[Paper Review] Dense subsets of boundaries of CAT(0) groups

Tetsuya Hosaka|ArXiv.org|Oct 24, 2005

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper establishes conditions under which the boundary of a CAT(0) group action is minimal, meaning every orbit is dense in the boundary. It proves that if a CAT(0) group $G$ acts geometrically on a CAT(0) space $X$ and contains an element $g_0$ with finite centralizer, non-connected complement of its fixed-point set, and convex, non-invariant components, then both the orbit $G\alpha$ and the set of infinite-order group elements' fixed points on the boundary are dense in $\partial X$. This result applies to Coxeter systems with finite centralizers of reflections.

ABSTRACT

In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$ and suppose that there exists an element $g_0\in G$ such that (1) $Z_{g_0}$ is finite, (2) $X\setminus F_{g_0}$ is not connnected, and (3) each component of $X\setminus F_{g_0}$ is convex and not $g_0$-invariant, where $Z_{g_0}$ is the centralizer of $g_0$ and $F_{g_0}$ is the fixed-point set of $g_0$ in $X$ (that is, $Z_{g_0}=\{h\in G| g_0h=hg_0\}$ and $F_{g_0}=\{x\in X| g_0x=x\}$). Then we show that each orbit $G α$ is dense in the boundary $\partial X$ (i.e.\ $\partial X$ is minimal) and the set $\{g^{\infty} | g\in G, o(g)=\infty\}$ is also dense in the boundary $\partial X$. We obtain an application for dense subsets on the boundary of a Coxeter system.

Motivation & Objective

• To determine sufficient conditions under which the boundary of a CAT(0) group action is minimal, i.e., every orbit is dense in the boundary.
• To investigate when the set of fixed points of infinite-order group elements is dense in the boundary of a CAT(0) space.
• To provide a criterion for minimality of the boundary using group-theoretic and geometric properties of a single element $g_0$ in the group.
• To apply the main theorem to Coxeter systems, particularly to the Davis complex, by analyzing reflections with finite centralizers.

Proposed method

• Uses a criterion based on the existence of a group element $g_0$ whose centralizer $Z_{g_0}$ is finite and whose fixed-point set $F_{g_0}$ has a non-connected complement in $X$.
• Applies the geometric structure of $X\setminus F_{g_0}$, assuming each component is convex and not $g_0$-invariant, to construct sequences of group elements with controlled displacement.
• Employs the limit set $L(A)$ of a subset $A\subset G$ as a key tool, relating it to the fixed-point set $\mathcal{F}_g$ on the boundary via Theorem 2.1.
• Uses a density criterion: if for large displacements $d(x_0, gx_0) > M$, there exists $\alpha$ in a set $A$ such that $d(gx_0, \operatorname{Im}\xi_\alpha) \leq N$, then $A$ is dense in $\partial X$.
• Analyzes the dynamics of the action of $g_0$ and its conjugates to show that the orbit of any boundary point is dense, using convexity and separation properties of components of $X\setminus F_{g_0}$.
• Constructs geodesic rays converging to $h^\infty$ for $h$ of infinite order, showing their image intersects neighborhoods of group translates, ensuring density of the set $\{g^\infty \mid o(g)=\infty\}$.

Experimental results

Research questions

• RQ1 Under what group-theoretic and geometric conditions is the boundary of a CAT(0) group action minimal?
• RQ2 When is the set of fixed points of infinite-order group elements dense in the boundary of a CAT(0) space?
• RQ3 Can the minimality of the boundary be guaranteed by properties of a single element $g_0$ in the group?
• RQ4 Does the existence of a reflection $r$ in a Coxeter system with finite centralizer imply minimality of the boundary of the Davis complex?

Key findings

• If $G$ acts geometrically on a CAT(0) space $X$ and there exists $g_0\in G$ with finite centralizer, non-connected $X\setminus F_{g_0}$, and convex, non-invariant components, then $\partial X$ is minimal.
• The set $\{g^\infty \mid g\in G, o(g)=\infty\}$ is dense in $\partial X$ under the same conditions.
• For a Coxeter system $(W,S)$, if there exists $s\in S$ with finite centralizer in $W$, then $\partial\Sigma(W,S)$ is minimal and the set of fixed points of infinite-order elements is dense in the boundary.
• The result applies to reflections in the Davis complex: if a reflection $r$ has finite centralizer, then $\partial\Sigma(W,S)$ is minimal.
• The proof shows that for any $\beta\in\partial X$, and any $R>0$, $\epsilon>0$, there exists $\alpha\in A$ such that $d(\xi_\beta(R), \operatorname{Im}\xi_\alpha) < \epsilon$, proving density of $A$.
• The construction uses the fact that $gF_{g_0} \cap gg_0gF_{g_0} = \emptyset$ for large $d(x_0, gx_0)$, leading to infinitely many distinct components and hence infinite order of $gg_0$.