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Properties of convergence groups and spaces. (English) Zbl 0983.57029
Let \(G\) be a group of homeomorphisms of a compact Hausdorff space \(X\). Then \(G\) is a (discrete) convergence group if for any infinite family of distinct \(g_m\in G\) there are points \(x,y\in X\) and a (sub)sequence \(g_j\) such that \(g_j(z)\to x\) [resp. \(g_j^{-1}(z)\to y]\), locally uniformly on \(X-y\) [resp. \(X-x]\). This concept, introduced by Gehring and Martin and further investigated by Tukia, imitates a crucial property of groups of Möbius transformations of spheres, and it turns out that many of the properties of Möbius groups do not depend on an analytic structure but rather on such a dynamic topological condition (first of all the distinction into elliptic, parabolic and loxodromic elements). “The aim of this paper is to illustrate the considerable restrictions that accompany the convergence action of a (nonelementary) group \(G\) on a compact Hausdorff space \(X\). Under mild hypotheses, the limit set of \(G\) must be metrizable of the cardinality of the reals. There is some control on the type and number of normal subgroups. Endomorphisms of \(G\) with finite kernel are automorphisms. There is a class of finitely generated convergence groups with solvable word problem.” Various examples are discussed, among them a convergence action on a non-metrizable space. The triple space is discussed and used to prove various topological properties of the limit set.

MSC:
57S30 Discontinuous groups of transformations
20F65 Geometric group theory
20F38 Other groups related to topology or analysis
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