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The centralizer of an infinite-order element in a hyperbolic group is virtually cyclic

Statement

Let $G$ be a hyperbolic group and let $g \in G$ have infinite order. Then its centralizer

C_{G} (g) = {h \in G : h g = g h}

contains a cyclic subgroup of finite index.

Facts & Assumptions

Given: A hyperbolic group $G$ and an infinite-order element $g \in G$ .

[A1]

There is $L \geq 0$ such that every geodesic segment from $1$ to $g^{n}$ lies in the $L$ -neighborhood of the powers of $g$ .

[A2]

A geodesic quadrilateral in a $δ$ -hyperbolic Cayley graph is $2 δ$ -thin, and every metric ball in a locally finite Cayley graph is finite.

[L1]

Infinite-order elements are undistorted (Infinite-order elements of hyperbolic groups are undistorted).

Proof

technique · direct

1.1

By [L1] and [A1], the powers of $g$ form a quasi-axis: geodesics joining distant powers stay uniformly close to the power orbit.

L1A1

2.1

Let $s \in C_{G} (g)$ and choose $m$ so large that the two long sides of the quadrilateral with vertices $1, g^{m}, s g^{m}, s$ have points outside the $2 δ$ -neighborhoods of its short sides. By [A2], some point on $[1, g^{m}]$ is within $2 δ$ of $[s, s g^{m}]$ . Using [A1] on both long sides gives integers $i, j$ with $d (g^{i}, s g^{j}) \leq 2 L + 2 δ$ . Since $s$ commutes with $g$ , this says that the coset $⟨ g ⟩ s$ has a representative of word length at most $2 L + 2 δ$ .

A1A2step 1.1choosealgebra

3.1

The ball of radius $2 L + 2 δ$ is finite by [A2], so step 2.1 leaves only finitely many cosets of $⟨ g ⟩$ in $C_{G} (g)$ . Hence $⟨ g ⟩$ has finite index in $C_{G} (g)$ , and the centralizer is virtually cyclic.

A2step 2.1∎

Depends on

Used by

Abelian subgroups of hyperbolic groups are virtually cyclic Corollary

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Clara Löh, Geometric Group Theory, Section 6.5.2 (standard reference, not scraped)