# Fundamental group of compact globally symmetric spaces

The fundamental group of a globally symmetric space $$M$$ of compact type is known (see Loos , Borel ). The result can be formulated as follows: it is isomorphic to the quotient $$(*) \quad \pi_1(M) = \Gamma_M/\Gamma_0$$ where $$\Gamma_M$$ is the lattice of $$M$$, given by $$\Gamma_M = \{ \gamma \in \mathfrak{t}: \, \exp_{x_0}(\gamma) = x_0 \}$$ where $$\mathfrak{t}$$ is a maximal flat on the tangent space $$T_{x_0}M$$, and $$\Gamma_0 = \Gamma_{\widetilde{M}}$$, the lattice of the simply connected covering $$\widetilde{M}$$ of $$M$$ (which shares the same $$T_{x_0}M$$ and $$\mathfrak{t}$$ as $$M$$). The isomorphism is induced by the natural inclusion $$\Gamma_M \to \pi_1(M)$$ as homotopy classes of geodesic loops at $$x_0$$, $$\gamma \mapsto [\exp_{x_0}(t\gamma)]$$, $$t \in [0,1]$$.

Does $$(*)$$ holds more generally for a globally symmetric space $$M$$ which is only compact, not necessarily of compact type? The difference being that, in this case, $$\widetilde{M}$$ is not necessarily compact, but a product of simply connected euclidean and compact type symmetric spaces. Furthemore, the (connected component) of the isometry group need not be semisimple, but only compact. Finally, $$\pi_1(M)$$ can be infinite.

Nevertheless, the above constructions can be made for $$M$$ compact and give the correct fundamental group $$\mathbb{Z}$$ for $$M = U(2)$$: loosely speaking, only the semisimple part gets quotiented out.

Does $$(*)$$ holds in general for compact globally symmetric spaces? I could not find a proof in the literature, which only considers this question for compact type.

-Remark: This is loosely related to this question, however the question of identifying the fundamental group as lattice quotients was not raised there.

: Borel, Semisimple Groups and Riemannian Symmetric Spaces, Hindustan Book Agency, 1998.

: Loos, Symmetric spaces, vol.II - Compact spaces and Classification, W. A. Benjamin ,1969.