On some exotic Schottky groups

TL;DR

This paper constructs a specific negatively curved manifold with isometry groups containing divergent free subgroups with parabolic elements that violate the parabolic gap condition, using Poincaré series and transfer operators.

Contribution

It introduces a novel construction of a negatively curved manifold with isometry groups exhibiting divergent free subgroups lacking the parabolic gap condition.

Findings

Existence of divergent free subgroups with parabolic elements violating the gap condition

Use of Poincaré series and transfer operators in the construction

New insights into the structure of isometry groups in negatively curved manifolds

Abstract

We construct a Cartan-Hadamard manifold with pinched negative curvature whose group of isometries possesses divergent discrete free subgroups with parabolic elements who do not satisfy the so-called "parabolic gap condition" . This construction relies on the comparaison between the Poincar\'e series of these free groups and the potential of some transfer operator which appears naturally in this context.