Normal form and parabolic dynamics for quadratically growing automorphisms of free groups

TL;DR

This paper introduces a normal form for quadratic growth automorphisms of free groups, revealing their dynamics as parabolic orbits in Outer space and characterizing growth behavior via a 2-level Dehn twist structure.

Contribution

It provides a new normal form for quadratic growth automorphisms using 2-level Dehn twists and analyzes their dynamics as parabolic orbits in Outer space.

Findings

Quadratic growth automorphisms have a normal form as 2-level Dehn twists.

Growth of conjugacy classes is linear if contained in a vertex group.

Automorphism dynamics are parabolic with limit points in a specific simplex.

Abstract

We present a normal form for outer automorphisms $\phi$ of a non-abelian free group $F_N$ which grow quadratically (measured through the maximal growth of conjugacy classes in $F_N$ under iteration of $\phi$). In analogy to the known normal form for linearly growing automorphisms as efficient Dehn twist, our normal form for $\phi$ is given in terms of a 2-level Dehn twist on a graph-of-groups $\cal{G}$ with $\pi_1 {\cal{G}} \cong F_N$, where a conjugacy class of $F_N$ grows at most linearly if and only if it is contained in a vertex group of $\cal{G}$. Our proof is based on earlier work of the second author and on a new cancellation result, which also allows us to show that the dynamics of the induced $\phi$-action on Outer space $CV_N$ consists entirely of parabolic orbits, with limit points all assembled in the simplex $\Delta_{\cal{G}} \subset \partial CV_N$ determined by $\cal{G}$.