Property (T) in k-gonal random groups

2021/04/04 by Montee, MurphyKate
#20F65 #20P05 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2104.01621

Abstract

The 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 d₀ ∈ (0, 1), if positive k-gonal random groups satisfy Property (T) with overwhelming probability for densities d >d₀, then so do nk-gonal random groups, for any n ∈ 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.

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