Relative twisting in Outer space

TL;DR

This paper introduces two new versions of relative twisting for outer automorphisms of free groups, providing conditions for their behavior in Outer space and analyzing the dynamics of certain automorphisms.

Contribution

It presents alternative definitions of relative twisting in Outer space and applies these to study the geometry and dynamics of outer automorphisms.

Findings

Conditions for folding paths to enter the thin part of Outer space

Construction of automorphisms with axes passing through graphs with short cycles

Analysis of asymptotic translation lengths of automorphisms

Abstract

Subsurface projection has become indispensable in studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free group. We use this to describe sufficient conditions for when a folding path enters the thin part of Culler-Vogtmann's Outer space. As an application of our condition, we produce a sequence of fully irreducible outer automorphisms whose axes in Outer space travel through graphs with arbitrarily short cycles; we also describe the asymptotic behavior of their translation lengths.