Web2.2. Discrete and Properly Discontinuous Groups 4 2.3. Fixed Points and Abelian Groups 7 3. Fundamental Regions 8 3.1. The Dirichlet Region 9 Conclusions 10 Acknowledgments 10 References 10 Introduction When considering the geometry of the hyperbolic plane, the most important group that arises is the group of orientation preserving isometries ... WebJun 5, 2024 · These are non-elementary finitely-generated Kleinian groups whose discontinuity sets are simply-connected domains. There is an extremely-complicated proof of the existence of such groups; meanwhile no …
Math 396. Quotients by group actions - Stanford …
WebXis properly discontinuous if for every two compact subsets Aand Bof Xthe number of automorphisms ˚2Gsuch that ˚(A) intersects Bis non-empty is nite. Theorem 18.3. Let X be a Riemann surface. Let GˆAut(X) be a subgroup of the automorphism group of X, whose action on X is properly discontinuous. Then the quotient topological space f: X! Let be a group acting on a set . The action is called faithful or effective if for all implies that . Equivalently, the morphism from to the group of bijections of corresponding to the action is injective. The action is called free (or semiregular or fixed-point free) if the statement that for some already implies that . In other words, no non-trivial element of fixes a point of . This is a much stronger pro… trafopower ag
Proper discontinuity and existence of a fundamental domain
WebOct 8, 2016 · Suppose that G × X → X is a properly discontinuous action of a countable group such that the fixed-point set of each nontrivial element of G is nowhere dense in X. Then this action admits a fundamental domain (obtained by the Dirichlet construction for a suitable G -invariant geodesic metric on X ). See Webis properly discontinuous, and that it isuniformif this associated action is cocompact; see e.g. [11], [25], and [3]. Bowditch has shown that uniform convergence groups provide an alternative characterization of word-hyperbolicity for a group [4]. WebSelect search scope, currently: articles+ all catalog, articles, website, & more in one search; catalog books, media & more in the Stanford Libraries' collections; articles+ journal articles & other e-resources trafo onaf