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.

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.

## Date:

Tue, 21/05/2019 - 16:00 to 17:00

## Location:

Ross 63