Growth of automorphisms of virtually special groups
Elia Fioravanti
TL;DR
The article develops a comprehensive framework for understanding the growth of outer automorphisms of virtually special groups via a canonical JSJ theory over centralisers. It proves that stretch factors are algebraic integers and automorphisms exhibit a dichotomy between polynomial and exponential growth, with coarse-median preserving maps yielding finitely many growth rates and a Nielsen–Thurston–type decomposition. It introduces an enhanced JSJ splitting invariant under Out$(G)$, proves accessibility over centralisers, and establishes boundary amenability and the Tits alternative for Out$(G)$. The results unify automorphism growth across broad families (notably RAAGs) and yield new structural corollaries, including a complexity-reduction homomorphism and consequences for hyperbolicity and rigidity. Overall, the work provides a robust toolkit for analyzing automorphisms of non-hyperbolic cubical groups and their asymptotic dynamics.
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 accessibile 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.