On $k$-free-like groups

TL;DR

This paper introduces the concept of $k$-free-like groups, explores their properties, and constructs numerous non-free examples that exhibit these properties for large $k$, linking group theory with percolation theory.

Contribution

It defines $k$-free-like groups, connects their properties to percolation thresholds, and constructs many non-free examples for large $k$, answering a question in the field.

Findings

Critical bond percolation probability tends to 1/(2k-1) for these groups.

Many non-free groups are shown to be $k$-free-like for large $k$.

The work links geometric group properties with percolation theory results.

Abstract

A $k$-free like group is a $k$-generated group $G$ with a sequence of $k$-element generating sets $Z_n$ such that the girth of $G$ relative to $Z_n$ is unbounded and the Cheeger constant of $G$ relative to $Z_n$ is bounded away from 0. By a recent result of Benjamini-Nachmias-Peres, this implies that the critical bond percolation probability of the Cayley graph of $G$ relative to $Z_n$ tends to $1/(2k-1)$ as $n\to \infty$. Answering a question of Benjamini, we construct many non-free groups that are $k$-free like for all sufficiently large $k$.