Conformal dimension and canonical splittings of hyperbolic groups

TL;DR

This paper establishes a criterion based on local cut points for a metric space to have conformal dimension one, and applies it to hyperbolic group boundaries, revealing a link between conformal dimension and group splittings.

Contribution

It introduces a new criterion for conformal dimension one and connects it to canonical splittings of hyperbolic groups.

Findings

Criterion for conformal dimension one based on local cut points

Application to hyperbolic group boundaries

Relationship between conformal dimension and group splittings

Abstract

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic groups and show an interesting relationship between conformal dimension and some canonical splittings of the group.