Numerical studies of Thompson's group F and related groups

TL;DR

This paper develops algorithms to compute and analyze the cogrowth series of Thompson's group F and related groups, providing numerical evidence and bounds that suggest Thompson's group F is non-amenable.

Contribution

It introduces polynomial-time algorithms for cogrowth series of specific groups and extends methods to analyze asymptotics, offering new evidence on Thompson's group F's amenability.

Findings

Computed 32 terms of the cogrowth series for Thompson's group F.

Provided improved lower bounds on the growth rate of the cogrowth series.

Numerical data suggests Thompson's group F is likely non-amenable.

Abstract

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups $\mathbb{Z}\wr \mathbb{Z},$ the lamplighter group, $(\mathbb{Z}\wr \mathbb{Z})\wr \mathbb{Z}$ and the Navas-Brin group $B.$ We have also given an improved algorithm for the coefficients of Thompson's group $F,$ giving 32 terms of the cogrowth series. We develop numerical techniques to extract the asymptotics of these various cogrowth series. We present improved rigorous lower bounds on the growth-rate of the cogrowth series for Thompson's group $F$ using the method from \cite{HHR15} applied to our extended series. We also generalise their method by showing that it applies to loops on any locally finite graph. Unfortunately, lower bounds less than 16 do not help in determining amenability. Again for Thompson's group $F$ we prove that, if the group is amenable, there cannot be a sub-dominant stretched exponential term in the asymptotics\footnote{ }. Yet the numerical data provides compelling evidence for the presence of such a term. This observation suggests a potential path to a proof of non-amenability: If the universality class of the cogrowth sequence can be determined rigorously, it will likely prove non-amenability. We estimate the asymptotics of the cogrowth coefficients of $F$ to be $$ c_n \sim c \cdot \mu^n \cdot \kappa^{n^\sigma \log^\delta{n}} \cdot n^g,$$ where $\mu \approx 15,$ $\kappa \approx 1/e,$ $\sigma \approx 1/2,$ $\delta \approx 1/2,$ and $g \approx -1.$ The growth constant $\mu$ must be 16 for amenability. These two approaches, plus a third based on extrapolating lower bounds, support the conjecture \cite{ERvR15, HHR15} that the group is not amenable.