# 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. – user641 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 . – user641 Dec 11 '11 at 6:34
@Steve: Very interesting reference! I need to read it carefully. – Nilo Dec 11 '11 at 7:22

A virtually $\mathbb{Z}^d$ group $G$ is also $\mathbb{Z}^d$-by-finite. Let us call $N$ a normal subgroup, of finite index in $G$, isomorphic to $\mathbb{Z}^d$.
We know that $Q = G/N$ is an arbitrary finite group and, in order to classify or characterize all such $G$, a possible approach could be made trying to solve the extension problem for $N$ and $Q$. What one have to do is to find, for all fixed $Q$, the homomorphisms $$\varphi\ :\ Q\ \rightarrow\ Aut(N)$$ in order to reconstruct the action of $Q$ on $N$. Since $Aut(\mathbb{Z}^d) \cong \mathsf{GL}(d,\mathbb{Z})$, one just has to consider the $k$-involutory elements of $\mathsf{GL}(d,\mathbb{Z})$.
It's easy to find some references with a classification of the finite subgroups of $\mathsf{GL}(d,\mathbb{Z})$ up to at least $d \le 7$ (and their number is finite $\forall\ d$) -if someone's interested, I can provide them.
Once we have such integer $k$-involutory matrices, we can hope to conclude something about $\varphi$ and then about $G$.