The classification of finite simple groups is one of the greatest mathematical achievememnts of the 21'st century. In contrast with that, a classification of infinite finitely presented groups is an undecidable algorithmic problem, due to classical results from 1950's. As a result, mathematicians study some special classes of infinite finitely presented groups, and try to classify them up to weaker equivalence relations than isomorphism.
Geometric group theory studies infinite groups by way of viewing them as certain geometric objects. In 1980's M.Gromov proposed to study a vast class of the so called word-hyperbolic groups, whose behaviour as geometric objects resembles that of the hyperbolic plane. Instead of classifying them up to isomorphism, one may try to classify their "behaviour at infinity", encoded in an object called Gromov boundary. Despite more than 30 years of efforts, there are still many quite basic open questions concerning the topological classification of Gromov boundaries of hyperbolic groups.
During the talk I will describe some recent developments concerning this problem. The first consists of a satisfactory description of the topology of the Gromov boundary of a free product of hyperbolic groups with amalgamation along finite groups. The other consists of showing that Gromov boundary of a hyperbolic group is a space belonging to some countable family of spaces called Markov compacta, which are describable in certain algorithmic way out of finite amount of initial data. The third one concerns the construction of a class of hyperbolic groups whose Gromov boundaries are homeomorphic to trees of manifolds (some homogeneous spaces appearing among Markov compacta).