Probabilistic Burnside groups

2023/06/20 by Gil Goffer, Goffer, Gil, Be’eri Greenfeld +1
Mathematics · #20F06 #20F65 #20F69 #20P05 #Advanced Topology and Set Theory #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2306.11204

openalex publication_date 2023/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that there exists a finitely generated group that satisfies a group law with probability 1 but does not satisfy any group law. More precisely, we construct a finitely generated group G in which the probability that a random element chosen uniformly from a finite ball in its Cayley graph, or via any non-degenerate random walk, satisfies the group law x^k=1 for some (fixed) integer k, tends to 1. Yet, G contains a non-abelian free subgroup, and therefore G does not satisfy any group law. In particular, this answers two questions of Amir, Blachar, Gerasimova, and Kozma.

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