Groups and Dynamics Seminar

Hyperbolic groupoids were developed by V. Nekrashevych as a way to study hyperbolic dynamical systems from an algebraic perspective, linking dynamic topological structures with algebraic notions. Though hyperbolic groupoids shed new light on such systems, they do not provide a complete algebraic characterization of hyperbolic systems as their definition relies upon metric structures of the groupoid unit space. However, when the groupoid is ample, the underlying space is totally disconnected and its metric structures can be encoded by a finite amount of combinatorial information. We introduce a new class of algebraic objects called hyperbolic inverse semigroups which we prove are mutually dual to hyperbolic ample groupoids, showing that hyperbolic systems on compact totally disconnected spaces can be realized as purely algebraic objects. This connection is analogous to how a Markov partition provides a topological conjugacy between a hyperbolic dynamical system and a shift of finite type exactly when the space is totally disconnected, except hyperbolic inverse semigroups generalize this to asynchronous hyperbolic presentations and provide algebraic tools for computation.