The Holonomy of a hyperbolic surface S in terms of differential geometry is either $SO(2)$ or $O(2)$ depending on Orientability. And a hyperbolic structure as a special (X,G)-structure: $\pi_1(S)⊂PSL(2,R)$. (An $(X,G)$ structure could be regarded as a flat X-bundle with a section transversal to the flat connection, so holonomy of the flat bundle is the holonomy of the structure.)
So there are two ways of describing the Holonomy of a Hyperbolic structure. But, are they both equally as valid in all cases? And what is the relationship between SO(2) and PSL(2,R)?
$SO(2)$ is the maximal compact subgroup of $PSL(2, \mathbb{R}).$