Growth of automorphisms of virtually special groups

TL;DR

This paper investigates the growth rates of automorphisms in virtually special groups, establishing polynomial or exponential growth, and introduces a Nielsen-Thurston type decomposition, with implications for automorphism groups and group structure.

Contribution

It provides new results on growth rates, automorphism decompositions, and structural properties of virtually special groups, including accessibility and JSJ decompositions.

Findings

Automorphisms grow polynomially or exponentially.

Stretch factors are algebraic integers.

Outer automorphism groups are boundary amenable and satisfy Tits alternative.

Abstract

We study the speed of growth of iterates of outer automorphisms of virtually special groups, in the Haglund-Wise sense. We show that each automorphism grows either polynomially or exponentially, and that its stretch factor is an algebraic integer. For coarse-median preserving automorphisms, we show that there are only finitely many growth rates and we construct an analogue of the Nielsen-Thurston decomposition of surface homeomorphisms. These results are new already for right-angled Artin groups. However, even in this particular case, the proof requires studying automorphisms of arbitrary special groups in an essential way. As results of independent interest, we show that special groups are accessible over centralisers, and we construct a canonical JSJ decomposition over centralisers. We also prove that, for any virtually special group $G$, the outer automorphism group ${\rm Out}(G)$ is boundary amenable, satisfies the Tits alternative, and has finite virtual cohomological dimension.