Statistics and growth in hyperbolic groups
Stephen Cantrell · Warwick Research Archive Portal (University of Warwick) · 2020
The work presented in this thesis is concerned with quantifying, in various different senses, how natural quantities associated to hyperbolic groups grow and distribute. Hyperbolic groups were introduced by Gromov in his seminal work [29] and are a fundamental object of study in geometric group theory. Given a group G with generating set S, the word metric (with respect to S) assigns to a group element g ∈ G its word length |g|, i.e. the length of the shortest word(s) that express g, with letters in S ∪ S −1 . A hyperbolic group is a finitely generated group that, when equipped with the word metric for any finite generating set, satisfies an abstract geometrical condition that mimics a property of the hyperbolic plane. That is, geodesic triangles in the Cayley graph of G are ‘thin’. This condition, although natural, seems at first to be somewhat superficial, yet the theory of hyperbolic groups is deep and interesting. For example, hyperbolic groups exhibit strong combinatorial properties and in particular have a solvable word problem: there exists an algorithm that decides whether two words (with letters in a fixed generating set) express the same group element. Furthermore, by the work of Cannon and Ghys and de le Harpe, hyperbolic groups are strongly Markov. That is, given a hyperbolic group G equipped with a finite generating set S, there exists a finite directed graph G that in some sense encodes the properties of G and S. Cannon proved that cocompact Kleinian groups are strongly Markov [10] and Ghys and de la Harpe showed that Cannon’s approach worked for all hyperbolic groups [25].