Margulis Lemma, entropy and free products

TL;DR

This paper extends Margulis' Lemma to manifolds with fundamental groups that are free products, providing bounds on systole and applications to finiteness and volume estimates.

Contribution

It proves a version of Margulis' Lemma for free product fundamental groups and establishes bounds on homotopy systole based on entropy and diameter.

Findings

Established Margulis' Lemma for free product groups without 2-torsion.

Provided lower bounds for homotopy systole in torsion-free cases.

Presented applications including finiteness theorems and volume estimates.

Abstract

We prove a Margulis' Lemma \`a la Besson Courtois Gallot, for manifolds whose fundamental group is a nontrivial free product A*B, without 2-torsion. Moreover, if A*B is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume entropy. We also provide examples and counterexamples showing the optimality of our assumption. Finally we give two applications of this result: a finiteness theorem and a volume estimate for reducible manifolds.