Surface subgroups of Kleinian groups with torsion

TL;DR

This paper proves that certain Kleinian groups with torsion always contain a surface subgroup, extending the understanding of their subgroup structure and confirming this property for all arithmetic Kleinian groups.

Contribution

It establishes that finitely generated Kleinian groups with non-cyclic torsion subgroups either are finite, virtually free, or contain a surface subgroup, including all arithmetic Kleinian groups.

Findings

Every finitely generated Kleinian group with a finite, non-cyclic subgroup contains a surface subgroup.

All arithmetic Kleinian groups contain a surface subgroup.

Classifies the subgroup structure of Kleinian groups with torsion.

Abstract

We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.