Conformal dimension and random groups

TL;DR

This paper establishes bounds on the conformal dimension of boundaries of small cancellation groups and applies these to random groups, showing they vary through many quasi-isometry classes as relator length increases.

Contribution

It provides new bounds on the conformal dimension for boundaries of small cancellation groups and applies these to analyze the diversity of random groups at various densities.

Findings

Bounds on conformal dimension for small cancellation groups.

Random groups at certain densities pass through infinitely many quasi-isometry classes.

Conformal dimension grows with relator length in the models studied.

Abstract

We give a lower and an upper bound for the conformal dimension of the boundaries of certain small cancellation groups. We apply these bounds to the few relator and density models for random groups. This gives generic bounds of the following form, where $l$ is the relator length, going to infinity. (a) $1 + 1/C < \Cdim(\bdry G) < C l / \log(l)$, for the few relator model, and (b) $1 + l / (C\log(l)) < \Cdim(\bdry G) < C l$, for the density model, at densities $d < 1/16$. In particular, for the density model at densities $d < 1/16$, as the relator length $l$ goes to infinity, the random groups will pass through infinitely many different quasi-isometry classes.