Flat solvmanifolds?

Question

I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds by Bieberbach's Theorem. I am almost sure I missed something trivial, but what is special about studying flat solvmanifolds?

Answer 1

I guess there could be many answers to this question. First, it is not quite correct to say that there is a complete classification of flat manifolds, i.e. Bieberbach's Theorem does not produce a list of manifolds in each dimension. I don't know for which $n$ the number of $n$-dimensional crystallographic groups is known, the following text provides numbers up to $n=6$ https://www.mathi.uni-heidelberg.de/~lee/TimSS16.pdf

Anyway, there are quite a few open questions about Bieberbach manifolds. Here is one, which I like:

Question. Is there a free complex action by a finite group on some complex torus $T^n$, $n\ge 4$ such that $H^2(T^n/G,\mathbb R)=\mathbb R$?

This is open according to http://www.numdam.org/article/AIF_1999__49_2_405_0.pdf

Concerning the paper of Morgan, it provides complete lists for the subclass of Bieberbach manifolds that are solve manifolds. Now, why could anyone care about solvmanifolds? This depends on person, but again some people do care. One reason is that since solvemanifolds are quotients of a Lie group by a lattice, one could study various translation invariant geometric structures on them and obtain exotic ones. See, for example, the following influential paper (it deals with nilpotent lie algebras) https://arxiv.org/abs/math/9808025

There is also a nice paper on complex solvmanifold that directly cites Morgan: https://projecteuclid.org/download/pdf_1/euclid.ojm/1146242998

I am not sure if this answers your question, but for me this looks like a motivation good enough.