Generalisations of hyperbolic groups and their properties

The notion of hyperbolic group was introduced in 1987 by Gromov, with inspiration from hyperbolic geometry and low dimensional topology and combinatorial group theory. The notion of a hyperbolic group has become central in geometric group theory with many properties being satisfied by hyperbolic groups. This has led to some exciting results in small cancellation theory as well as in density models for random groups.

In recent years, a lot of generalisations of hyperbolic groups have been studied. Such generalisations lead to the notion of relative hyperbolic groups, acylindrically hyperbolic groups or CAT(0) groups for example. It is a well known open problem to find a hyperbolic group that is not a CAT(0) group.

A lot of work has been done on acylindrically hyperbolic groups, where the action of the group is a weakening of the proper discontinuity of the action on a Gromov hyperbolic group. This class of groups includes hyperbolic groups, relative hyperbolic groups as well as Out(Fn) and the Mapping class group via its action on curve complexes. Acylindrically hyperbolic groups have several interesting properties. For example, they are SQ-universal meaning every countable group can be embedded in one of its quotient groups. Another interesting property is one proved by Dr. Sisto : for a finitely generated acylindrically hyperbolic group G, the probability that a simple random walk on G of length n produces a 'generalized loxodromic' element converges to 1 exponentially fast as n tends to infinity.

My PhD research will be looking at such, and other, generalisations of hyperbolic groups and what properties these classes of groups have.