Hyperbolic groupoids: definitions and duality

Details Hyperbolic groupoids: definitions and duality Hyperbolic groupoids: definitions and duality

2011-01-28

arxiv.org/abs/1101.5603v2

Mathematics

Dynamical Systems

68 pages, 9 figures

Scientific paper

We define a notion of a hyperbolic groupoid (pseudogroup) generalizing actions of Gromov hyperbolic groups on their boundaries. We show that the boundary of a Gromov hyperbolic groupoid has a natural local product structure and that actions of hyperbolic groupoids on their boundaries can be described axiomatically as generalized Smale spaces (which we call Smale quasi-flows). The original groupoid is equivalent to the projection of the corresponding Smale quasi-flow onto the stable direction of the local product structure. The projection onto the unstable direction is called the dual of the groupoid. Examples of pairs of mutually dual hyperbolic groupoids and associated Smale quasi-flows are described.

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