Nonvanishing of algebraic entropy for geometrically finite groups of isometries of Hadamard manifolds

TL;DR

This paper proves that geometrically finite groups of isometries in pinched Hadamard manifolds exhibit uniform exponential growth, establishing a key property of their algebraic entropy.

Contribution

It demonstrates that all such groups have nonvanishing algebraic entropy, a novel result linking geometric finiteness to exponential growth.

Findings

Geometrically finite groups have uniform exponential growth.

Algebraic entropy does not vanish for these groups.

Supports conjectures relating geometry and group dynamics.

Abstract

We prove that any geometrically finite (nonelementary) group of isometries of a pinched Hadamard manifold has uniform exponential growth.