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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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