Dynamics of the mapping class group action on the variety of PSL2ℂ characters

Souto, Juan; Storm, Peter

doi:10.2140/gt.2006.10.715

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Dynamics of the mapping class group action on the variety of $\mathrm{PSL}_2 \mathbb{C}$ characters

Juan Souto and Peter Storm

Geometry & Topology 10 (2006) 715–736

DOI: 10.2140/gt.2006.10.715

arXiv: math.GT/0504474

Abstract

We study the action of the mapping class group $Mod (S)$ on the boundary $\partial Q$ of quasifuchsian space $Q$ . Among other results, $Mod (S)$ is shown to be topologically transitive on the subset $C \subset \partial Q$ of manifolds without a conformally compact end. We also prove that any open subset of the character variety $X (π_{1} (S), {SL}_{2} ℂ)$ intersecting $\partial Q$ does not admit a nonconstant $Mod (S)$ –invariant meromorphic function. This is related to a question of Goldman.

Keywords

hyperbolic geometry, mapping class group

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 58D27

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Publication

Received: 14 January 2006

Accepted: 30 April 2006

Published: 11 July 2006

Proposed: Benson Farb

Seconded: Walter Neumann, Jean-Pierre Otal

Authors

Juan Souto
Department of Mathematics University of Chicago 5734 University Avenue Chicago IL 60637-1514 USA

Peter Storm
Department of Mathematics Stanford University 450 Serra Mall Stanford CA 94305-2125 USA

Volume 10, issue 2 (2006)