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

Goldman, William M¹

¹ Mathematics Department, University of Maryland, College Park, Maryland 20742, USA

Geometry & topology, Tome 7 (2003) no. 1, pp. 443-486.

Résumé

The group $\Gamma$ of automorphisms of the polynomial

is isomorphic to

For $t\in ℝ$, the $\Gamma$–action on ${\kappa }^{-1}\left(t\right)\cap ℝ$ displays rich and varied dynamics. The action of $\Gamma$ preserves a Poisson structure defining a $\Gamma$–invariant area form on each ${\kappa }^{-1}\left(t\right)\cap ℝ$. For $t<2$, the action of $\Gamma$ is properly discontinuous on the four contractible components of ${\kappa }^{-1}\left(t\right)\cap ℝ$ 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 $\overline{M}$. For $t=2$, the level set ${\kappa }^{-1}\left(t\right)\cap ℝ$ consists of characters of reducible representations and comprises two ergodic components corresponding to actions of $GL\left(2,\mathbb{Z}\right)$ on ${\left(ℝ∕\mathbb{Z}\right)}^2$ and ${ℝ}^2$ respectively. For $2<t\le 18$, the action of $\Gamma$ on ${\kappa }^{-1}\left(t\right)\cap ℝ$ 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 ${\kappa }^{-1}\left(t\right)\cap ℝ$ intersects $\Omega$ if and only if $t>18$, in which case the $\Gamma$–action on the complement $\left({\kappa }^{-1}\left(t\right)\cap ℝ\right)-\Omega$ is ergodic.

Détail

DOI : 10.2140/gt.2003.7.443

Keywords: surface, fundamental group, character variety, representation variety, mapping class group, ergodic action, proper action, hyperbolic structure with cone singularity, Fricke space, Teichmüller space

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