Hyperbolicity for semigroups

Hyperbolic groups can be characterized in several equivalent ways, including: (geometric) hyperbolicity of their Cayley graphs, having linear Dehn function, having finite Dehn presentations, and Gilman's characterization using context-free languages. This talk will survey how these characterizations can be used to generalize the concept of hyperbolicity to semigroups, with particular attention to the geometric and language-theoretic generalizations. These generalizations are not equivalent for semigroups, but there are certain nice classes of semigroups that are hyperbolic in every sense. The talk will conclude by discussing decision problems for semigroups that are hyperbolic in the linguistic sense, and how the word problem can be solved in polynomial time, improving on the previously-known exponential-time algorithm.