Round Trees and Conformal Dimension in Random Groups: low density to high density

Abstract

We investigate conformal dimension for the class of infinite hyperbolic groups in the Gromov density model ${\mathcal{G}}_{m,l}^d$ of random groups with $m\ge 2$ fixed generators, density $0<d<1/2$ and relator length $l\to \infty$. Our main result is a lower bound linear in $l$ at all densities $0<d<1/2$ achieved by building undistorted round trees coming directly from lower density Gromov random groups.

31 pages, 9 figures, comments welcome!