Relative hyperbolicity and asymptotic invariants of groups

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This project is devoted to the study of relatively hyperbolicgroups and their applications in low--dimensionaltopology. It consists of three main parts. First part deals withsmall cancellation theory over relatively hyperbolic groups.Namely we intend to use methods elaborated by the PI to constructnew groups with 'exotic' properties that solve some long--standingproblems in the classical theory of groups. The second part ismotivated by the Virtual Haken Conjecture and is based on ageneralization of Thurston's hyperbolic Dehn surgery theory in thecontext of relatively hyperbolic groups. The third part concernsnew results about group embeddings. The ultimate goal here is toobtain a relatively hyperbolic version of the Higman embeddingtheorem. Our approach to these problems is based on a new methodsrelated to asymptotic and combinatorial analysis of van Kampendiagrams.The notion of a relatively hyperbolic group was first proposed byGromov in 1987 to generalize many examples of algebraic andgeometric nature. Since then the theory has been elaborated fromdifferent points of views and now it is one of the most prominentparts of the group theory. The class of relatively hyperbolicgroups encompasses many examples of interest such as fundamentalgroups of finite volume hyperbolic manifolds, geometrically finiteconvergence groups, mapping class groups, fully residually freegroups, etc. Rather than just generalizing the theory ofhyperbolic groups to the relative case, this project is aimed atthe study of new and sometimes unexpected phenomena which areinvisible in the context of ordinary hyperbolicity. The closeinteraction between the theory of relatively hyperbolic groups andgeometry, low-dimensional topology, the theory of dynamicalsystems, provides a rich source of problems and motivations forour project.