Uniform growth of groups acting on Cartan-Hadamard spaces

TL;DR

This paper establishes a uniform lower bound on the algebraic entropy for non-virtually nilpotent groups acting on certain negatively curved manifolds, highlighting a dichotomy in group growth behavior.

Contribution

It proves a uniform lower bound on algebraic entropy for groups acting on pinched negatively curved manifolds, extending understanding of group growth in geometric contexts.

Findings

Groups acting on these manifolds are either virtually nilpotent or have entropy at least C(n,a)

The lower bound C(n,a) depends only on dimension and curvature bounds

Provides a dichotomy in the growth behavior of such groups

Abstract

Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \leq K \leq -1$. There exist a positive constant $C(n,a)$ such that for any finitely generated discrete group $\Gamma$ acting on $X$, then either $\Gamma$ is virtually nilpotent or the algebraic entropy $Ent (\Gamma) \geq C(n,a)$.