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

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

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

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 $GL\left(2,\mathbb{Z}\right)$ 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.