Conformal dimension via subcomplexes for small cancellation and random groups

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 lK 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<18 a.a.s. has conformal dimension ≍dl/|logd|. The upper bound for C′(18) groups has two main ingredients: ℓp-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<18, the density d is determined, up to a power, by the conformal dimension of the boundary and the Euler characteristic of the group.