On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups

TL;DR

This paper investigates the Hausdorff dimension of endpoint sets of algebraic laminations related to free group automorphisms, showing they have Hausdorff dimension zero for certain classes of automorphisms and laminations.

Contribution

It proves that the endpoint sets of attracting and ending laminations for exponentially growing, fully irreducible automorphisms have Hausdorff dimension zero.

Findings

Endpoint sets of attracting laminations have Hausdorff dimension zero.

Same conclusion holds for ending laminations under Cannon-Thurston maps.

Results extend to all trees in the Culler-Vogtmann outer space.

Abstract

Motivated by a classic theorem of Birman and Series about the set of complete simple geodesics on a hyperbolic surface, we study the Hausdorff dimension of the set of endpoints in $\partial F_r$ of some abstract algebraic laminations associated with free group automorphisms. For an exponentially growing outer automorphism $\phi\in Out(F_r)$ we show that the set of endpoints $\mathcal E_{L}\subseteq \partial F_r$ of any of the \emph{attracting laminations} $L$ of $\phi$ has Hausdorff dimension $0$ for any tree $T\in cv_r$ and any visual metric on the boundary $\partial T=\partial F_r$. If $\phi\in Out(F_r)$ is an atoroidal and fully irreducible, we deduce the same conclusion for the set of endpoints of the ending lamination $\Lambda_\phi$ of $\phi$ that gets collapsed by the Cannon-Thurston map $\partial F_r\to \partial G_\phi$ for the associated free-by-cyclic group $G_\phi=F_r\rtimes_\phi\mathbb Z$.