Discrete and free groups acting on locally finite trees

TL;DR

This paper introduces an algorithm to determine if a finitely generated subgroup of a tree's isometry group is discrete and free, based on a conjecture about minimal isometry tuples, with proven cases for small tuples.

Contribution

It presents a novel algorithm for subgroup analysis in trees and proves a key conjecture for small cases, extending Ihara's Theorem.

Findings

Algorithm successfully decides discreteness and freeness for certain subgroups.

Proves the conjecture for 2- and 3-tuples of isometries.

Establishes a link to a generalization of Ihara's Theorem.

Abstract

We present an algorithm to decide whether or not a finitely generated subgroup of the isometry group of a locally finite simplicial tree is both discrete and free. The correctness of this algorithm relies on the following conjecture: every `minimal' $n$-tuple of isometries of a simplicial tree either contains an elliptic element or satisfies the hypotheses of the Ping Pong Lemma. We prove this conjecture for $n=2,3$, and show that it implies a generalisation of Ihara's Theorem.