Strongly contracting axes for fully irreducible automorphisms of Generalized Baumslag-Solitar groups

TL;DR

This paper proves that axes of fully irreducible automorphisms in the deformation space of Generalized Baumslag-Solitar groups are strongly contracting, extending hyperbolic dynamics insights similar to those in Outer Space.

Contribution

It establishes the strong contraction property of axes for fully irreducible automorphisms in the deformation space of GBS groups, inspired by prior work on Outer Space.

Findings

Axes are strongly contracting in the deformation space.

Automorphisms act hyperbolically on the space.

Extension of hyperbolic dynamics to GBS groups.

Abstract

Similarly to the action of $Out(F_N)$ on Outer Space, the outer automorphism group of a Generalized Baumslag Solitar group acts on a deformation space endowed with the Lipschitz metric and the action of any fully irreducible automorphism with a train track representative is hyperbolic. Inspired by previous work by Algom-Kfir, we prove that the axis of such an automorphism is strongly contracting.