Algebraic Entropy and the Action of Mapping Class Groups on Character Varieties

TL;DR

This paper extends algebraic entropy to affine variety endomorphisms and computes it for mapping class group actions on character varieties, linking it to spectral radius and topological entropy.

Contribution

It introduces a generalized algebraic entropy for affine varieties and connects it with spectral radius in the context of mapping class group actions.

Findings

Algebraic entropy equals spectral radius for these actions

Calculations align with known topological entropy results

Provides a new framework for understanding group actions on character varieties

Abstract

We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov mapping class. Our calculations are compatible with all known calculations of the topological entropy of this action.