Geometry & Topology, Vol. 7 (2003) Paper no. 13, pages 443--486.

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

William M Goldman

Abstract. The group Gamma of automorphisms of the polynomial kappa(x,y,z) = x^2 + y^2 + z^2 - xyz -2 is isomorphic to PGL(2,Z) semi-direct product with (Z/2+Z/2). For t in R, Gamma-action on ktR = kappa^{-1}(t) intersect R displays rich and varied dynamics. The action of Gamma preserves a Poisson structure defining a Gamma-invariant area form on each ktR. For t < 2, the action of Gamma is properly discontinuous on the four contractible components of ktR and ergodic on the compact component (which is empty if t < -2). The contractible components correspond to Teichmueller spaces of (possibly singular) hyperbolic structures on a torus M-bar. For t = 2, the level set ktR consists of characters of reducible representations and comprises two ergodic components corresponding to actions of GL(2,Z) on (R/Z)^2 and R^2 respectively. For 2 < t <= 18, the action of Gamma on ktR is ergodic. Corresponding to the Fricke space of a three-holed sphere is a Gamma-invariant open subset Omega subset R^3 whose components are permuted freely by a subgroup of index 6 in Gamma. The level set ktR intersects Omega if and only if t > 18, in which case the Gamma-action on the complement ktR - Omega is ergodic.

Keywords. Surface, fundamental group, character variety, representation variety, mapping class group, ergodic action, proper action, hyperbolic structure with cone singularity, Fricke space, Teichmueller space

AMS subject classification. Primary: 57M05. Secondary: 20H10, 30F60.

DOI: 10.2140/gt.2003.7.443

E-print: arXiv:math.DG/0305096

Submitted to GT on 19 August 2001. (Revised 7 June 2003.) Paper accepted 10 July 2003. Paper published 18 July 2003.
Republished 21 August 2003 (the 18 July version did not take account of the referee's comments and was published in error).

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William M Goldman
Mathematics Department, University of Maryland
College Park, MD 20742 USA

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