Property (T) in k-gonal random groups

Abstract

AbstractThe k-gonal models of random groups are defined as the quotients of free groups on n generators by cyclically reduced words of length k. As k tends to infinity, this model approaches the Gromov density model. In this paper, we show that for any fixed $d_0 \in (0, 1)$ , if positive k-gonal random groups satisfy Property (T) with overwhelming probability for densities $d >d_0$ , then so do jk-gonal random groups, for any $j \in \mathbb{N}$ . In particular, this shows that for densities above 1/3, groups in 3k-gonal models satisfy Property (T) with probability 1 as n approaches infinity.