Stable Torsion Length

TL;DR

This paper investigates the stable torsion length in groups, proving it vanishes in crystallographic groups and providing algorithms to compute it in free products, including exact rational computations for finite groups, with applications to specific examples.

Contribution

It introduces algorithms for computing stable torsion length in free products, including an exact rational computation method for finite groups, and applies these to nontrivial examples.

Findings

Stable torsion length vanishes in crystallographic groups.

Algorithms for lower bounds and exact calculations in free products.

First exact computations of stable torsion length in nontrivial cases.

Abstract

The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the first exact computations of stable torsion length for nontrivial examples.