# Informal intro / motivation:

Suppose I have an infinite set of atoms arranged in a 2D periodic crystalline "sheet". By crystalline I simply mean that it is preserved by the action of integer linear combinations of a pair of linearly independent translations (which generate a subgroup $\Gamma\cong\mathbb{Z}^2$ of the 3D Euclidean group $E(3)$).

It seems to me that there exist deformations of this flat sheet into a cylindrical structure, where $\Gamma$ itself is deformed so that one of the translations becomes a rotation around an axis or more generally, a screw motion. I would like to know what tools are available to think about this.

# Attempt at math:

Here's my first attempt at formalizing what I want.

Let $\Gamma=\mathbb{Z}^m$ for some $m$ and let $E(d)$ be the group of isometries for $d$-dimensional Euclidean space. I would like to understand the space $\mathcal{R}$ of homomorphisms $\rho:\Gamma\rightarrow E(d)$ for, say, $d=3, m=2$, ($m\leq d$ generally is interesting to me).

The only thing I could think to do was to represent elements of $E(d)$ by $(d+1)\times(d+1)$-matrices in the standard way and ask about the space of pairs which commute, i.e. solutions to $ab=ba$. (Is that the correct subvariety to look at?) It seemed very messy to work with so I wanted to ask here for better ways to approach this.

What books / surveys should I look at to get started? I suspect I may have to learn about the general case of "non-cocompact" discrete subgroups of Lie groups (about which I know absolutely nothing).

As an example of things I'd like to understand, what is the dimension of $\mathcal{\tilde{R}}=\mathcal{R}/\text{Aut }E(d)$ (i.e. how rigid is $\mathbb{Z}^m$ inside $E(d)$)? I imagine that's probably pretty complicated due to singularities, so is there a simple way to compute the dimension of the space of deformations infinitesimally around a given homomorphism $\rho$?

The things I could find (e.g. this survey of Gromov and Pansu) from googling seem to restrict to the case of finite covolume $\Gamma$.

I do not have a lot of knowledge about these matters, so references at an introductory level (hopefully treating plenty of examples in a hands-on way) would be helpful.