Period growth and co-context-free groups

TL;DR

This paper investigates period growth in co-context-free groups, providing new bounds, algorithms for element analysis, and applying these methods to Thompson and Houghton groups to deepen understanding of their algebraic properties.

Contribution

It introduces refined bounds on word metrics and efficient algorithms for torsion detection and rotation number computation in Thompson groups, advancing the analysis of co-context-free groups.

Findings

Refined upper bounds on the word metric in Thompson V

Efficient algorithms for torsion detection in V

Algorithms for computing rotation numbers in T

Abstract

We study period growth in co-context-free groups, giving general results and looking at specific examples such as Thompson groups $T$ and $V$ and the Houghton groups $H_m$. Along the way, we give a refined upper bound on the word metric in Thompson $V$, as well as efficient algorithms to determine if elements of $V$ are torsion, and compute their order. We also adapt our algorithm to compute the rotation number of elements of $T$ and answer a question of D. Calegari.