Vastness properties of automorphism groups of RAAGs - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of topology Année : 2018

Vastness properties of automorphism groups of RAAGs

Résumé

Outer automorphism groups of RAAGs, denoted $Out(A_\Gamma)$, interpolate between $Out(F_n)$ and $GL_n(\mathbb{Z})$. We consider several vastness properties for which $Out(F_n)$ behaves very differently from $GL_n(\mathbb{Z})$: 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.

Dates et versions

hal-01375997 , version 1 (04-10-2016)

Identifiants

Citer

Vincent Guirardel, Andrew Sale. Vastness properties of automorphism groups of RAAGs. Journal of topology, 2018, 11 (1), pp.30-64. ⟨10.1112/topo.12047⟩. ⟨hal-01375997⟩
173 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More