Hyperbolic quotients of projection complexes

TL;DR

This paper explores the hyperbolic properties of quotients of projection complexes resulting from group actions, demonstrating conditions under which these quotients are hyperbolic and retain non-elementary WPD actions, leading to acylindrical hyperbolicity.

Contribution

It establishes new conditions for the hyperbolicity of quotient complexes and shows that certain group actions are preserved under these quotients, advancing understanding of group actions on hyperbolic spaces.

Findings

Quotient complexes are δ-hyperbolic under certain conditions.

Non-elementary WPD actions are preserved in quotients.

Quotient groups are shown to be acylindrically hyperbolic.

Abstract

This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions that the quotient complex is $\delta$-hyperbolic. Additionally, under certain circumstances, we show that if the original action on the projection complex was a non-elementary WPD action, then so is the action of the quotient group on the quotient of the projection complex. This implies that the quotient group is acylindrically hyperbolic.