Vastness properties of automorphism groups of RAAGs

Abstract

Outer automorphism groups of RAAGs, denoted $Out(A_{\Gamma })$, interpolate between $Out(F_n)$ and $G{L}_n(\mathbb{Z})$. We consider several vastness properties for which $Out(F_n)$ behaves very differently from $G{L}_n(\mathbb{Z})$: virtually mapping onto all finite groups, SQ-universality, virtually having an infinite dimensional space of homogeneous quasimorphisms, and not being boundedly generated. We give a neccessary and sufficient condition in terms of the defining graph $\Gamma$ for each of these properties to hold. Notably, the condition for all four properties is the same, meaning $Out(A_{\Gamma })$ will either satisfy all four, or none. In proving this result, we describe conditions on $\Gamma$ that imply $Out(A_{\Gamma })$ is large. Techniques used in this work are then applied to the case of McCool groups, defined as subgroups of $Out(F_n)$ that preserve a given family of conjugacy classes. In particular we show that any McCool group that is not virtually abelian virtually maps onto all finite groups, is SQ-universal, is not boundedly generated, and has a finite index subgroup whose space of homogeneous quasimorphisms is infinite dimensional.

Some simplifications thanks to referee's suggestions. 37 pages, 4 figures

References in corpus (1)

• McCool groups of toral relatively hyperbolic groups

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