Growth of automorphisms of virtually special groups

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.

💡 Research Summary

This paper investigates the growth behavior of outer automorphisms of virtually special groups, a broad class of non‑positively curved groups introduced by Haglund and Wise. The authors introduce a natural notion of length for group elements—conjugacy length with respect to a fixed finite generating set—and define the growth rate of an automorphism φ as the bi‑Lipschitz equivalence class of the sequence n ↦ ‖φⁿ(g)‖ for each g∈G. The stretch factor str(φ) is defined as the supremum over g of the exponential growth rate lim supₙ ‖φⁿ(g)‖¹⁄ⁿ.

The first main result (Theorem A) shows that every outer automorphism of a virtually special group either grows polynomially or exponentially. Moreover, the stretch factor is always a weak Perron number, i.e. an algebraic integer whose modulus dominates that of all its Galois conjugates. If str(φ)=1 the automorphism has at most polynomial growth; if str(φ)>1 then the exponential growth is realized on a subgroup that is either a surface group, a free product, or a group possessing a free abelian direct factor. Thus the only sources of exponential growth are analogues of pseudo‑Anosov homeomorphisms, fully irreducible automorphisms of free products, and skew‑product automorphisms of direct products.

The second major theorem (Theorem B) deals with coarse‑median preserving automorphisms—this class includes untwisted automorphisms of right‑angled Artin groups (RAAGs), all automorphisms of right‑angled Coxeter groups, and automorphisms of Gromov‑hyperbolic groups. For such φ the set of possible exponential growth rates is finite: there exist finitely many weak Perron numbers λ₁,…,λₘ>1 and an integer P such that for every g∈G the sequence n ↦ ‖φⁿ(g)‖ is either bi‑Lipschitz equivalent to nᵖ·λᵢⁿ for some i and 0≤p≤P, or eventually bounded by nᴾ. Moreover, for each growth rate o the collection K(o) of subgroups whose elements grow at most at rate o has only finitely many G‑conjugacy classes of maximal members, and each such maximal subgroup is quasi‑convex in G. This provides a Nielsen‑Thurston‑type decomposition for coarse‑median preserving automorphisms.

A crucial technical achievement is the construction of a canonical JSJ decomposition over centralisers for any special group (Theorem D). The authors first prove an accessibility result (Theorem E) stating that any reduced graph‑of‑groups splitting of a special group over centralisers has a uniformly bounded number of edges. Using this, they build an Aut(G)‑invariant splitting whose edge groups are either centralisers or cyclic, and whose vertex groups are either quadratically hanging subgroups with trivial fibre or “rigid” quasi‑convex subgroups that are elliptic in all splittings over centralisers relative to the family S(G) of maximal virtually direct‑product subgroups. This decomposition replaces the classical JSJ theory for hyperbolic groups, which fails for many special groups because centralisers are not slender and can intersect in complicated ways.

The JSJ decomposition enables an inductive analysis of growth: the top growth rate of an automorphism of a rigid vertex group is always realized on a lower‑complexity singular subgroup (typically a surface group or a free product). The base of the induction consists of surface groups and free products, where train‑track and Nielsen‑Thurston techniques apply. For coarse‑median preserving automorphisms, the Bestvina–Paulin construction yields actions on ℝ‑trees whose arc stabilisers are genuine centralisers, guaranteeing that point stabilisers are quasi‑convex (by Theorem E). Consequently the “slow” subgroups—those on which φ grows sub‑exponentially—are finitely generated and quasi‑convex, allowing the full decomposition of Theorem B.

Finally, the paper derives several structural properties of the outer automorphism group Out(G) for any virtually special group (Theorem C): Out(G) is boundary amenable (hence satisfies the Novikov conjecture on higher signatures), virtually torsion‑free with finite virtual cohomological dimension, and satisfies the Tits alternative while containing no Baumslag–Solitar subgroups BS(m,n) with |m|≠|n|. These results extend known facts for right‑angled Artin and Coxeter groups and for Gromov‑hyperbolic groups, and they are among the few properties known to hold uniformly for all virtually special groups.

The paper is organized as follows. Section 3 proves the accessibility theorem over centralisers. Section 4 constructs the enhanced JSJ decomposition. Section 5 uses this decomposition to establish the properties of Out(G) listed in Theorem C. Section 6 contains the proof of the basic growth dichotomy and the algebraic nature of stretch factors (Theorem A). Section 7 proves the refined growth classification for coarse‑median preserving automorphisms (Theorem B). Throughout, the authors blend techniques from Bass–Serre theory, median geometry, and the theory of ℝ‑tree actions, providing a comprehensive framework that unifies and extends previous results on automorphisms of free groups, surface groups, and RAAGs.