Mathematische Annalen

, Volume 364, Issue 3–4, pp 937–982 | Cite as

Conformal dimension via subcomplexes for small cancellation and random groups

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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 \(2 + o(1)\) 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 \(d<\frac{1}{8}\) a.a.s. has conformal dimension \(\asymp dl / |\log d|\). The upper bound for \(C'(\frac{1}{8})\) 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 (Geom Funct Anal 22(1):213–239, 2012) to create larger ‘round trees’ in the Cayley complex of such groups. As a corollary, in the density model at \(d<\frac{1}{8}\), the density d is determined, up to a power, by the conformal dimension of the boundary and the Euler characteristic of the group.

Mathematics Subject Classification

Primary 20F65 Secondary 20F06 20F67 20P05 57M20 

Notes

Acknowledgments

I 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 referee(s) for many helpful comments. The author was partially supported by EPSRC Grant “Geometric and analytic aspects of infinite groups”.

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK

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