On trivial words in finitely presented groups

TL;DR

This paper introduces a numerical approach to analyze the cogrowth of finitely presented groups, validating results with known cases and extending to Baumslag-Solitar groups, proving their cogrowth rates are algebraic.

Contribution

It presents a new numerical method for studying cogrowth, including the first computation for Baumslag-Solitar groups and a proof of their cogrowth rates being algebraic.

Findings

Numerical method effectively estimates cogrowth in finitely presented groups.

Cogrowth series for Baumslag-Solitar groups $ ext{BS}(N,N)$ are computed.

Cogrowth rates for these groups are proven to be algebraic numbers.

Abstract

We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to the set of such groups by finding the cogrowth series for Baumslag-Solitar groups $\mathrm{BS}(N,N) = < a,b | a^N b = b a^N >$ and prove that their cogrowth rates are algebraic numbers.