Fixed points on boundary of hyperbolic group

Question

Let G be a word-hyperbolic group with torsion and let ∂G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on ∂G?

Answer 1

Let $G$ be a hyperbolic group with the Cayley graph $X$. Let $F<G$ be a finite subgroup and $L\subset \partial G$ be the fixed-point set of $F$. I will assume that $F$ is the maximal finite subgroup with the fixed-point set $L$.

Lemma. The normalizer $H$ of $F$ in $G$ has the property that:

1. $H$ is quasiconvex in $G$.

2. The Gromov boundary of $H$ maps to $L$ under the natural embedding $\partial H\to \partial G$,

Proof. I will first give a proof assuming that $L$ contains at least 2 points and then explain what to do in the 1-point case. Let $C\subset X$ denote the quasiconvex hull of $L$, i.e., the union of all complete geodesics which are asymptotic (in both directions) to points of $L$. Then there exists $r<\infty$ so that $$ d(x, gx)\le r, \forall g\in F, \forall x\in C. $$ Let $H<G$ denote the stabilizer of $C$ in $G$. Then (by maximality of $F$) $H$ normalizes $F$. Conversely, if $g\in G$ belongs to the normalizer of $F$ then $g(L)=L$ and, hence, $g(C)=C$.

Suppose now that $H$ does not act cocompactly on $C$. Then there exists a sequence of vertices $x_n\in C$ such that $$ x_i\ne H x_j, \forall i\ne j. $$ Let $g_n\in G$ be elements representing the vertices $x_n$. Consider the subgroups $$ F_i= g_i^{-1} F g_i<G. $$ These subgroups have the property that $$ \max_{g\in F_i} |g|\le r, $$ where $|g|$ is the word metric on $G$ and $r$ is the constant defined above. The number of elements of $G$ whose norms are bounded above is finite. Hence, after passing to a subsequence in $(x_n)$ , we can assume that $F_i=F_j$ for all $i, j$. In particular, $$ (g_i g_j^{-1}) F (g_ig_j^{-1})^{-1} = F. $$ Therefore, $h_{ij}=g_i^{-1} g_j$ preserves $C$. By construction, $$ h_{ij}(x_j)=x_i, $$ which is a contradiction. Thus, $H$ acts cocompactly on $C$. Since $C$ is quasiconvex, so is $H$. The limit set of $C$ in $\partial G$ is $H$-equivariantly homeomorphic to the ideal boundary of (the hyperbolic group) $H$, since $H$ acts on $C$ cocompactly and $C$ the inclusion map $C\to X$ is a q.i. embedding.

This finishes the proof except for the possibility that $L$ is a singleton. I claim that this cannot happen. The argument is similar to the main proof. Suppose that $L$ consists of a single point $z$. Consider a geodesic ray $\rho$ in $X$ asymptotic to $z$. Let $v_i$ denote the vertices on $\rho$ and let $g_i\in G$ be the corresponding group elements. Consider the sequence of rays $$ \rho_i= g_i^{-1}(\rho). $$ These rays contain the vertex $e\in G\subset X$ and, hence, subconverge to a complete geodesic $\gamma$ in $X$. The conjugate groups $F_i=g_i^{-1} F g_i$ move $e$ by uniformly bounded amount, hence, after passing to another subsequence, we can assume they are all equal to a subgroup $F'<G$. Hence, $F'$ is conjugate to $F$ and quasi-preserves the limit geodesic $\gamma$. It is now clear that $\gamma$ is asymptotic to two fixed points of $F'$. Hence, $F$ also has at least 2 ideal fixed points. Contradiction. qed

Thus, you can give purely algebraic conditions for the free action of $F$ on the Gromov boundary of $G$. In particular, we have a short exact sequence $$ 1\to F\to H\to Q \to 1. $$ By passing to a finite index subgroup $Q'<Q$, we obtain another (infinite) quasiconvex subgroup $H'<H$ containing $F$ so that $H'$ centralizes $F$. Therefore, in order to exclude (nontrivial) finite subgroups of $G$ with nonempty fixed-point sets in $\partial G$, it suffices to assume that every infinite quasiconvex subgroup in $G$ has trivial centralizer.