Algebraically recurrent random walks on groups

TL;DR

This paper explores the conditions under which random walks on finitely generated groups almost surely generate the entire group as a semigroup, advancing understanding of algebraic properties of such stochastic processes.

Contribution

It introduces initial insights into when random walk paths generate groups as semigroups, a novel perspective linking algebraic structure and probabilistic behavior.

Findings

Identifies conditions for random walks to generate groups as semigroups

Provides initial theoretical framework for algebraic properties of random walks

Lays groundwork for further research in probabilistic group theory

Abstract

Initial steps are presented towards understanding which finitely generated groups are almost surely generated as semigroups by the path of a random walk on the group.