#### 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).

Notes on file formats
William M Goldman

Mathematics Department, University of Maryland

College Park, MD 20742 USA

Email: wmg@math.umd.edu

GT home page

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