Chaos and integrability in $\operatorname{SL}(2,\mathbb R)$-geometry

We review the integrability of the geodesic flow on a threefold $\mathcal M^3$ admitting one of the three group geometries in Thurston's sense. We focus on the $\operatorname{SL}(2,\mathbb R)$ case. The main examples are the quotients $\mathcal M^3_\Gamma=\Gamma\backslash \operatorname{PSL}(2,\mathbb R)$, where $\Gamma \subset \operatorname{PSL}(2,\mathbb R)$ is a cofinite Fuchsian group. We show that the corresponding phase space $T^*\mathcal M_\Gamma^3$ contains two open regions with integrable and chaotic behaviour, with zero and positive topological entropy, respectively.
As a concrete example we consider the case of the modular threefold with the modular group $\Gamma=\operatorname{PSL}(2,\mathbb Z)$. In this case $\mathcal M^3_\Gamma$ is known to be homeomorphic to the complement of a trefoil knot $\mathcal K$ in a 3-sphere. Ghys proved the remarkable fact that the lift of a periodic geodesic on the modular surface to $\mathcal M^3_\Gamma$ produces the same isotopy class of knots as that which appears in the chaotic version of the celebrated Lorenz system and was studied in detail by Birman and Williams. We show that these knots are replaced by trefoil knot cables in the integrable limit of the geodesic system on $\mathcal M^3_\Gamma$.
Bibliography: 60 titles.