# The fundamental group of semisimple group orbits

It is known that the fundamental group of a compact semisimple Lie group is finite. Can this fact be generalized to compact orbits of semisimple Lie groups? i.e. Let $G$ be a semisimple Lie group such that $G/H$ is compact where $H$ is a closed subgroup. Then is it true that $\pi_1(G/H)$ is finite?

This is an immediate corollary of the fact (Borel) that every semisimple Lie group $$G$$ admits a cocompact lattice (i.e a discrete subgroup $$\Gamma$$ such that $$G/\Gamma$$ is compact). $$\Gamma$$ is closed (it's discrete), and also infinite (unless $$G$$ is compact). Furthermore $$\Gamma = \pi_1(G/\Gamma)$$ since $$\Gamma$$ acts on $$G$$ freely and properly discontinuously.
Yet why not keep it more concrete. Take the hyperbolic plain $$X=\mathbb{H}^2$$. It admits an octogonal regular tilling with degree 4 vertices (a fact which can be found in any textbook on hyperbolic geometry). The surface of genus 2: $$S_2$$ can be obtained by gluing the edges of the regular octagon in a particular way, and thus it inherits a hyperbolic Riemannian metric for which $$X$$ is the universal cover, and each tile is a fundamental domain. The fundamental group $$\Gamma$$ of $$S_2$$ acts freely and properly discontinuously on its universal cover $$X$$ by Deck isometrics. Thus $$\Gamma$$ is a discrete subgroup of the group of isometries of $$X$$ which is just $$SL(2,\mathbb{R})$$.