Hyperbolic structures on groups

TL;DR

This paper introduces and studies the set of hyperbolic structures on groups, exploring their properties, relationships with hyperbolically embedded subgroups, and how they encode group actions on hyperbolic spaces.

Contribution

It initiates the systematic study of the posets of hyperbolic and acylindrically hyperbolic structures on groups, analyzing their properties and how they relate to group actions.

Findings

Characterized the cardinality and extremal elements of hyperbolic structure posets.

Connected hyperbolic structures to hyperbolically embedded subgroups.

Explored how hyperbolic structures are determined by loxodromic elements.

Abstract

For every group $G$, we introduce the set of hyperbolic structures on $G$, denoted $\mathcal{H}(G)$, which consists of equivalence classes of (possibly infinite) generating sets of $G$ such that the corresponding Cayley graph is hyperbolic; two generating sets of $G$ are equivalent if the corresponding word metrics on $G$ are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded $G$-actions on hyperbolic spaces. We are especially interested in the subset $\mathcal{AH}(G)\subseteq \mathcal{H}(G)$ of acylindrically hyperbolic structures on $G$, i.e., hyperbolic structures corresponding to acylindrical actions. Elements of $\mathcal{H}(G)$ can be ordered in a natural way according to the amount of information they provide about the group $G$. The main goal of this paper is to initiate the study of the posets $\mathcal{H}(G)$ and $\mathcal{AH}(G)$ for various groups $G$. We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of $G$, and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.