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 $X^{\prime }$ of $〈X〉$ such that either $〈X〉$ is purely hyperbolic and $X^{\prime }$ is a free basis of $〈X〉$, or $X^{\prime }$ 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 $X^{\prime }$ 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.