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The modular group action on real $SL(2)$–characters of a one-holed torus

William M Goldman

Geometry & Topology 7 (2003) 443–486

arXiv: math.DG/0305096

Abstract

The group Γ of automorphisms of the polynomial

κ(x,y,z) = x2 + y2 + z2 xyz 2

is isomorphic to

PGL(2, ) (2 2).

For t , the Γ–action on κ1(t) displays rich and varied dynamics. The action of Γ preserves a Poisson structure defining a Γ–invariant area form on each κ1(t) . For t < 2, the action of Γ is properly discontinuous on the four contractible components of κ1(t) and ergodic on the compact component (which is empty if t < 2). The contractible components correspond to Teichmüller spaces of (possibly singular) hyperbolic structures on a torus M¯. For t = 2, the level set κ1(t) consists of characters of reducible representations and comprises two ergodic components corresponding to actions of GL(2, ) on ()2 and 2 respectively. For 2 < t 18, the action of Γ on κ1(t) is ergodic. Corresponding to the Fricke space of a three-holed sphere is a Γ–invariant open subset Ω 3 whose components are permuted freely by a subgroup of index 6 in Γ. The level set κ1(t) intersects Ω if and only if t > 18, in which case the Γ–action on the complement (κ1(t) ) Ω is ergodic.

Keywords
surface, fundamental group, character variety, representation variety, mapping class group, ergodic action, proper action, hyperbolic structure with cone singularity, Fricke space, Teichmüller space
Mathematical Subject Classification 2000
Primary: 57M05
Secondary: 20H10, 30F60
References
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Publication
Received: 19 August 2001
Revised: 7 June 2003
Accepted: 10 July 2003
Published: 18 July 2003
Proposed: Walter Neumann
Seconded: Benson Farb, Martin Bridson
Authors
William M Goldman
Mathematics Department
University of Maryland
College Park
Maryland 20742
USA