# F.g. groups with a finite index abelian subgroup

It is well known that a virtually cyclic group is either finite, or finite-by-(infinite cyclic) or finite-by-(infinite dihedral).

I want to know if there is some similar description for f.g. virtually abelian groups, or even for simpler groups (e.g. virtually (free abelian of rank 2)). In the affirmative case, are the proofs easy/short? References are welcome too.

Thank you!

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Groups which are virtually $\mathbb{Z}^2$ are precisely those groups which act properly and cocompactly on the plane $\mathbb{R}^2$. So really it would be enough to classify compact Euclidean orbifolds of dimension 2. For example, if your group is torsion-free, then your quotient is a manifold, and so the group is either the fundamental group of a torus or a Klein bottle. – Steve D Dec 11 '11 at 5:52
@Steve: I'm really looking for an algebraic description of groups that are not necessarily torsion-free, but I'd be interested in learning how this geometric description can be achieved. Maybe this can give me some idea on the algebraic side in general. Do you think so? Thanks. – Nilo Dec 11 '11 at 6:25
This really is an algebraic description via van Kampen's theorem. You are basically looking for orbifolds with Euler characteristic zero. A good source for all of this is Peter Scott's "Geometries of 3-Manifolds" article, which is available for free on his website: math.lsa.umich.edu/~pscott/8geoms.pdf . – Steve D Dec 11 '11 at 6:34
@Steve: Very interesting reference! I need to read it carefully. – Nilo Dec 11 '11 at 7:22