Probabilistic Burnside groups

TL;DR

This paper constructs a finitely generated group that almost always satisfies a specific law under random sampling but contains a free subgroup, thus not satisfying any law, answering open questions in group theory.

Contribution

It introduces a group that probabilistically satisfies a law but does not satisfy any law, bridging probabilistic and algebraic properties in group theory.

Findings

Group law satisfied with probability 1 under random sampling

Contains a non-abelian free subgroup

Answers open questions in the field

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.