Geometric and Functional Analysis

, Volume 22, Issue 1, pp 213–239 | Cite as

Conformal Dimension And Random Groups



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.

  1. (a)

    \({1 + 1/C < \mathcal{C}{\rm dim}(\partial_{\infty}G) < Cl/{\rm log}(l)}\) , for the few relator model, and

  2. (b)

    \({1 + l/(C\, {\rm log}(l)) < \mathcal{C}{\rm dim}(\partial_{\infty}G) < Cl}\) , 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.

Keywords and phrases

Conformal dimension random groups 

2010 Mathematics Subject Classification

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


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Mathematical InstituteOxfordUK

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