I'm interested in the 3-dimensional model geometries; specifically $\widetilde{SL}(2,\mathbb{R})$. I'm looking for a good (see, easily visualizable) example of a compact manifold formed as a quotient of a group acting on $\widetilde{SL}(2,\mathbb{R})$.
I'm trying to show that the unit tangent bundle (i.e., the unit vectors on the tangent bundle, so a circle bundle) on the closed 2-dimensional genus 2 surface is such a manifold. I call this manifold $UF_2$. In order to do that, I think I need to show that $\pi_1(UF_2)=\left<a,b,c,d,s\big|[a,b][c,d]=s^k\right>$ where $[a,b]=aba^{-1}b^{-1}$ and $k\in\mathbb{Z}$ and $k\not=0$.
I am using Van Kampen's theorem to calculate $\pi_1(UF_2)$ and I finally got a result. I think that $\pi_1(UF_2)=\left<a,b,c,d,s\big|[a,b][c,d]=s^2\right>$, but I'm not that confident in the result.
My application of Van Kampen's theorem involves splitting $UF_2$ into a unit tangent bundle on 2 different punctured tori. But really, I did a lot of hand waving. I couldn't even write down how I arrived at the result $k=2$. So I'm asking:
1 Can anyone confirm, using any method, that $\pi_1(UF_2)=\left<a,b,c,d,s\big|[a,b][c,d]=s^2\right>$?
2 Can anyone confirm that $UF_2$ is a quotient of $\widetilde{SL}(2,\mathbb{R})$ by a isometry group?
3 Can anyone write down an explicit method for calculating $\pi_1(UF_2)$?
Thank you all
Any closed surface $\Sigma$ of genus $g \ge 2$ admits a hyperbolic metric, and hence is a quotient $\mathbb{H}/\Gamma$ of the upper half plane $\mathbb{H}$ by a discrete group $\Gamma$ of isometries acting on $\mathbb{H}$ which can be identified, as an abstract group, with the fundamental group $\pi_1(\Sigma)$.
The isometry group of $\mathbb{H}$ is $PSL_2(\mathbb{R})$, so $\Gamma$ is naturally a discrete subgroup of $PSL_2(\mathbb{R})$. This full isometry group acts freely and transitively on the unit tangent bundle of $\mathbb{H}$, and so the unit tangent bundle of $\mathbb{H}$ can itself be identified with $PSL_2(\mathbb{R})$. Moreover, this identification identifies the unit tangent bundle of $\Sigma$ with the quotient $PSL_2(\mathbb{R})/\Gamma$, which of course can also be written as a quotient of $\widetilde{SL}_2(\mathbb{R})$.
Using this description, although other approaches are also possible, it follows that the fundamental group of the unit tangent bundle is a certain extension
$$1 \to \mathbb{Z} \to \pi_1(UT(\Sigma)) \to \pi_1(\Sigma)) \to 1$$
of $\pi_1(\Sigma) \cong \Gamma$ by $\mathbb{Z}$, coming from the action of $\Gamma$ on the fundamental groupoid of $PSL_2(\mathbb{R})$.