Homological shadows of attracting laminations

TL;DR

This paper introduces a homological perspective on attracting laminations in free groups, describing convergence to convex polytopes with rational vertices and algebraic measures, especially when automorphisms have finite order action on abelianization.

Contribution

It extends the concept of attracting laminations by defining a homological version involving convex polytopes and measures, particularly for automorphisms with finite order action.

Findings

Homological attracting objects are convex polytopes with rational vertices.

Convergence involves measures supported at points with algebraic coordinates.

Applicable when automorphism action on abelianization has finite order.

Abstract

Given a free group $F_n$, a fully irreducible automorphism $f \in \aut$, and a generic element $x \in F_n$, the elements $f^k(x)$ converge in the appropriate sense to an object called an attracting lamination of $f$. When the action of $f$ on $\frac{F_n}{[F_n, F_n]}$ has finite order, we introduce a homological version of this convergence, in which the attracting object is a convex polytope with rational vertices, together with a measure supported at a point with algebraic coordinates.