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. (a) 1 + 1/C < Cdim(∂∞G) < Cl/ log(l), for the few relator model, and (b) 1 + l/(C log(l)) < Cdim(∂∞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.