Random sampling of trivials words in finitely presented groups

TL;DR

This paper introduces a new Monte Carlo algorithm for randomly sampling trivial words in finitely presented groups, linking statistical properties to group amenability.

Contribution

The paper presents a novel Metropolis Monte Carlo algorithm for sampling trivial words, connecting the distribution to the group's cogrowth and amenability.

Findings

Successfully implemented on various groups including Baumslag-Solitar and Thompson's group F.

Established a relationship between word sampling distribution and group cogrowth.

Demonstrated the algorithm's applicability to both amenable and non-amenable groups.

Abstract

We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann distribution \begin{align*}\pi(w) &= (|w|+1)^{\alpha} \beta^{|w|} \cdot Z^{-1} \end{align*} where $|w|$ is the length of a word $w$, $\alpha$ and $\beta$ are parameters of the algorithm, and $Z$ is a normalising constant. It follows that words of the same length are sampled with the same probability. The distribution can be expressed in terms of the cogrowth series of the group, which then allows us to relate statistical properties of words sampled by the algorithm to the cogrowth of the group, and hence its amenability. We have implemented the algorithm and applied it to several group presentations including the Baumslag-Solitar groups, some free products studied by Kouksov, a finitely presented amenable group that is not subexponentially amenable (based on the basilica group), and Richard Thompson's group $F$.