Conformal dimension of hyperbolic groups that split over elementary subgroups

TL;DR

This paper investigates the conformal dimension of the boundary at infinity of Gromov hyperbolic groups that split over elementary subgroups, providing a formula and characterizations that answer a previously open question.

Contribution

It establishes a formula for the conformal dimension of such groups and characterizes when this dimension is attained, answering a question by Bonk and Kleiner.

Findings

Conformal dimension equals the maximum of vertex group dimensions or 1 for non-virtually free groups.

Characterization of groups with conformal dimension 1.

Provides conditions under which the conformal dimension is attained.

Abstract

We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without $2$-torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.