Acylindrical actions on projection complexes

TL;DR

This paper simplifies the construction of projection complexes, introduces a stronger Behrstock inequality, and establishes acylindricity results for actions on these complexes and related quasi-trees.

Contribution

It provides a streamlined construction of projection complexes, a sharper inequality, and new acylindricity results for their actions and associated quasi-trees.

Findings

Simplified construction of projection complexes.

Established a sharper Behrstock inequality.

Proved acylindricity of actions on projection complexes and related quasi-trees.

Abstract

We simplify the construction of projection complexes due to Bestvina-Bromberg-Fujiwara. To do so, we introduce a sharper version of the Behrstock inequality, and show that it can always be enforced. Furthermore, we use the new setup to prove acylindricity results for the action on the projection complexes. We also treat quasi-trees of metric spaces associated to projection complexes, and prove an acylindricity criterion in that context as well.