Round Trees and Conformal Dimension in Random Groups: low density to high density

TL;DR

This paper establishes a linear lower bound on the conformal dimension of infinite hyperbolic groups in the Gromov density model across all densities below 1/2, using undistorted round trees.

Contribution

It introduces a method to derive linear lower bounds on conformal dimension for random groups by constructing undistorted round trees from lower density models.

Findings

Linear lower bounds on conformal dimension for all densities d<1/2

Construction of undistorted round trees from Gromov random groups

Applicable across a range of densities in the Gromov density model

Abstract

We investigate conformal dimension for the class of infinite hyperbolic groups in the Gromov density model $\mathcal{G}^d_{m,l}$ of random groups with $m \geq 2$ fixed generators, density $0 < d < 1/2$ and relator length $l \to \infty$. Our main result is a lower bound linear in $l$ at all densities $0 < d < 1/2$ achieved by building undistorted round trees coming directly from lower density Gromov random groups.