Conformal dimension via subcomplexes for small can

Authors: John M Mackay

Publish Date: 2015/06/13

Volume: 364, Issue: 3-4, Pages: 937-982

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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 + o1 asymptotically almost surely aas In fact if the number of relators grows like lK in the length l of the relators then aas such a random group has conformal dimension 2+K+ o1 In Gromov’s density model a random group at density dfrac18 aas has conformal dimension asymp dl / log d The upper bound for Cfrac18 groups has two main ingredients ell pcohomology 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 Geom Funct Anal 221213–239 2012 to create larger ‘round trees’ in the Cayley complex of such groups As a corollary in the density model at dfrac18 the density d is determined up to a power by the conformal dimension of the boundary and the Euler characteristic of the groupI gratefully thank Marc Bourdon for describing to me his work with Bruce Kleiner and Piotr Przytycki for many interesting conversations about random groups and walls I also thank the referees for many helpful comments The author was partially supported by EPSRC Grant “Geometric and analytic aspects of infinite groups”

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