Applications of weak attraction theory in Out($F_n$)

TL;DR

This paper applies weak attraction theory and pingpong techniques to analyze the dynamics of exponentially growing outer automorphisms of free groups, establishing conditions under which their powers generate free groups with fully irreducible, hyperbolic elements.

Contribution

It introduces a new application of weak attraction theory combined with pingpong methods to understand the composition of powers of fully irreducible automorphisms in Out($F_n$).

Findings

Existence of an integer M such that for all m,n >= M, automorphisms generate a free group of rank two.

Elements outside conjugates of generators are fully irreducible and hyperbolic.

Provides a new proof of a theorem in mapping class groups and a relativized version of the main theorem.

Abstract

Given a free group of rank r >= 3 and two exponentially growing outer automorphisms {\psi} and {\phi} with dual lamination pairs {\Lambda^\pm}_{\psi} and {\Lambda^\pm}_{\phi} associated to them, which satisfy a notion of independence described in this paper, we will use the pingpong techniques developed in recent works of Handel and Mosher to show that there exists an integer M > 0, such that for every m, n >= M, {\psi}^m and {\phi}^n will freely generate a group of rank two and every element of this free group which is not conjugate to a power of the generators will be fully irreducible and hyperbolic. The main corollary is a complete understanding of composition of powers of fully irreducible elements. As bonus we will reprove a theorem in MCG of surfaces with one boundary component and also have a new proof of a theorem of Kapovich and Lustig. The final section also has a relativised version of the main theorem which was recently used my Handel and Mosher to prove the second bounded cohomology alternative for Out($F_n$) in arXiv:1511.06913