Recognition and constructive membership for purely hyperbolic groups acting on trees

We present an algorithm which takes as input a finite set $X$ of automorphisms of a simplicial tree, and outputs a generating set $X^{\mathrm{\prime }}$ of $⟨X⟩$ such that either $⟨X⟩$ is purely hyperbolic and $X^{\mathrm{\prime }}$ is a free basis of $⟨X⟩$, or $X^{\mathrm{\prime }}$ 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 $X^{\mathrm{\prime }}$ 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.