What you have there is the "dodecahedron universe" notion that managed to make a brief splash in the popular press a couple of years ago. Basically, the investigators presented evidence that the universe has a strange topology, wrapping in on itself so that it could be fairly small (as far as universes go). The evidence was refuted almost as soon as it was published, and this particular notion is now effectively dead. The search for unusual topologies continues; but it's a really long shot.

Ned Wright has a good concise write up in his News of the Universe archive. Look for "A twelve-sided Universe? - Probably not".

Another approach is to look at the curvature of the universe. Neglecting topological complications, whether the universe is finite or not depends on whether curvature is positive or negative -- assuming homogeneity, which is another big assumption. Basically, positive curvature means a universe roughly analogous to a finite hypersphere; and negative curvature means a universe roughly analogous to an infinite hyperboloid; and zero means a "flat" universe roughly analogous to what we'd normally think of as unbounded space in which parallel lines remain at the same separation indefinitely.

A first rate description is at this Cosmology 101 lecture by James Schombert.

The curvature is so close to flat that measurement can barely discriminate between positive and negative curvature models. But recent observations do indicate a slight positive curvature, which suggests a finite but enormously large universe. This turns up in, for example, astro-ph/0501171, in which curvature is considered even though finiteness is not mentioned. Look for Ω_k < 0; which confusingly means positive curvature.

Strange topologies in which different regions of space connect in weird ways would still allow a finite universe with positive curvature; but negative curvature would be a strong indication of finiteness without any need for strange topologies.

If distant regions of the universe far beyond horizons of visibility have significantly different curvature properties, then all bets are off again. This is by no means a settled question.

I plumb for finite; but there is a fair amount of meta-physics bound up in any expression of definite opinion. Some scientists have philosophical preferences for finite; and others for infinite; and so far the universe is being coy on the matter.

Cheers -- Sylas