Combinatorial group theory began with Dehn's study of surface groups, where he used arguments from hyperbolic geometry to solve the word/conjugacy problems. In 1984, Cannon generalized those ideas to all "hyperbolic groups", where he was able to give a solution to the word/conjugacy problem, and to show that their growth function satisfies a finite linear recursion. The key observation that led to his discoveries is that the global geometry of a hyperbolic group is determined locally: first, one discovers the local picture of G, then the recursive structure of G by means of which copies of the local structure are integrated. The talk will be about our result with Eike generalizing Cannons result to hyperbolic-like geodesics in any f.g group (and hence recovering Cannon's result). This will have the following consequences: 1) a finite linear recursion (and hence a closed form ) to the growth of hyperbolic-like geodesics in any f.g group, 2) Using work of Bestvina, Osin and Sisto, our result imply that any f.g group containing a "contracting geodesic" must be acylindrically hyperbolic.
Tue, 21/05/2019 - 16:00 to 17:00