Sub-dominant cogrowth behaviour and the viability of deciding amenability numerically

TL;DR

This paper critically examines a numerical method for determining group amenability via cogrowth rates, identifies potential errors, and proposes modifications for more accurate initial term estimates.

Contribution

It introduces a modified numerical approach that improves the accuracy of cogrowth sequence estimates for finitely generated groups.

Findings

Identified two sources of error in the existing method

Proposed a modification enabling accurate initial cogrowth estimates

Demonstrated improved accuracy with the new approach

Abstract

We critically analyse a recent numerical method due to the first author, Rechnitzer and van Rensburg, which attempts to detect amenability or non-amenability in a finitely generated group by numerically estimating its asymptotic cogrowth rate. We identify two potential sources of error. We then propose a modification of the method that enables it to easily compute surprisingly accurate estimates for initial terms of the cogrowth sequence.