Conformal dimension via subcomplexes for small cancellation and random groups

TL;DR

This paper establishes new bounds on the conformal dimension of small cancellation groups and applies these results to analyze the conformal dimension of random groups in various models, revealing relationships with group parameters.

Contribution

It introduces novel bounds on conformal dimension for small cancellation groups and applies these to random groups, linking conformal dimension with group parameters and density models.

Findings

Random groups have conformal dimension 2+o(1) asymptotically almost surely.

In the density model at d<1/8, conformal dimension is approximately proportional to dl/|log d|.

New bounds are derived using $ extit{ ext{l}}_p$-cohomology and walls in Cayley complexes.

Abstract

We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators grows like l^K in the length l of the relators, then a.a.s. such a random group has conformal dimension 2+K+o(1). In Gromov's density model, a random group at density d<1/8 a.a.s. has conformal dimension $\asymp dl / |\log d|$. The upper bound for C'(1/8) groups has two main ingredients: $\ell_p$-cohomology (following Bourdon-Kleiner), and walls in the Cayley complex (building on Wise and Ollivier-Wise). To find lower bounds we refine the methods of [Mackay, 2012] to create larger `round trees' in the Cayley complex of such groups. As a corollary, in the density model at d<1/8, the density d is determined, up to a power, by the conformal dimension of the boundary and the Euler characteristic of the group.