Hyperbolic groupoids and duality

TL;DR

This paper introduces hyperbolic groupoids, extending Gromov hyperbolic groups, and establishes a duality theory connecting a hyperbolic groupoid with its boundary action, enriching the understanding of hyperbolic structures.

Contribution

It defines hyperbolic groupoids and proves a duality theorem linking each hyperbolic groupoid to a boundary action, generalizing known hyperbolic group concepts.

Findings

Hyperbolic groupoids include actions on boundaries and foliations.

Existence of a dual hyperbolic groupoid acting on the Gromov boundary.

Duality is involutive, with (G')' equivalent to G.

Abstract

We define hyperbolic groupoids, generalizing the notion of a Gromov hyperbolic group. Examples of hyperbolic groupoids include actions of Gromov hyperbolic groups on their boundaries, pseudogroups generated by expanding self-coverings, natural pseudogroups acting on leaves of stable (or unstable) foliation of an Anosov diffeomorphism, e.t.c.. We show that for every hyperbolic groupoid G there is a naturally defined dual groupoid G' acting on the Gromov boundary of a Cayley graph of G, which is also hyperbolic and such that (G')' is equivalent to G.