Weak asymptotic hereditary asphericity for free product and HNN extension of groups

Abstract

Asymptotic hereditary asphericity (AHA) is a coarse property of metric spaces and groups, introduced by T Januszkiewicz and J Swicatkowski in [Geom. Topol. 11 (2007) 727–758]. Conjecturally, this property is closed under amalgamated free products and HNN extensions over finite subgroups. We prove this conjecture for a slightly weaker property, weak asymptotic hereditary asphericity (AHA(-)), which is still strong enough for the purposes which AHA was used for in [Geom. Topol. 11 (2007) 727–758].