Spaces and groups with conformal dimension greater than one

TL;DR

This paper proves that certain metric spaces and hyperbolic group boundaries with specific connectivity properties have conformal dimension greater than one, providing a quantitative answer to a question in geometric group theory.

Contribution

It establishes a lower bound on the conformal dimension for spaces with annular linear connectivity and applies this to hyperbolic group boundaries without local cut points.

Findings

Conformal dimension > 1 for annularly linearly connected spaces

Hyperbolic group boundaries without local cut points have conformal dimension > 1

Quantitative bounds on conformal dimension are provided

Abstract

We show that if a complete, doubling metric space is annularly linearly connected then its conformal dimension is greater than one, quantitatively. As a consequence, we answer a question of Bonk and Kleiner: if the boundary of a one-ended hyperbolic group has no local cut points, then its conformal dimension is greater than one.