Mathematical Sciences: Limit Sets of Kleinian Groups and Hyperbolic Groups

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Bonahon will study limit sets of Kleinian groups and of hyperbolic groups. A Kleinian group is a discrete group of isometries of hyperbolic 3-space. Bonahon intends to study the topological type of the limit set of a Kleinian group and show that it is determined by the algebraic structure of the group together with certain geometric invariants of the associated quotient space. He also wants to study the asymptotic behavior of the leaves of foliations of hyperbolic 3-space which are invariant under a Kleinian group (namely lifts of foliations of hyperbolic 3-manifolds) and to analyze how fast these leaves approach the limit set of the group. Following Gromov's terminology, a hyperbolic group is a finitely generated group which has certain growth properties similar to those of a cocompact Kleinian group. Such a hyperbolic group has a well defined limit set, also called its boundary. Bonahon wants to analyze how certain topological properties of the limit set of a hyperbolic group are related to algebraic properties of the group, involving in particular its outer automorphism group and decompositions into amalgamated products. Advances in the study of 3-dimensional manifolds in recent years, following work of Thurston, has rather surprisingly shown the extreme relevance of hyperbolic geometry to the understanding of the structure of 3-manifolds.