Why does hyperbolic universe must be infinite?

Question

I remember back in the day, I took a course in cosmology where I was taught that there are three possible types of universes based on their curvature, the flat universe which may be compact or infinite, the spherical which is compact and the hyperbolic which is infinite.
I don't understand why a hyperbolic universe must be infinite, I know many manifolds which admit a hyperbolic structure and are compact, in fact most manifolds are hyperbolic, an example is, all the surfaces of genus greater than or equal to 2.
Now I am not a physicist and don't have good physics intuition, so I want to ask why does hyperbolic universe means infinite universe?
Have I misunderstood something?

Eletie · Answer

You're correct that in general a hyperbolic geometry does not imply the manifold is infinite: it could have some non-trivial topology and be compact (e.g. 3-torus). In FLRW cosmology we usually assume$^1$ the topology is simply-connected, in which case negative curvature does imply an infinite universe, but of course this isn't the only option.
e.g. see What are the allowed topologies for a FRW metric?
Edit: 
$^1$I'd also add that if one picks some of these more interesting compact topologies with constant negative curvature for the FLRW metric, then there are no continuous group isometries (no global Killing vector fields) and the spacetime is not isotropic at every point. So there are physical reasons why people often make the assumption that hyperbolic implies infinite, but it should be stated explicitly.

Why does hyperbolic universe must be infinite?

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