#### Volume 7, issue 1 (2003)

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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 $\Gamma$ of automorphisms of the polynomial

$\kappa \left(x,y,z\right)={x}^{2}+{y}^{2}+{z}^{2}-xyz-2$

is isomorphic to

$PGL\left(2,ℤ\right)⋉\left(ℤ∕2\oplus ℤ∕2\right).$

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 $GL\left(2,ℤ\right)$ on ${\left(ℝ∕ℤ\right)}^{2}$ and ${ℝ}^{2}$ respectively. For $2, 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.

##### 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