Volume 5, issue 2 (2005)

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Bootstrapping in convergence groups

Eric L Swenson

Algebraic & Geometric Topology 5 (2005) 751–768

arXiv: math.GR/0508172

Abstract

We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group H which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set ΛH is the closure of a “tree of circles" (adjacent circles meeting in pairs of points). We alter the action of H on its limit set such that H no longer acts as a convergence group, but the stabilizers of the circles remain unchanged, as does the action of a circle stabilizer on said circle. This is done by first separating the circles and then gluing them together backwards.

Keywords
convergence group, bootstrapping, Peano continuum
Mathematical Subject Classification 2000
Primary: 20F34
Secondary: 57N10
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Publication
Received: 16 June 2004
Accepted: 24 June 2005
Published: 23 July 2005
Authors
Eric L Swenson
Mathematics Department
Brigham Young University
Provo UT 84604
USA