Recognition and constructive membership for purely hyperbolic groups acting on trees

TL;DR

This paper introduces an algorithm to analyze groups acting on trees, determining hyperbolicity, freeness, and membership, with practical implementation and performance evaluation.

Contribution

It presents a novel algorithm based on Nielsen's reduction to solve the constructive membership problem for purely hyperbolic groups acting on trees.

Findings

Algorithm successfully identifies purely hyperbolic groups and free bases.

Implementation in Magma demonstrates practical efficiency.

Decides discreteness and freeness of groups acting on trees.

Abstract

We present an algorithm which takes as input a finite set $X$ of automorphisms of a simplicial tree, and outputs a generating set $X'$ of $\langle X \rangle$ such that either $\langle X \rangle$ is purely hyperbolic and $X'$ is a free basis of $\langle X \rangle$, or $X'$ contains a non-trivial elliptic element. As a special case, the algorithm decides whether a finitely generated group acting on a locally finite tree is discrete and free. This algorithm, which is based on Nielsen's reduction method, works by repeatedly applying Nielsen transformations to $X$ to minimise the generators of $X'$ with respect to a given pre-well-ordering. We use this algorithm to solve the constructive membership problem for finitely generated purely hyperbolic automorphism groups of trees. We provide a Magma implementation of these algorithms, and report its performance.