Vastness properties of automorphism groups of RAAGs. Issue 1 (19th January 2018)

• Record Type: Journal Article
• Authors: Guirardel, Vincent; Sale, Andrew
• Is Part Of: Journal of topology. Volume 11:Issue 1(2018)
• Page Start: 30
• Page End: 64
• Publication Date: 2018-01-19
• Subjects: 20F65 (primary); Topology -- Periodicals; 514.05
• DOI: 10.1112/topo.12047
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract: Outer automorphism groups of right‐angled Artin groups, denoted Out ( A Γ ), interpolate between Out ( F n ) and GL n ( Z ) . We consider several vastness properties for which Out ( F n ) behaves very differently from GL n ( Z ) : virtually mapping onto all finite groups, SQ‐universality, virtually having an infinite dimensional space of homogeneous quasi‐morphisms, and not being boundedly generated. We give a necessary and sufficient condition in terms of the defining graph Γ for each of these properties to hold. Notably, the condition for all four properties is the same, meaning Out ( A Γ ) will either satisfy all four or none. In proving this result, we describe conditions on Γ that imply Out ( A Γ ) 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 quasi‐morphisms is infinite dimensional.