Bootstrapping in convergence groups

Swenson, Eric L

doi:10.2140/agt.2005.5.751

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Bootstrapping in convergence groups

Eric L Swenson

Algebraic & Geometric Topology 5 (2005) 751–768

DOI: 10.2140/agt.2005.5.751

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 $Λ H$ 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

Volume 5, issue 2 (2005)