Entropy of automorphisms, homology and the intrinsic polynomial structure of nilpotent groups

TL;DR

This paper investigates how the entropy of automorphisms of residually nilpotent groups is primarily determined by their action on abelianization, with applications to surface homeomorphisms and a resolution of a question by Sullivan.

Contribution

It establishes that the entropy of automorphisms of nilpotent groups is governed by their induced action on abelianization, and applies this to pseudo-Anosov homeomorphisms of surfaces.

Findings

Entropy of automorphisms relates to abelianization entropy.

Bounded entropy for automorphisms of nilpotent quotients of surface groups.

Answers Sullivan's question on entropy bounds for surface automorphisms.

Abstract

We study the word length entropy of automorphisms of residually nilpotent groups, and how the entropy of such group automorphisms relates to the entropy of induced automorphisms on various nilpotent quotients. We show that much like the structure of a nilpotent group is dictated to a large degree by its abelianization, the entropy of an automorphism of a nilpotent group is dictated by its entropy on the abelianization. We give some applications to the study of pseudo-Anosov homeomorphisms of surfaces. In particular, we show that if $\psi$ is a non--homological pseudo-Anosov homeomorphism of a surface $\Sigma$ with dilatation $K$ and $N$ is any nilpotent quotient of any finite index characteristic subgroup of $\pi_1(\Sigma)$ to which $\psi$ descends, the entropy of $\psi$ viewed as an automorphism of $N$ is bounded away from $K$. This answers a question of D. Sullivan.