Infinite groups acting faithfully on the outer automorphism group of a right-angled Artin group

Abstract

We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A Γ ), for certain right-angled Artin groups A Γ . This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A Γ )). This demonstrates a marked departure from the known behavior of Out(Out(A Γ )) when A Γ is free or free abelian, as in these cases Out(Out(A Γ )) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite order members of certain Out(Aut(A Γ )).

Between buildings and free factor complexes: A Cohen–Macaulay complex for Out(RAAGs)

“…It seems that our definition of parabolic subgroups and the corresponding coset complex capture well the aspects of Out(𝐴 Γ ) that come from similarities of this group with GL 𝑛 (ℤ) and Out(𝐹 𝑛 ): Firstly, our definitions recover the Tits building as CC(GL 𝑛 (ℤ), (GL 𝑛 (ℤ))) and the free factor graphs that appear in the work of Bregman and Fullarton [7], if the standard ordering on the graph Γ is trivial. In that case, Out 0 (𝐴 Γ ) is a semi-direct product of a free abelian group generated by partial conjugations and the (finite) group of inversions.…”

“…In particular, if $$O$$ does not contain any transvection, the complex $$\mathcal {CC}$$ is trivial, while this need not be the case for the RAAG Outer space and its boundary. This, for example, occurs for RAAGs defined by focused graphs that appear in the work of Bregman and Fullarton [7], if the standard ordering on the graph $$\Gamma$$ is trivial. In that case, $$\operatorname{Out}^0(A_\Gamma )$$ is a semi‐direct product of a free abelian group generated by partial conjugations and the (finite) group of inversions.…”

Subspace arrangements, BNS invariants, and pure symmetric outer automorphisms of right-angled Artin groups

“…As well as the above virtually free examples, there are examples of RAAGs whose outer automorphism groups are finite [7,9] or virtually free abelian [3]. There are also some more interesting examples; for instance Out(F 2 × F 2 ) is commensurable with F 2 × F 2 itself.…”

Relative automorphism groups of right‐angled Artin groups

“…There is a popular mantra that as RAAGs interpolate between free and free‐abelian groups, their outer automorphism groups should interpolate between and . Putting this idea into practice is harder: for example, in many cases is a finite group and there are examples where is infinite but virtually abelian . However, there are common properties shared by as varies over all graphs.…”