Divisible cube complexes and finite-order automorphisms of RAAGs (Q6597540)

Let \(G\) be a group and \(\varphi \in \mathrm{Aut}(G)\), a recurring theme in group theory is to derive information about the subgroup \(\mathrm{Fix}(\varphi)=\{ g \in G \mid g^{\varphi}=g \}\) by knowing \(\varphi\) (and about \(G\) by knowing \(\mathrm{Fix}(\varphi)\) and \(\varphi\)).\N\NIn the paper under review the author gives a geometric characterisation of those groups that arise as fixed subgroups of finite-order untwisted automorphisms of right-angled Artin groups (\(\mathsf{RAAGs}\)). The main result is Theorem A: The following properties are equivalent for a group \(H\): (1) \(H\) is the fundamental group of a divisible, compact, special cube complex; (2) there exist a right-angled Artin group \(A_{\Gamma}\) and a finite-order, pure, untwisted outer automorphism \([\varphi] \in \mathrm{Out}(A_{\Gamma})\) such that \(H \simeq \mathrm{Fix}(\varphi)\), for some representative \(\varphi\) of \([\varphi]\).\N\NAn important corollary of Theorem A is that surface groups arise as fixed subgroups of finite order automorphisms of \(\mathsf{RAAGs}\), as do all commutator subgroups of right-angled Coxeter groups. Such fixed subgroups are the earliest known examples that are not themselves isomorphic to \(\mathsf{RAAGs}\).\N\NUsing a variation of canonical completions, the author also shows that every special group arises as the fixed subgroup of an automorphism of a finite-index subgroup of a \(\mathsf{RAAG}\).