Proper actions on finite products of hyperbolic spaces

TL;DR

This paper introduces and characterizes the property (PH) for groups acting on finite products of hyperbolic spaces, extending previous concepts and exploring implications for various classes of groups.

Contribution

It provides a complete characterization of groups with property (PH) and (PH') via lineal actions, and analyzes how these properties behave under extensions and product constructions.

Findings

Characterization of groups with property (PH) and (PH')

Extension of property (PH) through bounded Euler class

Examples including 3-manifold groups and automorphism groups

Abstract

A group $G$ is said to have property (PH') if there exist finitely many hyperbolic spaces $X_1,\cdots,X_n$ on which $G$ acts coboundedly such that the diagonal action of $G$ on the product $\prod_{i=1}^nX_i$ equipped with $\ell^1$-metric is proper. A group $G$ has property (PH) if it virtually has property (PH'). This notion is a generalization of property (QT) introduced by Bestvina-Bromberg-Fujiwara \cite{BBF21}. In this paper, we initiate the study of property (PH) of groups and give a complete characterization of groups with property (PH') or (PH) from lineal actions. In addition, by considering a central extension of groups $1\to Z\to E\to G\to 1$, we prove that $E$ has property (PH) (resp. (QT)) if and only if $G$ has property (PH) (resp. (QT)) and the Euler class of the extension is bounded. We also derive similar results for amalgamated direct products and graph products. As corollaries, we characterize when 3-manifold groups have property (PH) and obtain more interesting examples with property (QT) including the central extension of residually finite hyperbolic groups, the mapping class group of any finite-type surface and the outer automorphism group of torsion-free one-ended hyperbolic groups.