BibTeX
@MISC{Maskit74classificationof,
author = {Bernard Maskit},
title = {Classification of Kleinian groups},
year = {1974}
}
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Abstract
We present here a complete classification of those Kleinian groups which have an invariant region of discontinuity and which, in their action on hyperbolic 3-space, have a finite-sided fundamental polyhedron. This classification is complete in the same sense that finitely-generated Fuchsian groups of the first kind are completely classified, i.e., there is a countable collection of topologically distinct classes, each such class can be described by a finite set of numbers called the signature; all the groups belonging to any one topological class appear (infinitely often) in the deformation space (defined using quasi-conformal mappings) of any one group in the class; this deformation space can be parametrized as a complex manifold. Our results can also be regarded as a classification of all uniformizations of any finite Riemann surface (i.e., a closed Riemann surface from which a finite number of points have been deleted), where the uniformizing group has a finite-sided fundamental polyhedron (see §9). The proofs of the theorems are based on the combination theorems [12], the planarity theorem [13], Bers' technique of variation of parameters using quasi-conformal mappings [4], and Marden's isomorphism theorem [11]. Details will appear elsewhere. 1. A Kleinian group is a discrete subgroup of PSL(2; C) which acts discontinuously at some point of C = C U {oo}- The set of points at which G acts dis-continuously is denoted by Û = 0{G); its complement, the limit set, is denoted by A = A{G). The components of 0 are called components of G. G is a function group if there is a component A which is kept invariant by G.
Keyphrases
kleinian group deformation space quasi-conformal mapping invariant region finite-sided funda-mental polyhedron complex manifold discrete subgroup function group planarity theorem isomorphism theorem first kind finite-sided fundamental polyhedron finite set limit set topological class appear uniformizing group complete classification closed riemann surface finitely-generated fuchsian group distinct class finite number countable collection finite riemann surface
