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• W1671385284 abstract "In this article we survey and describe various aspects of the geometry and arithmetic of Kleinian groups - discrete nonelementary groups of isometries of hyperbolic 3-space. In particular we make a detailed study of two-generator groups and discuss the classication of the arithmetic generalised triangle groups (and their near relatives). This work is mainly based around my collaborations over the last two decades with Fred Gehring and Colin Maclachlan, both of whom passed away in 2012. There are many others involved as well. Over the last few decades the theory of Kleinian groups has ourished because of its intimate connections with low dimensional topology and geometry. We give little of the general theory and its connections with 3-manifold theory here, but focus on two main problems: Siegel's problem of identifying the minimal covolume hyperbolic lattice and the Margulis constant problem. These are both universal constraints on Kleinian groups { a feature of discrete isometry groups in negative curvature and include results such as Jrgensen's inequality, the higher dimensional version of Hurwitz's 84g 84 theorem and a number of other things. We will see that big part of the work necessary to obtain these results is in getting concrete descriptions of various analytic spaces of two-generator Kleinian groups, somewhat akin to the Riley slice." @default.
• W1671385284 created "2016-06-24" @default.
• W1671385284 creator A5048251698 @default.
• W1671385284 creator A5081252109 @default.
• W1671385284 date "2013-11-11" @default.
• W1671385284 modified "2023-09-22" @default.
• W1671385284 title "The Geometry and Arithmetic of Kleinian Groups" @default.
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