Kleinian Groups in Higher Dimensions

Abstract

Abstract. This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space H n for n ≥ 4. Our main emphasis is on the topological and geometric aspects of higher-dimensional Kleinian groups and their contrast with the discrete groups of isometry of H 3 .To the memory of Sasha Reznikov IntroductionThe goal of this survey is to give an overview (mainly from the topological perspective) of the theory of Kleinian groups in higher dimensions. The survey grew out of a series of lectures I gave in the University of Maryland in the Fall of 1991. An early (much shorter) version of this paper appeared as the preprint [109]. In this survey I collect well-known facts as well as less-known and new results. Hopefully, this will make the survey interesting to both non-experts and experts. We also refer the reader to Tukia's short survey [217] of higher-dimensional Kleinian groups.There is a vast variety of Kleinian groups in higher dimensions: It appears that there is no hope for a comprehensive structure theory similar to the theory of discrete groups of isometries of H 3 . I do not know a good guiding principle for the taxonomy of higher-dimensional Kleinian groups. In this paper the higher-dimensional Kleinian groups are organized according to the topological complexity of their limit sets. In this setting one of the key questions that I will address is the interaction between the geometry and topology of the limit set and the algebraic and topological properties of the Kleinian group. This paper is organized as follows. In Section 2 we consider the most basic concepts of the theory of Kleinian groups, e.g. domain of discontinuity, limit set, geometric finiteness, etc. In Section 3 we discuss various ways to construct Kleinian groups and list the tools of the theory of Kleinian groups in higher dimensions. In Section 4 we review the homological algebra used in the paper. In Section 5 we state topological rigidity results of Farrell and Jones and the coarse compact core theorem for higher-dimensional Kleinian groups. In Section 6 we discuss various notions of equivalence between Kleinian groups: From the weakest (isomorphism) to the strongest (conjugacy). In Section 7 we consider groups with zero-dimensional limit sets; such groups are relatively well-understood. Convex-cocompact groups with 1-dimensional limit sets are discussed in Section 8. Although the topology of the limit sets of such groups is well-understood, their group-theoretic structure is a mystery. We know very little about Kleinian groups with higher-dimensional limit sets, thus we restrict the discussion to Kleinian groups whose limit sets are topological spheres (Section 9). We Date: February 2, 2008. 1 then discuss Ahlfors finiteness theorem and its failure in higher dimensions (Section 10). We then consider the representation varieties of Kleinian groups (Section 11). Lastly we discuss algebraic and topological constraints on Kleinian groups in higher dimensions (Section 12).Acknowledgments. During this work ...

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