4
$\begingroup$

Suppose $\Gamma$ is a cocompact lattice of $PSL_2(\mathbb{R})$. Then $\mathbb{H}^2/\Gamma$ has a natural structure of orbifold. My questions are:

  1. What is $\pi_1(\mathbb{H}^2/\Gamma)$?
  2. What is $\pi_1^{orb}(\mathbb{H}^2/\Gamma)$?

I know that if $\Gamma$ is torsion free they both coincide with $\Gamma$.

Improve this question
$\endgroup$
1
  • 5
    $\begingroup$ But the answers are trivial to state: $\pi_1^{orb}(\mathbb{H}^2/\Gamma)\cong\Gamma$, and $\pi_1(\mathbb{H}^2/\Gamma)$ is naturally isomorphic to the quotient of $\Gamma$ by the normal subgroup generated by its torsion elements. $\endgroup$
    – HJRW
    Commented Oct 14, 2021 at 11:32

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .