Tree-irreducible automorphisms of free groups

TL;DR

This paper introduces a new class of automorphisms of free groups called tree-irreducible automorphisms, which generalize fully irreducible automorphisms and include automorphisms induced by pseudo-Anosov homeomorphisms, with significant dynamical properties.

Contribution

The paper defines tree-irreducible automorphisms, explores their properties, and shows they form fundamental building blocks for automorphisms of free groups, extending the class of known automorphisms.

Findings

Tree-irreducible automorphisms include all iwips and automorphisms from pseudo-Anosov homeomorphisms.

They exhibit North-South dynamics on the Thurston compactification of Outer space.

A blow-up construction produces new tree-irreducible automorphisms from iwips.

Abstract

We introduce a new class of automorphisms $\varphi$ of the non-abelian free group $F_N$ of finite rank $N \geq 2$ which contains all iwips (= fully irreducible automorphisms), but also any automorphism induced by a pseudo-Anosov homeomorphism of a surface with arbitrary many boundary components. More generally, there may be subgroups of $F_N$ of rank $\geq 2$ on which $\varphi$ restricts to the identity. We prove some basic facts about such {\em tree-irreducible} automorphisms, and show that, together with Dehn twist automorphisms, they are the natural basic building blocks from which any automorphism of $\FN$ can be constructed in a train track set-up. We then show: {\bf Theorem:} {\it Every tree-irreducible automorphism of $F_N$ has induced North-South dynamics on the Thurston compactification $\bar{\rm CV}_N$ of Outer space.} Finally, we define a "blow-up" construction on the vertices of a train track map, which, starting from iwips, produces tree-irreducible automorphisms which in general are not iwip.