnects $M$ into two genus zero components) with the dihedral group $D_4$ of order 8, see Theorem~\ref{mainthm} for a precise statement. For this reason, surfaces invariant by $\Z_2\times D_4$ are called $D_{4h}$-symmetric. Our assumption on $D_{4h}$-symmetry in Theorem~\ref{mainthm} allows to drop the hypothesis of Conjecture~\ref{conj1} on all boundary components of $\partial \Omega_1^+=\partial \Omega_1^-$ intersecting a great circle of $S$.

In order to prove Theorem~\ref{mainthm}, we first describe
the fundamental piece of an embedded, $D_{4h}$-symmetric minimal surface of genus two, and show that under conjugation, an appropriate reflection of
this conjugate fundamental piece is bounded by a geodesic pentagon in $\mathbb{S}^3$ with right angles at all its vertices. We then study the space of such geodesic, right-angled polygons, which is a two-parameter family $\mathcal P_{l, \omega}$, $(l, \omega) \in [(0, \pi)\times (-\pi/2,\pi/2)]\setminus \{ (\pi/2,0)\}$, and related one-parameter family $\mathcal{P}_{\sigma}$,
$\sigma \in [0,\pi/2]$ that describes the parameter choice $(l,\omega)=
(\pi/2,0)$ for the first family. We next show that these pentagons are the boundary of (embedded) area-minimizing disks $\Sigma _{l , \omega}$ (resp.
$\Sigma_{\sigma}$), and explore the geometric properties of $\Sigma _{l, \omega},\Sigma_\sigma$. Among these properties, we are particularly interested in understanding the behavior of the mapping that associates to each $(l, \omega)$ (resp. $\sigma$) the length $L$ of a reflective geodesic $\g_{l,\omega}\subset
\S_{l,\omega}$ (resp. $\g_{\sigma}\subset \S_{\sigma}$). The level set
$L=\pi/2$ of this function consists of the only potential examples that after conjugation and Schwarz reflection, might give rise to fundamental pieces of our desired $D_{4h}$-minimal surfaces of genus two. We then analyze the closing problem
for the surfaces arising in the level set $L=\pi/2$, in the sense of
determining under which conditions the conjugate piece $\Sigma_{l,\omega}^*$ of $\Sigma_{l,\omega}$ with $L=\pi/2$, produces after Schwarz
reflection an embedded, $D_{4h}$-symmetric minimal surface. We will prove that this closing problem has a unique solution, from which we will deduce the desired uniqueness result. See Section~\ref{sec2.6} for a detailed
description of the distribution of the paper.

In order to prove Theorem~\ref{mainthm}, the authors: (1) "first describe the fundamental piece of an embedded, $D_{4h}$-symmetric minimal surface of genus two," and show that (after conjugation) "an appropriate reflection of this conjugate fundamental piece is bounded by a geodesic pentagon in $\mathbb{S}^3$ with right angles at all its vertices." (2) They "then study the space of such geodesic, right-angled polygons," a two-parameter family $\mathcal P_{l, \omega}$, $(l,\omega)\in[(0,\pi)\times(-\pi/2,\pi/2)]\setminus\{(\pi/2,0)\}$, and a related one-parameter family $\mathcal P_{\sigma}$ for the parameter choice $(l,\omega)=(\pi/2,0)$. (3) They "next show that these pentagons are the boundary of (embedded) area-minimizing disks $\Sigma_{l,\omega}$ (resp. $\Sigma_{\sigma}$)," and study geometric properties of these disks. (4) A key quantity is the map sending $(l,\omega)$ (resp. $\sigma$) to "the length $L$ of a reflective geodesic $\gamma_{l,\omega}\subset \Sigma_{l,\omega}$ (resp. $\gamma_{\sigma}\subset \Sigma_{\sigma}$)." The "level set $L=\pi/2$" gives "the only potential examples" that can yield the desired fundamental pieces after conjugation and Schwarz reflection. (5) They "then analyze the closing problem" for surfaces in this level set, i.e. determine when the conjugate piece $\Sigma^*_{l,\omega}$ with $L=\pi/2$ produces, after Schwarz reflection, an embedded $D_{4h}$-symmetric minimal surface; they "prove that this closing problem has a unique solution," and "from which" they deduce the desired uniqueness result.

(\cite[conjecture~1.20]{KaKuMGSt})
\label{conj1}
$\xi_{g,1}$ is the unique closed, embedded minimal surface $M$ of genus $g\geq 2$ in $\mathbb{S}^3$ whose isometry group contains a Klein subgroup $\Z_2\times \Z_2=\langle \phi,\phi'\rangle$ such that $\phi,\phi'$ are reflections across orthogonal spheres $S,S'\subset \mathbb{S}^3$ (orientation-reversing on $M$), the closures $\Omega^+,\Omega^- = \phi(\Omega^+)$ of the components of $M\setminus S$
have genus zero, and all the boundary components of $\partial \Omega^+=\partial \Omega^-$ intersect the great circle $S\cap S'$.

Question: Let \(g\ge 2\), and let \(\xi_{g,1}\) denote Lawson's closed embedded minimal surface of genus \(g\) in \(\mathbb S^3\). Suppose \(M\subset \mathbb S^3\) is a closed embedded minimal surface of genus \(g\) whose isometry group contains a Klein four subgroup \(\mathbb Z_2\times \mathbb Z_2=\langle \phi,\phi'\rangle\), where \(\phi\) and \(\phi'\) are reflections across orthogonal spheres \(S,S'\subset \mathbb S^3\) and each acts orientation-reversingly on \(M\). Let \(\Omega^+\) and \(\Omega^- = \phi(\Omega^+)\) be the closures of the two components of \(M\setminus S\). Assume that both \(\Omega^+\) and \(\Omega^-\) have genus zero, and that every connected component of \(\partial\Omega^+=\partial\Omega^-\) meets the great circle \(S\cap S'\). Which statement is conjectured to hold?

The only such surface is the Lawson surface \(\xi_{g,1}\); equivalently, \(\xi_{g,1}\) is the unique closed embedded minimal surface of genus \(g\) in \(\mathbb S^3\) satisfying all of the symmetry and boundary-intersection conditions above.

Choices:
B. For every genus \(g\ge 2\), any such surface \(M\) is congruent either to the Lawson surface \(\xi_{g,1}\) or to one of a one-parameter family of closed embedded minimal surfaces obtained by varying the reflective geodesic length along the level set \(L\le \pi/2\) subject to the same Klein four symmetry and boundary-intersection conditions.
C. Any such surface \(M\) is congruent to a Lawson surface of the form \(\xi_{g',1}\) for some genus \(g'\ge 2\).
D. The only such surface is the Lawson surface \(\xi_{g,1}\), and this remains true even if one drops the requirement that every connected component of \(\partial\Omega^+=\partial\Omega^-\) meet the great circle \(S\cap S'\), retaining only the Klein four symmetry and the genus-zero condition on \(\Omega^\pm\).
E. For every genus \(g\ge 2\), there exists a unique closed embedded minimal surface \(M\subset \mathbb S^3\) with the stated Klein four symmetry and genus-zero decomposition property, and this surface is determined effectively by the condition that the associated right-angled geodesic pentagon lies on the level set \(L=\pi/2\), without any further closing condition.

replace unique closing solution on L=pi/2 by family on L<=pi/2

need extra closing condition beyond L=pi/2

The stem does not explicitly state the correct conclusion, but it strongly cues it by introducing Lawson's surface \(\xi_{g,1}\) and then asking which conjectural uniqueness statement holds under exactly those hypotheses.

This is largely a recognition task about the precise formulation of a conjecture. The correct option mostly restates the natural conjectural conclusion attached to the setup rather than requiring application of a theorem to a new situation.

Some reasoning is needed to distinguish the exact conjecture from nearby stronger, weaker, or tampered variants, but the task is still primarily recall/discrimination rather than substantial mathematical generation.

The distractors are plausible and mathematically structured: they vary strength, genus specificity, hypothesis-dropping, and existence/uniqueness mechanism in ways that reflect realistic failure modes.

A reasonably well-constructed recognition-style MCQ with strong distractors, but it leans toward restating a known conjecture and offers only moderate genuine reasoning pressure.


\begin{thm}\label{thm: main theorem}
    Let $X = G/\Gamma$  and $F = \{g_t : t\in\R\}\subset G$, where
    \begin{itemize}
    \item $G$ is  a     Lie group with discrete center,
    \item $\Gamma$ is a non-uniform
    lattice in $G$,
\item $g_t$ is as in \eqref{gt}, where $a_0 \in \fg\nz$ is $\ad$-diagonalizable over $\Co$;
 \item the action of $F$  on $(X,m_X)$ is mixing.
\end{itemize}
    Let $y \in E(F,\infty)$. Then for any non-empty open $U\subset X$ we have
    \begin{equation}\label{eq: dim X - dim Z + 1}
        \dim \big(U\cap A(F_+,y) \cap E(F_+,\infty)\big) \geq \dim X - \dim Z + 1,
    \end{equation}
    where $Z \subset G$ is the {neutral subgroup of $G$ with respect to $F$} .
\end{thm}

\begin{thm}\label{main-lemma} \cite[Theorem 1.3]{KR1}
Let $X$ and $F$ be as above, and let $y \in E(F,\infty)$; then
\begin{equation}\label{dim1}
     \dim\big(\{ \alpha \in \R : x_\alpha \in  A(F_+,y) \cap E(F_+,\infty)\}\big) = 1.
\end{equation}
\end{thm}

\begin{defn}\label{horosph}[Expanding, contracting and neutral subgroups]\label{defn: expanding, contracting and neutral groups}
    We let $H$, $H^-$ and $Z$ be the connected Lie subgroups of $G$ associated to the algebras $\fh$, $\fh^-$ and $\fn$ respectively. We fix $m_H$, a Haar measure on $H$. We further consider the connected subgroup $\tilde{H}$ associated to the Lie algebra $\fn \oplus \fh^-$ and a left Haar measure $m_{\tilde{H}}$ on $\tilde{H}$, scaled so that $m_G$ is a direct product of $m_H$ and $m_{\tilde{H}}$, see \cite[Ch.~VII, \S 9, Proposition 13]{B} or \cite[Theorem 8.32]{Kn} for justification.
\end{defn}

\label{intro}

 Let $X$ be a   metric  space, and let $F$ be {either of the (semi) groups $\R$ or $\Z$ or $\R_{\ge 0}$ or $\Z_{\ge 0}$ represented as a family of} self-maps of $X$. We will write $F = \{g_t\}$. For $g\in F$ we will denote its action on $X$ by
$(g,x) \mapsto g
x$.
 Now fix a subset $Y\subset  X$ and define the set of points of $X$ {\sl approaching} $Y$ by $F$ as
    \begin{equation*}
        A(F,Y) := \big\{ x \in X : \exists \text{ an unbounded sequence }(g_{t_k})_{k \in \N} \subset F \text{ with }
        {\lim_{k \to \infty} g_{t_k}         x \in Y}
 \big\}.
    \end{equation*}
    Its complement,  the set of points of $X$ {\sl escaping} {$Y$} by $F$ will be denoted by $E(F,{Y})$, namely
    \begin{equation*}
        E(F,{Y}) := \bigcap_{y \in Y}\big\{ x\in X : y
        \notin\overline{(F\smallsetminus F_0) x}
         \text{ for some bounded }F_0\subset F\big\}.
    \end{equation*}
 If $X$ is not bounded, we make the convention that $x\in A(F,\infty)$ when the orbit $F
 x$ is unbounded.

 {Take $y\in Y$ and   consider $A(F,y):= A(F,\{y\})$ and $E(F,y):= E(F,\{y\})$.}  A natural question one could ask  is how large the sets $A(F,y)$   and   $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be  answered. Namely, under the assumption of existence of an ergodic measure $\mu$ of full support the set  $A(F,y)$ has full measure for any $y\in X\cup \{\infty\}$. Additionally,  for many hyperbolic systems one can prove  that the $\mu$-null sets $E(F,y)$  are \textsl{thick},  that is, their intersection with any nonempty open set has full Hausdorff dimension.  For results of this kind see e.g.\ \cite{Da3} for toral endomorphisms, \cite{U, Do} for Anosov flows, and {\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \hs.
Furthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g.  \cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).
Consequently in these cases  it follows that both $A(F,y)$   and   $E(F,y)$ are thick.
Moreover, both full-measure and winning conditions are stable with respect to countable intersections; thus
for any choice of countably many semigroups $F_i$ and points $y_i \in X\cup \{\infty\}$  for which the above conclusions can be established, it holds that both $\bigcap_i A(F_i,y_i)$   and   $\bigcap_i E(F_i,y_i)$  are thick.

\smallskip
This paper addresses the following question: what if one considers
a mixed case, that is, investigate the intersection \eq{mixed}{A(F,y) \cap E(F,z),} where $y,z\in X\cup \{\infty\}$? Problems of this type are amenable neither to the full-measure argument, nor to the technique based on Schmidt games. Also it is clear that some conditions on $y,z$ should be imposed; for example if $y=z$ it is clear that the intersection  \eqref{eq:mixed} is empty.

\smallskip
In what follows we restrict our attention to $F\cong \R$, also denoting $F_+ := \{g_t: t\ge 0\}$.
Assume that the $F$-action on $X$ is continuous. Then let us record {a natural}
obstruction to
the set  \eq{mixedplus}{A(F_+,y) \cap E(F_+,z)} being non-empty:

\begin{lem}\label{lem: obstruction}
    If {$y \in X$  and $z \in X \cup \{\infty\}$} are such that $y \in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then
    \begin{equation*}
        A(F_+,y) \subset A(F_+,z).
    \end{equation*}
\end{lem}

\begin{proof}
Let $x \in A(F_+,y)$, and let $(s_\ell)_{\ell \in \N}$ be an unbounded sequence in $\R_{\geq 0}$ with
\begin{equation*}
    \lim_{\ell \to \infty} g_{s_\ell}x = y.
\end{equation*}
Let $W \subset X$ be an open neighborhood of $z$. In the case, when $z = \infty$, we take $W$ to be an open set with bounded complement.
By the hypothesis, there exists a time $t \in \R$ for which $g_t y \in W$. By the continuity of the action we have
\begin{equation*}
    \lim_{\ell \to \infty} g_{s_\ell}g_tx = g_t y \in W.
\end{equation*}
Note that the sequence $(s_\ell + t)_{\ell \in \N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \in A(F_+, z)$.
\end{proof}
\ignore{{Suppose} we have a point $x \in A(F_+,y) \cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \in \N} \subset \R$ with
\begin{equation}\label{eq: y_0 to y_1}
    \lim_{k \to \infty} g_{t_k} y= z
\end{equation}
(in the case $z = \infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).
Since $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \infty$ we can let $W$ be the complement of a compact subset of $X$) and   $t_0>0$ such that
\begin{equation}\label{eq: tx notin W}
    g_tx \notin W \text{  for all }t> t_0.
\end{equation}
Using the fact that $x \in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \in \N} \subset \R_{\geq 0}$ with
\begin{equation}\label{eq: x to y_0}
    \lim_{k \to \infty} g_{s_k} x = y.
\end{equation}
The convergence in \eqref{eq: y_0 to y_1} guarantees us some $p \in \N$ with
\begin{equation*}
    {g_{t_{p}}     y \in W}.
\end{equation*}
The continuity of the action and the convergence in \eqref{eq: x to y_0} guarantees us that
\begin{equation*}
    \lim_{k \to \infty} g_{t_p + s_k} x = g_{t_p}    y \in W.
\end{equation*}
However, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \eqref{eq: tx notin W}.}

{In view of the above lemma it makes sense to look for conditions guaranteeing that the set  \eqref{eq:mixedplus} is reasonably large
under the assumption that $y \in E(F,z)$.
In the present paper we will  do it in the following set-up: we let $X = G/\Gamma$, where $G$ is  a real Lie group
and $\Gamma \subset G$ is a lattice,
and    consider the $\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:
    \begin{equation}\label{gt}
        \big\{ g_t := \exp(ta_0) \in G: t \in \R \big\},
    \end{equation}
    where ${a_0}$ is an     element of     {$\fg:=\operatorname{Lie}(G)$, the Lie algebra of $G$}.
    We shall fix once and for all an inner product on $\fg$ to induce a right-invariant Riemannian metric $\dist_G$ on $G$.
 Note that  the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\Gamma$ by setting
\begin{equation*}
    \dist_X(g_1\Gamma, g_2\Gamma) := \inf\big\{\dist_G(g_1\gamma, g_2): \gamma \in \Gamma \big\}.
\end{equation*}
This is the set-up in which we will attempt to study sets of the form  \eqref{eq:mixedplus}. We will denote by $m_X$  the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\Gamma$ is equal to $1$. Further,
   {in this paper} we will    assume that $X$ is noncompact and $z = \infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points.

  Recently the second- and third-named authors proved the following result. Take \linebreak  $G = \SL_2(\R)$, $\Gamma = \SL_2(\Z)$ and $X = G/\Gamma$. For $t,\alpha\in\R$ consider
\begin{equation*}\label{eq: gtsl2}
        g_t := \left[ {\begin{array}{cc} e^t & 0 \\
    0 & e^{-t}  \end{array}}\right]\in G,\ F = \{g_t: t\in\R\},
    \end{equation*}
    and  $${x_\alpha  := \left[ {\begin{array}{cc} 1 & \alpha \\
    0 & 1  \end{array}}\right]\Gamma\in X.}
    \ignore{the latter being identified with the lattice in $\R^2$ generated by $\left[ {\begin{array}{cc} 1 \\
    0  \end{array}}\right]$ and $ \left[ {\begin{array}{cc}  \alpha \\
    1 \end{array}}\right]$.}$$

{Take $y\in Y$ and consider $A(F,y):= A(F,\{y\})$ and $E(F,y):= E(F,\{y\})$.} A natural question one could ask is how large the sets $A(F,y)$ and $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be answered. Namely, under the assumption of existence of an ergodic measure $\mu$ of full support the set $A(F,y)$ has full measure for any $y\in X\cup \{\infty\}$. Additionally, for many hyperbolic systems one can prove that the $\mu$-null sets $E(F,y)$ are \textsl{thick}, that is, their intersection with any nonempty open set has full Hausdorff dimension. For results of this kind see e.g.\ \cite{Da3} for toral endomorphisms, \cite{U, Do} for Anosov flows, and {\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \hs.
Furthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g. \cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).
Consequently in these cases it follows that both $A(F,y)$ and $E(F,y)$ are thick.
Moreover, both full-measure and winning conditions are stable with respect to countable intersections; thus
for any choice of countably many semigroups $F_i$ and points $y_i \in X\cup \{\infty\}$ for which the above conclusions can be established, it holds that both $\bigcap_i A(F_i,y_i)$ and $\bigcap_i E(F_i,y_i)$ are thick.

\begin{lem}\label{lem: obstruction}
 If {$y \in X$ and $z \in X \cup \{\infty\}$} are such that $y \in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then
 \begin{equation*}
 A(F_+,y) \subset A(F_+,z).
 \end{equation*}
\end{lem}

\begin{proof}
Let $x \in A(F_+,y)$, and let $(s_\ell)_{\ell \in \N}$ be an unbounded sequence in $\R_{\geq 0}$ with
\begin{equation*}
 \lim_{\ell \to \infty} g_{s_\ell}x = y.
\end{equation*}
Let $W \subset X$ be an open neighborhood of $z$. In the case, when $z = \infty$, we take $W$ to be an open set with bounded complement.
By the hypothesis, there exists a time $t \in \R$ for which $g_t y \in W$. By the continuity of the action we have
\begin{equation*}
 \lim_{\ell \to \infty} g_{s_\ell}g_tx = g_t y \in W.
\end{equation*}
Note that the sequence $(s_\ell + t)_{\ell \in \N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \in A(F_+, z)$.
\end{proof}
\ignore{{Suppose} we have a point $x \in A(F_+,y) \cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \in \N} \subset \R$ with
\begin{equation}\label{eq: y_0 to y_1}
 \lim_{k \to \infty} g_{t_k} y= z
\end{equation}
(in the case $z = \infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).
Since $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \infty$ we can let $W$ be the complement of a compact subset of $X$) and $t_0>0$ such that
\begin{equation}\label{eq: tx notin W}
 g_tx \notin W \text{ for all }t> t_0.
\end{equation}
Using the fact that $x \in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \in \N} \subset \R_{\geq 0}$ with
\begin{equation}\label{eq: x to y_0}
 \lim_{k \to \infty} g_{s_k} x = y.
\end{equation}
The convergence in \eqref{eq: y_0 to y_1} guarantees us some $p \in \N$ with
\begin{equation*}
 {g_{t_{p}} y \in W}.
\end{equation*}
The continuity of the action and the convergence in \eqref{eq: x to y_0} guarantees us that
\begin{equation*}
 \lim_{k \to \infty} g_{t_p + s_k} x = g_{t_p} y \in W.
\end{equation*}
However, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \eqref{eq: tx notin W}.}

{In view of the above lemma it makes sense to look for conditions guaranteeing that the set \eqref{eq:mixedplus} is reasonably large
under the assumption that $y \in E(F,z)$.
In the present paper we will do it in the following set-up: we let $X = G/\Gamma$, where $G$ is a real Lie group
and $\Gamma \subset G$ is a lattice,
and consider the $\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:
 \begin{equation}\label{gt}
 \big\{ g_t := \exp(ta_0) \in G: t \in \R \big\},
 \end{equation}
 where ${a_0}$ is an element of {$\fg:=\operatorname{Lie}(G)$, the Lie algebra of $G$}.
 We shall fix once and for all an inner product on $\fg$ to induce a right-invariant Riemannian metric $\dist_G$ on $G$.
 Note that the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\Gamma$ by setting
\begin{equation*}
 \dist_X(g_1\Gamma, g_2\Gamma) := \inf\big\{\dist_G(g_1\gamma, g_2): \gamma \in \Gamma \big\}.
\end{equation*}
This is the set-up in which we will attempt to study sets of the form \eqref{eq:mixedplus}. We will denote by $m_X$ the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\Gamma$ is equal to $1$. Further,
 {in this paper} we will assume that $X$ is noncompact and $z = \infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points.

Recently the second- and third-named authors proved the following result. Take \linebreak $G = \SL_2(\R)$, $\Gamma = \SL_2(\Z)$ and $X = G/\Gamma$. For $t,\alpha\in\R$ consider
\begin{equation*}\label{eq: gtsl2}
 g_t := \left[ {\begin{array}{cc} e^t & 0 \\
 0 & e^{-t} \end{array}}\right]\in G,\ F = \{g_t: t\in\R\},
 \end{equation*}
 and $${x_\alpha := \left[ {\begin{array}{cc} 1 & \alpha \\
 0 & 1 \end{array}}\right]\Gamma\in X.}
 \ignore{the latter being identified with the lattice in $\R^2$ generated by $\left[ {\begin{array}{cc} 1 \\
 0 \end{array}}\right]$ and $ \left[ {\begin{array}{cc} \alpha \\
 1 \end{array}}\right]$.}$$

\begin{prop}\label{prop: equidistirbution for small balls}
 Let $V \subset H$ be bounded measurable with $m_H(\partial V) =0$. Let $K \subset X$ be compact and let $0<\sigma<\rho(K)$.
 Let $L \subset X$ be another compact set, and let $\eta >0$. Then there exists $t_0 := t_0(V, K, \sigma, L, \eta) >0$ such that, for all $t > t_0$, base point $x \in L$ and center $y \in K$,
 \begin{equation*}
 m_H\left\lbrace h \in V: g_thx \in B_X(y,\sigma) \right\rbrace > m_H(V) m_G\big(B_G(\sigma)\big) - \eta.
 \end{equation*}
\end{prop}
\begin{proof}
 The proof is a modification of that of \cite[Proposition 2.4]{Kl2}.
 By partitioning $V$ into finitely many smaller subsets with boundary of measure zero and noting the invariance of $m_H$, one can without loss of generality assume that $e_H\in V$, $m_H(V)<1$ and $\textup{diam}(V) < \rho(L)$. The latter condition implies that there is an open ball $U \subset G$ centered at $e_G$ and containing $V$ such that for all $x \in L$ the map
 $\pi_x$ is injective on $U$, and hence
 \begin{equation*}
 m_X(Ux) = m_G(U).
 \end{equation*}
Similarly we have, for any $0 < \alpha < 1$ and $y \in K$,
 \begin{equation*}
 m_X\big(B_X(y,\alpha \sigma)\big) = m_G\big(B_G(\alpha \sigma)\big).
 \end{equation*}
 Fix $\alpha <1$ such that
 \begin{equation}\label{eq: alpha ball close}
 m_G\big(B_G(\alpha \sigma)\big) > m_G\big(B_G(\sigma)\big) - \frac{\eta}{2}.
 \end{equation}
 Now let $\tilde{H}$ and $m_{\tilde{H}}$
 be as in Definition~\ref{horosph}.
Choose a nonempty open ball $\tilde V \subset \tilde{H}$ such that $\tilde VV \subset U$ and such that for all $\tilde v \in \tilde V$ and $t>0$ we have
 \begin{equation*}
 \dist_G\big(e_G,\Phi_t(\tilde v)\big) < (1-\alpha)\sigma. \end{equation*}
 Now for $y \in K$, $x \in L$ and $t>0$ denote
 \begin{equation*}
 W := \left\lbrace h \in V : g_t hx \in B_X(y,\sigma)\right\rbrace\ \text{ and }\ U' := \tilde V V.
 \end{equation*}
 \begin{claim}\label{claim: 99 containment}
 The set $U'x \cap g_t^{-1} B_X(y, \alpha \sigma)$ is contained in $\tilde V W x$.
 \end{claim}
 \begin{proof} Let $\tilde v \in \tilde V$, $h \in V$ be such that $g_t\tilde vhx \in B_X(y, \alpha \sigma)$. We have
 \begin{equation*}
 \begin{split}
 \dist_X(g_t h x, y) &\leq \dist_X(g_t hx, \Phi_t(\tilde v)g_thx) + \dist_X(g_t\tilde vhx, y) \\
 &< (1-\alpha)\sigma + \alpha \sigma = \sigma.
 \end{split}
 \end{equation*}
 Thus $g_thx \in B_X(y,\sigma)$.
 \end{proof}
 We now use Proposition \ref{thm: KM mixing} with \begin{equation*}
 \Xi := \left\lbrace \mathds{1}_{U'x} : x \in L\right\rbrace,\
\Psi := \{\mathds{1}_{B_X(y,\alpha \sigma}) : y \in K\},
 \end{equation*}
 and with $\eta$ replaced by $\eta m_{\tilde{H}}(\tilde V)/2$.
 This gives $t_0 >0$ such that, for all $t>t_0$, $x \in L$ and $y \in K$, we have
 \begin{equation*}
 \left|m_X\big(U'x \cap g_t^{-1}B_X(y,\alpha \sigma)\big) - m_G(U')m_G\big(B_G(\alpha \sigma)\big) \right| < \eta m_{\tilde{H}}(\tilde{V})/2.
 \end{equation*}
 In particular, using the choice of $m_{\tilde{H}}$ and Claim \ref{claim: 99 containment}, we compute
 \begin{equation*}
 \begin{split}
 m_{\tilde{H}}(\tilde{V})m_H(W) &=m_X(\tilde VWx) \\
 &\geq m_X\big(U'x \cap g_t^{-1}B_X(y,\alpha \sigma)\big) \\
 &\geq m_G(U') m_G\big(B_G(\alpha \sigma)\big) - \eta m_{\tilde{H}}(\tilde V)/2 \\
 &= m_{\tilde{H}}(\tilde V)m_H(V) m_G\big(B_G(\alpha \sigma)\big) - \eta m_{\tilde{H}}(\tilde V)/2.
 \end{split}
 \end{equation*}
 Canceling $m_{\tilde{H}}(\tilde V)$, using \eqref{eq: alpha ball close} and the fact that $m_H(V) < 1$, we get
 \begin{equation*}
 \begin{split}
 m_{H}(W) &\geq m_H(V) m_G\big(B_G(\alpha \sigma)\big) - \eta/2 \\
 &\geq m_H(V) m_G\big(B_G( \sigma)\big) - m_H(V)\eta/2 - \eta/2 \\
 &\geq m_H(V)m_G\big(B_G( \sigma)\big) - \eta
 \end{split}
 \end{equation*}
 which gives the desired result.
\end{proof}

\begin{defn}[Strongly tree-like collections and Cantor sets]\label{defn: tree like collection}\label{defn: tree-like collection}
 We say a collection of subsets $\ca{E} \subset 2^H$ is \textsl{strongly tree-like} if:
 \begin{itemize}
 \item Each $E \in \ca{E}$ is a compact set with nonempty interior.
 \item We have a partition
 \begin{equation*}
 \ca{E} = \bigcup_{k \in \Z_{\geq 0}} \ca{E}_k
 \end{equation*}
 with each $\ca{E}_k$ being finite and $\ca{E}_0$ being a singleton.
 \item If $E' \in \ca{E}_k$ with $k \in \N$, there is a unique $E \in \ca{E}_{k-1}$ with $E' \subset E$.
 \item For each $E \in \ca{E}_k$ with $k \in \Z_{\geq 0}$, there exists $E' \in \ca{E}_{k+1}$ with $E' \subset E$.
 \item If $E_1, E_2 \in \ca{E}_k$ are distinct, then $m_H(E_1 \cap E_2)=0$.
 \item If we define
 \begin{equation}\label{eq: diam}
 d_k := \sup\{\on{diam}_H(E) : E \in \ca{E}_k\}, \text{ then } \lim_{k \to \infty} d_k = 0.
 \end{equation}
 \end{itemize}
 We write
 \begin{equation*}
 \cup \ca{E}_k := \bigcup_{E \in \ca{E}_k} E
 \end{equation*}
 and define the {\sl limit set} of the collection to be $E_\infty := \bigcap_{k \in \N} \cup \ca{E}_k$.
\end{defn}
Associated to a tree-like collection $\ca{E}$ we also define, for $E \in \ca{E}_k$,
\begin{equation*}
 \on{density}(\ca{E}_{k+1},E) := \frac{m_H((\cup \ca{E}_{k+1}) \cap E)}{m_H(E)}.
\end{equation*}
Further, for each $k \in \Z_{\geq 0}$, we set
\begin{equation}\label{eq: density}
 \Delta_k := \inf\left\lbrace \operatorname{density}(\mathcal{E}_{{k+1}}, E) : E \in \mathcal{E}_k \right\rbrace
\end{equation}
\begin{thm}[Lemma 2.1 in \cite{U}]\label{thm: urbanski}
 For a tree-like collection $\ca{E}$ and the resulting limit set $E_\infty$, we have the dimension estimate
 \begin{equation*}
 \dim H - \dim E_\infty \leq \limsup_{k \to \infty} \frac{\sum_{j=0}^{k} \log \Delta_j}{\log d_k}.
 \end{equation*}
\end{thm}

Conversely, let us assume that $x_\alpha \in E(F_+, K_i)$. Then, by the compactness of $K_i$ and by Mahler's compactness criterion, there exists an $r$ with $0< r < \sqrt{r_0}$ and a time $s_0 >0$ such that for all $s > s_0$,
 \begin{equation*}
 g_s x_\alpha \notin K_i(r).
 \end{equation*}
 In particular,
 \begin{equation*}
 x_\alpha \cap \left\lbrace (z,w)\in \Co^2: |z| \leq e^{-s}r, |w| \leq e^{s}r \right\rbrace \neq \{(0,0)\}.
 \end{equation*}
 In particular, $\alpha \in \DI_i$ with constant $\frac{r^2}{r_0}$.
\end{proof}
This dynamical perspective on $\DI_i$ allows one to effortlessly establish certain properties of $\DI_i$ using existing techniques in ergodic theory and dynamics: the question of measure and dimension. (The interested reader may consult \cite[Theorem 1.5]{KW2} and \cite[Theorem 2.8]{AGK} where very general dynamical analogs of these questions are considered.)
In particular, we may use the above dynamical results to conclude that $\DI_i$ has measure zero but full Hausdorff dimension.
With respect to this current article, we are primarily concerned with the relation between Dirichlet improvability and bad approximability (cf. \cite[Theorems 1 and 2]{DS}).
Consider the set of badly approximable complex numbers with respect to $\Z[i]$; those $\alpha \in \Co$ for which there is a constant $c >0$ such that for all nonzero $q \in \Z[i]$,
\begin{equation*}
 |q| \min_{p \in \Z[i]} |q\alpha - p| >c.
\end{equation*}
We write this set as $\BA_i$.
\begin{ques}\label{ques: BA DI i}
 What is the relationship between Dirichlet improvability and bad approximabilty in the case of the Gaussian integers? Is it true that $\BA_i$ is a subset of $\DI_i$?
\end{ques}
We are able to provide a partial answer to this question as follows. However, it seems that a complete answer might require a further study of the set $K_i$ in Proposition \ref{prop: DIi dynamical}.

Here and hereafter "dim" stands for \hd.
 In fact, even though it was not mentioned in \cite{KR1}, the proof actually demonstrates the thickness of the set in the left-hand side of \eqref{dim1}. Also,  by the standard "slicing" argument (see \S \ref{sec: reduction to H}), it can be easily derived from Theorem \ref{main-lemma}
that  for any $y \in E(F,\infty)$ the set $A(F_+,y) \cap E(F_+,\infty)$ is thick.
The proof of the above theorem used the technique of continued fractions, and in fact the result can be interpreted within the framework of Diophantine approximation. Namely it constructs numbers that are badly approximable but not $\nu$-Dirichlet improvable, where $\nu$ is an irreducible norm on $\R^2$. See \cite{KR, KR1, KRS, AD} for more detail, and \cite{MS} for some new results in that direction.

\smallskip
One of the main results of this paper is a generalization of Theorem \ref{main-lemma} to a wider class of flows on \hs s.

\begin{thm}\label{thm: main theorem}
    Let $X = G/\Gamma$  and $F = \{g_t : t\in\R\}\subset G$, where
    \begin{itemize}
    \item $G$ is  a     Lie group with discrete center,
    \item $\Gamma$ is a non-uniform
    lattice in $G$,
\item $g_t$ is as in \eqref{gt}, where $a_0 \in \fg\smallsetminus\{0\}$ is $\ad$-diagonalizable over $\Co$;
 \item the action of $F$  on $(X,m_X)$ is mixing.
\end{itemize}
    Let $y \in E(F,\infty)$. Then for any non-empty open $U\subset X$ we have
    \begin{equation}\label{eq: dim X - dim Z + 1}
        \dim \big(U\cap A(F_+,y) \cap E(F_+,\infty)\big) \geq \dim X - \dim Z + 1,
    \end{equation}
    where $Z \subset G$ is the {neutral subgroup of $G$ with respect to $F$} .
\end{thm}
See Definition \ref{defn: expanding, contracting and neutral groups} below for a precise definition of $Z$.
In view of Moore's Theorem \cite{Mo}, the mixing assumption is satisfied when $G$ is a connected semi-simple
    Lie group
    with finite center and with no compact factors, and $\Gamma\subset G$ is irreducible. However mixing is not restricted to the semi-simple case, see e.g.\  \cite{BM, Kl}.
Note that when $G =  \SL_2(\R)$ (that is, under the assumptions of Theorem \ref{main-lemma}) we know that {$\dim Z=\dim F = 1$}, and thus the set $A(F_+,y) \cap E(F_+,\infty)$ is thick. Moreover, in the latter case one can generalize Theorem~\ref{main-lemma}  as follows.

  \begin{thm}\label{thm: dense-orbits}
Let $G$, $\Gamma$ and $F$ be as in Theorem \ref{thm: main theorem}, and assume in addition that \linebreak {$\dim Z =
1$}.
Then for any {compact}
F-invariant
subset $B$ of $E(F,\infty)$ there exists a thick set of $x \in E(F_+,\infty)$ such that the closure of the (bounded) trajectory $F_+x$ contains $B$.
 \end{thm}

In addition to $G =  \SL_2(\R)$, the assumptions of the above theorem hold when \linebreak $G =  \SL_2(\R)\ltimes \R^2$; the choice $\Gamma =  \SL_2(\Z)\ltimes \Z^2$ yields the space of two-dimensional unimodular grids, cf.\ \cite{Kl}.
  \begin{cor}\label{cor: dim-drop}
Let $G$, $\Gamma$, $X$ and $F$ be as in Theorem \ref{thm: dense-orbits}; then for any $\varepsilon > 0$ there exists a thick set of $x\in X$ such that the orbit closure $\overline{F^+x}$ is compact and has Hausdorff dimension greater than $\dim X - \varepsilon$.
 \end{cor}
 \begin{proof}
 When $
X$ and $F$ are as in Theorem \ref{thm: main theorem}, it follows from
 \cite[Theorem 1.1]{KM} that, given $\varepsilon>0$, one can find a large enough compact set $K \subset X$ such that the compact $F$-invariant set
 \begin{equation*}
     B = \{x \in X: Fx \subset K\}
 \end{equation*}
 has Hausdorff dimension greater than $\dim X - \varepsilon$.
Under the additional hypothesis of $\dim Z = 1$, an application of Theorem \ref{thm: dense-orbits} to $B$ then gives the result.
 \end{proof}

 \noindent{\bf Acknowledgements.}
    The second-named author is grateful to ETH (Z\"urich) for its hospitality during a sabbatical stay in 2022, and to Barak Weiss for helpful discussions. The third-named author thanks N.\ Chandgotia, T.\ Mesikepp and Y.\ Peres for holding public-access office hours at TIFR-CAM, Peking University and Tsinghua University respectively.

The post-theorem discussion says that (although not stated in \cite{KR1}) “the proof actually demonstrates the thickness of the set in the left-hand side of \eqref{dim1}.” It also notes that “by the standard ‘slicing’ argument (see \S \ref{sec: reduction to H}), it can be easily derived from Theorem \ref{main-lemma} that for any $y\in E(F,\infty)$ the set $A(F_+,y)\cap E(F_+,\infty)$ is thick.” Finally, it describes the method/interpretation: “The proof of the above theorem used the technique of continued fractions,” and “can be interpreted within the framework of Diophantine approximation,” namely constructing “numbers that are badly approximable but not $\nu$-Dirichlet improvable, where $\nu$ is an irreducible norm on $\R^2$.”

The post-theorem discussion says that (although not stated in

Kenyon, R. and Ralston, J., \emph{[Theorem 1.3]} (KR1)

) “the proof actually demonstrates the thickness of the set in the left-hand side of
\begin{equation}\label{dim1}
     \dim\big(\{ \alpha \in \R : x_\alpha \in  A(F_+,y) \cap E(F_+,\infty)\}\big) = 1.
\end{equation}
.” It also notes that “by the standard ‘slicing’ argument (see \S \ref{sec: reduction to H}), it can be easily derived from the main theorem that for any $y\in E(F,\infty)$ the set $A(F_+,y)\cap E(F_+,\infty)$ is thick.” Finally, it describes the method/interpretation: “The proof of the above theorem used the technique of continued fractions,” and “can be interpreted within the framework of Diophantine approximation,” namely constructing “numbers that are badly approximable but not $\nu$-Dirichlet improvable, where $\nu$ is an irreducible norm on $\R^2$.”

\label{main-lemma} \cite[Theorem 1.3]{KR1}
Let $X$ and $F$ be as above, and let $y \in E(F,\infty)$; then
\begin{equation}\label{dim1}
     \dim\big(\{ \alpha \in \R : x_\alpha \in  A(F_+,y) \cap E(F_+,\infty)\}\big) = 1.
\end{equation},

Question:
Let
\(X=\mathrm{SL}_2(\mathbb R)/\mathrm{SL}_2(\mathbb Z)\), let
\(g_t=\begin{pmatrix}e^t&0\\0&e^{-t}\end{pmatrix}\), and let
\(F=\{g_t:t\in\mathbb R\}\) with \(F_+=\{g_t:t\ge 0\}\). For \(\alpha\in\mathbb R\), write
\(x_\alpha=u_\alpha\,\mathrm{SL}_2(\mathbb Z)\) where
\(u_\alpha=\begin{pmatrix}1&\alpha\\0&1\end{pmatrix}\). For \(y\in X\), define
\(A(F_+,y)=\{x\in X: g_{t_n}x\to y\text{ for some unbounded sequence }t_n\ge 0\}\), and let \(E(F_+,\infty)\) denote the set of points whose positive \(F_+\)-orbit does not approach the point at infinity in the one-point compactification of \(X\) (equivalently, whose positive orbit is bounded in \(X\)). If \(y\in E(F,\infty)\), i.e. the full \(F\)-orbit of \(y\) is bounded, which statement holds about the Hausdorff dimension of
\[
\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}?
\]

Its Hausdorff dimension is exactly \(1\):
\[
\dim\big(\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}\big)=1.
\]

Choices:
B.
Its Hausdorff dimension is exactly \(0\); in fact,
\[
\dim\big(\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}\big)=0.
\]
C. Its Hausdorff dimension is at least \(0\) and at most \(1\); in particular, the set is a subset of \(\mathbb R\) and may be nonempty, but no sharper dimension conclusion is asserted here.
D.
Its Hausdorff dimension is exactly \(1\) provided \(y\in E(F_+,\infty)\); that is,
\[
\dim\big(\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}\big)=1
\]
for every point whose positive \(F_+\)-orbit is bounded.
E.
The set has full Lebesgue measure in \(\mathbb R\), and hence in particular Hausdorff dimension \(1\):
\[
m\big(\{\alpha\in\mathbb R: x_\alpha\in A(F_+,y)\cap E(F_+,\infty)\}\big)=1.
\]

thickness/full-dimension conclusion replaced by zero-dimension exceptional-set heuristic

dropped the sharp equality \(=1\), keeping only the trivial ambient-dimension bound

hypothesis weakened from full-orbit boundedness \(y\in E(F,\infty)\) to only forward boundedness \(y\in E(F_+,\infty)\)

dimension/thickness conclusion strengthened to full Lebesgue measure

The stem does not state the conclusion or embed the correct dimension claim. It gives the full setup and hypotheses, but the correct answer is not leaked beyond the natural theorem-style framing.

This is close to a theorem-recall item: the stem presents the hypotheses and asks for the conclusion. However, it is not a pure restatement because the options introduce meaningful nearby variants (weaker, stronger, and hypothesis-altered claims).

Some reasoning is needed to reject plausible alternatives such as the weaker trivial bound, the stronger full-measure claim, and the subtly weakened hypothesis in choice D. Still, the item mainly tests recognition of the sharp theorem statement rather than substantial derivation.

The distractors are plausible and mathematically targeted: zero dimension reflects an exceptional-set heuristic, choice C is a weaker true-but-uninformative bound, choice D weakens the hypothesis in a subtle way, and choice E improperly strengthens dimension to full measure.

A solid theorem-discrimination MCQ with strong distractors and little answer leakage, though it leans more toward precise recall/recognition than deep generative reasoning.


Johnson \cite{JohnsonLantern} proved that the Torelli group is normally generated by a bounding pair map of genus 1. In the authors' previous paper joint with Lanier \cite{Normal}, we
studied the subgroup normally generated by a bounding pair map of genus $n$ in the mapping class group $\M$ of a genus $g$ surface with 1 bondary component. Let $\ch:\I\to H_1(S_g^1;\Z)$ denote the Chillingworth homomorphism.
Let $\Ch[2n]$ denote the subgroup of $\I$ consisting of $f$ such that $\ch(f)=0\pmod{2n}.$
 Let $d_{4n}: \Ch [2n]\to\Z/4n\Z$ denote the Casson--Morita's $d$ map modulo $4n$. We proved:

\begin{abstract}
Justin Lanier and the authors recently determined the group normally generated by a single bounding pair map of genus $n$. We related this subgroup with the Chillingworth subgroup and the Casson--Morita's $d$ map. In this paper, we extend the results to the case when $n=0$. Let $\M$ be the mapping class group, $\Ch$ be the Chillingworth subgroup and $d$ be the Casson--Morita's $d$-map. We show that $Ker(d)=[\Ch,\M]$ and it is generated by a single homological genus 0 bounding pair map. We also construct an element $H_0\in \Ch$, and show that $\Ch$ is normally generated by this single element $H_0$.
\end{abstract}

\vspace{-.15in}

\noindent Intuitively, $B_0$ has homological genus 0 because it is the difference (and not the sum) of two genus 1 bounding pair maps. In Section 2, we will formally define the homological genus of a fake bounding pair map $T_aT_b^{-1}$ where $a$ and $b$ are homologous curves that can possibly intersect. Denote the Chillingworth subgroup by $\Ch:=\ker (\ch:\I\to H_1(S_g^1;\Z))$. Let $d:\Ch\to\Z$ denote the Casson-Morita's $d$ map restricted to $\Ch$. Kosuge introduced $B_0$ in order to show that $\ker(d)$ is normally generated by $B_0$ together with $[\J,\M]$ and a genus 1 separating twist \cite[Theorem B]{Chilling}. We will show that Kosuge's normal generating set only needs one element. In fact, we prove more:
\begin{theorem}
\label{main1}
 When $g\ge 6$, the normal subgroup of $\M$ generated by $B_0$ is
 \begin{equation}
 \label{eq:B0=}
 \langle\langle B_0\rangle\rangle=[\Ch ,\M ]=\ker(d).
 \end{equation}
\end{theorem}

Our next theorem will show that $\Ch$ is normally generated by a single mapping class $H_0:=T_{a_1}T_{a_3}T_{a_2}^{-2}$ where $a_1,a_2,a_3$ are as in Figure \ref{fig:a1a2a3} below:
\begin{figure}[h]
\centering\includegraphics[width=0.6\linewidth]{a1a2a3.jpg}
 \caption{The number $g-3$ denotes the genus of the subsurface.}
 \label{fig:a1a2a3}
\end{figure}
\begin{theorem}\label{main2}
For $g\ge 6$, we have that
\[\langle \langle H_0\rangle \rangle = \Ch\]
\end{theorem}

Let $\W(0):=\langle\langle B_0\rangle\rangle$. We also prove the following homological results.
\begin{theorem}
 \label{thm:homological results}
 For $g\ge 6$, we have that
 \begin{enumerate}
 \item $H_1(\W(0);\Z)_{\M}=0,$\ \ \ \ \ \ \ \ \ \ $H^1(\W(0);\Z)^{\M}=0.$
 \item $H_1(\Ch;\Z)_{\M}\cong \Z, \ \ \text{ }\ \ \ \ \ \ \ \ H^1(\Ch;\Z)^{\M}= \langle d/8\rangle\cong \Z.$\\
 $H^1(\Ch;\Z/m\Z)^{\M}= \langle d/8\rangle\cong \Z/m\Z$ for any $m\in\Z_{>0}$.
 \end{enumerate}
\end{theorem}
Kosuge \cite[Theorem C]{Chilling} computed $H_1(\Ch;\mathbb{Q})$ as $\M$-modules when $g\ge 6$. Our Theorem \ref{thm:homological results} provides the $\M$-invariant part over $\Z$ and $\Z/m\Z$.

The Chillingworth homomorphism
$\ch:\I\to H$
has $\ker(\ch) = \Ch$ and $\im(\ch) = 2H.$ Therefore, we have a short exact sequence of groups
\begin{equation}
 \label{eq: SES Ch I H}
 0\to \Ch\to\I\xrightarrow{\ch/2} H\to 0
\end{equation}
which induces the following five-term exact sequence:
\begin{equation}
 \label{eq: five over Z}
 0\to H^1(H;\Z)\to H^1(\I;\Z)\to H^1(\Ch;\Z)^H\xrightarrow{
 \delta} H^2(H;\Z)\to H^2(\I;\Z).
\end{equation}
Let $d:\Ch\to\Z$ denote the restriction of the Casson-Morita's $d$ map to the Chillingworth subgroup. By \cite[Proposition 19 and Proposition 20]{Chilling}]), its restriction to $\Ch$ is a $\M$-invariant group homomorphism with image $d(\Ch)=8\Z$. Thus $d/8$ is an $\M$-invariant homomorphism on $\Ch$. We have the following.
\begin{proposition}
 \label{prop: delta(d)}
 The connecting homomorphism $\delta:H^1(\Ch;\Z)^H\to H^2(H;\Z)$ in (\ref{eq: five over Z}) maps $d/8$ to $j$.
\end{proposition}
\begin{proof}
In general, for $A$ an abelian group and for a short exact sequence of groups
\[
0\to N\to G\to Q\to 0,\]
the connecting homomorphism $\delta: H^1(N;A)^Q\to H^2(Q;A)$ can be defined as the following: Let $f: N\to A$ be a $Q$-invariant group homomorphism. Take any extension $f': G\to A$ of $f$ to $G$. Define $\delta (f):Q\times Q\to A$ by
\[\delta(f)(q_1,q_2)=f'(\tilde{q}_1\tilde{q}_2)-f'(\tilde{q}_1)-f'(\tilde{q}_2)\]
where $\tilde{q}_i$ is any lift of $q_i$ to $G$. Then $\delta(f)$ is a 2-cocyle representing the cohomology class $\delta[f]$. See \emph{e.g.} Proposition (1.6.6) and Theorem (2.4.3) in \cite{neukirch2013cohomology} for a proof of this general fact.

Recall that the Casson-Morita's $d$ homomorphism is a map $d:\I\to \Z$ such that for any two elements $\phi,\psi \in \I$,
\begin{equation}
 \label{eq: d properties}
 d ( \phi \psi ) = d ( \phi ) + d ( \psi ) +\ii(\ch(\phi),\ch(\psi )).
\end{equation}
Please see Morita \cite{morita2}, Proposition 5.1 for more discussion. Define $\phi:\Ch[4m]\to\Z/m\Z$ by
 $$\phi(f):=d(f)/8\pmod {m}.$$
When $1\le 2m\le g-2$, we have the following results in \cite{Normal}:
\begin{enumerate}
 \item \cite[Proposition 4.10]{Normal} $d(\Ch)=d(\Ch[4m])=8\Z$.
 \item \cite[Lemma 4.2]{Normal}
 The homomorphism $\phi$ is a $\M$-invariant group homomorphism.
 \item \cite[Theorem 1.4]{Normal} The homomorphism $\phi$ induces an isomorphism
 $$H_1(\Ch[4m];\Z)_{\M}\cong \Z/m\Z.$$
\end{enumerate}
By the universal coefficient theorem and (3) above, we have that
$$H^1(\Ch[4m];\Z/m\Z)^\M =\langle\phi\rangle \cong \Z/m\Z. $$
Hence, the proof of Theorem \ref{thm:Abmodm} will be complete if we can show that the vertical map $R$ in (\ref{eq:diag3}) is an isomorphism, which we will prove in the following claim.

\begin{proposition}
\label{prop: T1 in commutator}
For $g\ge 6$, we have that
\[{T_1\in [\Ch,\M].}\]
\end{proposition}
\begin{proof}
Let $A:=\Ch/[\Ch,\M]$. Since $\Ch$ is normally generated by $B_0,T_1,T_2$, we know that $A$ is generated by $B_0,T_1,T_2$. Since $B_0, T_1^2\in [\Ch,\M]$, we know that $A$ as an abelian group is generated by $T_1,T_2$ and that $T_1$ is a 2-torsion. Since we have a homomorphism $d:A\to\Z$ such that $d(T_2)=8\ne 0$ by \cite[Theorem 5.3]{morita2}. We have that $T_2$ must be of infinite order in $A$. Hence, by the classification of finitely generated abelian group, if $T_1\notin [\Ch,\M]$, we have that $A\cong \Z\oplus \Z/2$. However, this implies that
\[
H^1(\Ch;\Z/2\Z)^{\M}\cong\text{Hom}(A;\Z/2) \cong \Z/2\oplus \Z/2
\]
This contradicts Theorem \ref{thm:Abmodm} when $m=2$ where we need $g-2\ge 4$. We thus obtain that $T_1\in [\Ch,\M]$ when $g\ge 6$.
\end{proof}

\begin{figure}[h]
\centering\includegraphics[width=0.6\linewidth]{a1a2a3.jpg}
    \caption{The number $g-3$ denotes the genus of the subsurface.}
    \label{fig:a1a2a3}
\end{figure}

\begin{theorem}
    \label{thm:homological results}
    For $g\ge 6$,  we have that
    \begin{enumerate}
        \item $H_1(\W(0);\Z)_{\M}=0,$\ \  \ \ \ \ \ \ \ \  $H^1(\W(0);\Z)^{\M}=0.$
        \item     $H_1(\Ch;\Z)_{\M}\cong \Z, \ \ \text{ }\ \ \ \ \ \ \ \ H^1(\Ch;\Z)^{\M}= \langle d/8\rangle\cong \Z.$\\
        $H^1(\Ch;\Z/m\Z)^{\M}= \langle d/8\rangle\cong \Z/m\Z$ for any $m\in\Z_{>0}$.
    \end{enumerate}
\end{theorem}

In this paper, we consider an analog of Theorem \ref{thm:old} in the case when $n=0$. The equality (\ref{eq:Wn}) obviously fails when $n=0$ because any genus 0 bounding pair map is trivial. Instead, we consider a \emph{homological genus 0 bounding pair map} $B_0:=T_aT_b^{-1}$ introduced recently by Kosuge \cite[Theorem 22]{Chilling}. See Figure \ref{fig:B0} below.
\begin{figure}[h]
        \centering
                \vspace{-20pt}
\includegraphics[width=0.5\linewidth]{B0-3.jpg}
        \vspace{-20pt}
        \caption{A homological genus 0 bounding pair map $B_0=T_aT_b^{-1}$.}
                \vspace{-5pt}
        \label{fig:B0}
    \end{figure}

\noindent Intuitively, $B_0$ has homological genus 0 because it is the difference (and not the sum) of two genus 1 bounding pair maps. In Section 2, we will formally define the homological genus of a fake bounding pair map $T_aT_b^{-1}$ where $a$ and $b$ are homologous curves that can possibly intersect. Denote the Chillingworth subgroup by $\Ch:=\ker (\ch:\I\to H_1(S_g^1;\Z))$. Let $d:\Ch\to\Z$ denote the Casson-Morita's $d$ map restricted to $\Ch$. Kosuge introduced $B_0$ in order to show that $\ker(d)$ is normally generated by $B_0$ together with $[\J,\M]$ and a genus 1 separating twist \cite[Theorem B]{Chilling}. We will show that Kosuge's normal generating set only needs one element. In fact, we prove more:
\begin{theorem}
\label{main1}
 When $g\ge 6$, the normal subgroup of $\M$ generated by $B_0$  is
 \begin{equation}
         \label{eq:B0=}
            \langle\langle B_0\rangle\rangle=[\Ch ,\M ]=\ker(d).
 \end{equation}
\end{theorem}

As a corollary, Theorem \ref{main1} implies that a genus 1 separating twist, which we denote by $T_1$, is a product of conjugates of $B_0$. However, our proof is indirect. We ask:
\begin{question} What is an explicit way to write a genus 1 separating twist $T_1$ as a product of conjugates of $B_0$?
\end{question}
We remark that any answer to this question can only work for genus 1 separating twists because any separating twist of  genus $n>1$ does not belong to $\langle\langle B_0\rangle\rangle$.

Our next theorem will show that $\Ch$ is normally generated by a single mapping class $H_0:=T_{a_1}T_{a_3}T_{a_2}^{-2}$ where $a_1,a_2,a_3$ are as in Figure \ref{fig:a1a2a3} below:
\begin{figure}[h]
\centering\includegraphics[width=0.6\linewidth]{a1a2a3.jpg}
    \caption{The number $g-3$ denotes the genus of the subsurface.}
    \label{fig:a1a2a3}
\end{figure}
\begin{theorem}\label{main2}
For $g\ge 6$, we have that
\[\langle \langle H_0\rangle \rangle = \Ch\]
\end{theorem}

Let $\W(0):=\langle\langle B_0\rangle\rangle$. We also prove the following homological results.
\begin{theorem}
    \label{thm:homological results}
    For $g\ge 6$,  we have that
    \begin{enumerate}
        \item $H_1(\W(0);\Z)_{\M}=0,$\ \  \ \ \ \ \ \ \ \  $H^1(\W(0);\Z)^{\M}=0.$
        \item     $H_1(\Ch;\Z)_{\M}\cong \Z, \ \ \text{ }\ \ \ \ \ \ \ \ H^1(\Ch;\Z)^{\M}= \langle d/8\rangle\cong \Z.$\\
        $H^1(\Ch;\Z/m\Z)^{\M}= \langle d/8\rangle\cong \Z/m\Z$ for any $m\in\Z_{>0}$.
    \end{enumerate}
\end{theorem}
Kosuge \cite[Theorem C]{Chilling} computed $H_1(\Ch;\mathbb{Q})$ as $\M$-modules when $g\ge 6$. Our Theorem \ref{thm:homological results} provides the $\M$-invariant part over $\Z$ and $\Z/m\Z$.

\p{An important ingredient}
A key step in the proof of Theorem \ref{main1} is to prove that a curve complex $C_0^\alpha$, to be described below, is connected. In Section 2, we will define the \emph{homological genus} $g(a,b)$ of two homologous curves $a$ and $b$, not necessarily disjoint. For   a  nonseparating curve $\alpha$, we define a complex $C_0^\alpha$ to have:
\begin{itemize}
\item vertex: a curve $c$ such that $g(\alpha,c)=0$.
\item edge: two vertices $c,d\in C_0^a$ are connected by an edge if and only if $T_cT_d^{-1}$ is conjugate to $B_0$ in $\M$.
\end{itemize}
In other words, there is an edge between $c,d\in  C_0^\alpha$ if and only if there is a mapping class $f$ such that $\{f(c),f(d)\}=\{a,b\}$ for $a,b$ in Figure \ref{fig:B0}. \begin{theorem}\label{C_0}
Suppose that $g\ge 6$.
\begin{enumerate}
    \item The curve complex $C_0^\alpha$ is path-connected.
    \item The groups $\Ch$ and $\W(0)$ act on  $C_0^\alpha$  and both actions are transitive.
\end{enumerate}
\end{theorem}
\begin{corollary}[$\Ch$–orbits of nonseparating curves]
\label{cor: ch orbit nonseparating}
Suppose that $g\ge 6$. For any two nonseparating, homologous curves $c$ and $d$,  the following conditions are equivalent:
\begin{itemize}
\item
$c$ and $d$ are equivalent under $\Ch$, or equivalently, there exists an $f\in \Ch$ such that $T_c=fT_df^{-1}$.
\item
$T_cT_d^{-1}\in \Ch$.
\item
the homological genus $g(c,d)$ is 0.
\end{itemize}
\end{corollary}
This corollary analogous to a theorem of Church \cite[Theorem 1.1]{Church}, which provides equivalent conditions for two nonseparating curves to be equivalent under the Johnson kernel $\J$. In the same paper, Church also gave equivalent conditions for two separating curves to be equivalent under $\J$. Hence, we ask:
\begin{question}
    What are the conditions, similar to those in Corollary \ref{cor: ch orbit nonseparating}, for two separating curves to be equivalent under $\Ch$?
\end{question}

Moreover, we do not know whether the hypothesis in Theorem \ref{main1} that $g\ge 6$ is necessary, although our proof requires this condition.
\begin{question}
Can Theorem \ref{main1} be extended to surfaces with genus $g<6$?
\end{question}

On Page 255 of \cite{JohnsonConjugacy}, Johnson asked whether it is possible to define the Birman--Craggs homomorphisms algebraically without using 4-dimensional topology. We are in a similar situation. When we prove $\Ch/[\Ch,\M]\cong \Z$ (or even just that $\Ch\neq [\Ch,\M]$), we need to use the Casson-Morita's $d$-map, which involves $4$-dimensional topology. Is there an algebraic way to see that $\Ch/[\Ch,\M]\cong \Z$?

\p{Strategy and outline}
We use the lantern relations in various ways. In Section 2, we prove a rigidity result about two homologous curves intersecting at two points. This topological result helps us to recognize mapping classes that are conjugates of $B_0$. It then allows us to prove that $[\Ch,\M]$ contains $B_0$ and $T_1^2$.
In Section 3, we use a homological argument about the cohomology of groups to obtain the key lemma showing that $T_1\in [\Ch,\M]$. This part of proof is indirect and  uses the results in \cite{Normal} in an unexpected way. As consequences, we prove the first equality of Theorem \ref{main1}, Theorem \ref{main2}, and Theorem \ref{thm:homological results}.
In Section 4, we prove the connectivity of $C_0^\alpha$ and the last equality of Theorem \ref{main1}. The proof of the connectivity of $C_0^\alpha$ uses Putman's trick multiple times on multiple different curve complexes that we construct. We are curious if one can find a more direct proof.

\p{Acknowledgments} The authors would like to thank Dan Margalit and Justin Lanier for the conversations that set this work in motion. The second author is supported by the National Natural Science Foundation of China under the Young Scientists Fund No. 12101349.

\vspace*{4ex}

In the “Strategy and outline” for proving Theorem~\ref{main1} (the analog of Theorem~\ref{thm:old} for $n=0$), the authors say they “use the lantern relations in various ways.”

They outline the proof as follows:
\begin{itemize}
\item \textbf{Section 2:} Prove “a rigidity result about two homologous curves intersecting at two points.” This “helps us to recognize mapping classes that are conjugates of $B_0$,” and “allows us to prove that $[\Ch,\M]$ contains $B_0$ and $T_1^2$.”
\item \textbf{Section 3:} Use “a homological argument about the cohomology of groups to obtain the key lemma showing that $T_1\in [\Ch,\M]$.” They emphasize this is “indirect” and “uses the results in \cite{Normal} in an unexpected way.” From this they obtain “the first equality of Theorem~\ref{main1},” and also prove Theorem~\ref{main2} and Theorem~\ref{thm:homological results}.
\item \textbf{Section 4:} Prove “the connectivity of $C_0^\alpha$ and the last equality of Theorem~\ref{main1}.” The connectivity proof “uses Putman’s trick multiple times on multiple different curve complexes that we construct.”
\end{itemize}

In the “Strategy and outline” for proving the analog for $n=0$ of
\begin{theorem}[Theorem 1.1 and 1.2 in \cite{Normal}]
\label{thm:old}
        When $1\le n\le g-2$, the normal subgroup  of $\M$ generated by  a genus $n$ bounding pair map $BP_n$ is
        \begin{equation}
        \label{eq:Wn}
             \langle\langle BP_n\rangle\rangle
            =[\Ch [2n],\M ]=\ker(d_{4n}).
        \end{equation}
\end{theorem}
(the analog of this theorem for $n=0$), the authors say they “use the lantern relations in various ways.”

They outline the proof as follows:
\begin{itemize}
\item \textbf{Next:} Prove “a rigidity result about two homologous curves intersecting at two points.” This “helps us to recognize mapping classes that are conjugates of $B_0$,” and “allows us to prove that $[\Ch,\M]$ contains $B_0$ and $T_1^2$.”
\item \textbf{Next:} Use “a homological argument about the cohomology of groups to obtain the key lemma showing that $T_1\in [\Ch,\M]$.” They emphasize this is “indirect” and “uses the results in \cite{Normal} in an unexpected way.” From this they obtain “the first equality” of the main theorem,
\begin{theorem}
\label{main1}
 When $g\ge 6$, the normal subgroup of $\M$ generated by $B_0$  is
 \begin{equation}
         \label{eq:B0=}
            \langle\langle B_0\rangle\rangle=[\Ch ,\M ]=\ker(d).
 \end{equation}
\end{theorem}

and also prove the following theorems:
\begin{theorem}\label{main2}
For $g\ge 6$, we have that
\[\langle \langle H_0\rangle \rangle = \Ch\]
\end{theorem}

\begin{theorem}
    \label{thm:homological results}
    For $g\ge 6$,  we have that
    \begin{enumerate}
        \item $H_1(\W(0);\Z)_{\M}=0,$\ \  \ \ \ \ \ \ \ \  $H^1(\W(0);\Z)^{\M}=0.$
        \item     $H_1(\Ch;\Z)_{\M}\cong \Z, \ \ \text{ }\ \ \ \ \ \ \ \ H^1(\Ch;\Z)^{\M}= \langle d/8\rangle\cong \Z.$\\
        $H^1(\Ch;\Z/m\Z)^{\M}= \langle d/8\rangle\cong \Z/m\Z$ for any $m\in\Z_{>0}$.
    \end{enumerate}
\end{theorem}
\item \textbf{Next:} Prove “the connectivity of $C_0^\alpha$ and the last equality” of the main theorem. The connectivity proof “uses Putman’s trick multiple times on multiple different curve complexes that we construct.”
\end{itemize}

Question: Let $S_g^1$ be a genus $g$ surface with one boundary component, let $\M$ be its mapping class group, and let $\I$ be the Torelli group. For an integer $n$ with $1\le n\le g-2$, let $BP_n$ be a genus $n$ bounding pair map, let $\ch:\I\to H_1(S_g^1;\mathbb Z)$ be the Chillingworth homomorphism, let $\Ch[2n]=\{f\in\I\mid \ch(f)\equiv 0\pmod{2n}\}$, and let $d_{4n}:\Ch[2n]\to \mathbb Z/4n\mathbb Z$ be the Casson--Morita $d$-map modulo $4n$. Which statement holds for the normal subgroup of $\M$ generated by $BP_n$?

It is exactly
\[
\langle\!\langle BP_n\rangle\!\rangle=[\Ch[2n],\M]=\ker(d_{4n}).
\]

Choices:
B.
It is exactly
\[
\langle\!\langle BP_n\rangle\!\rangle=[\Ch[2n],\M]=\ker(d_{2n}).
\]
C.
It is contained in
\[
\ker(d_{4n}).
\]
D.
For each such $n$, it is exactly
\[
\langle\!\langle BP_n\rangle\!\rangle=[\Ch[2n],\M],
\]
and this subgroup has finite index in \(\ker(d_{4n})\).
E.
It is exactly
\[
\langle\!\langle BP_n\rangle\!\rangle=[\Ch,\M]=\ker(d_{4n}).
\]

The stem does not state the conclusion outright; it only defines the objects involved. There is no explicit answer leakage beyond introducing the notation needed to state the theorem.

This is essentially a theorem-recall item: the correct option is the exact structural statement of the result, with only slight perturbations in the distractors. It functions as a direct restatement rather than a non-tautological application.

There is some pressure to distinguish exact equality from mere containment and to track the precise modulus (e.g. 4n vs 2n), but the item mainly tests recall of the theorem rather than generating or deriving the conclusion from reasoning.

The distractors are mathematically plausible and target realistic errors: wrong modulus, weakening equality to containment, replacing equality by finite-index containment, and confusing Ch[2n] with Ch. They are distinct and well-calibrated.

A technically well-constructed theorem-identification MCQ with strong distractors and no answer leakage, but it mainly assesses precise recall rather than genuine generative reasoning.