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Bootstrapping in convergence groups. (English) Zbl 1083.20039
The purpose of the paper is to present conditions under which a group $$G$$ of homeomorphisms of a compact, connected and locally connected metric space $$X$$ (a Peano continuum, also assumed to be without cut points) acts as a convergence group on $$X$$, by reducing the problem to subgroups which are known to act as convergence groups on smaller sets. More precisely, given a $$G$$-invariant collection of closed subsets of $$X$$ whose stabilizers in $$G$$ act as convergence groups on these sets then, under certain additional conditions, $$G$$ acts as a convergence group on $$X$$. For example, the main result applies when $$X$$ is an $$n$$-sphere and the subgroups have limit sets which are $$(n-1)$$-spheres. On the other hand, an example is given which shows that such a result cannot be expected in general.
MSC:
 20F65 Geometric group theory 57M07 Topological methods in group theory 57N10 Topology of general $$3$$-manifolds (MSC2010) 57S30 Discontinuous groups of transformations
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