Growth of intersection numbers for free group automorphisms

TL;DR

This paper analyzes how the intersection number between trees in Outer space grows under iteration of a fully irreducible automorphism of a free group, revealing asymptotic behavior and geometric insights.

Contribution

It provides the first detailed asymptotic analysis of intersection numbers for free group automorphisms and explores the geometry of the Guirardel core in this context.

Findings

Asymptotic growth rate of intersection numbers under automorphism iteration

Qualitative description of Guirardel core geometry for large n

Insights into the dynamics of fully irreducible automorphisms

Abstract

For a fully irreducible automorphism \phi of the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'\phi^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'\phi^n for n large.