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 $E(3)$).

crystalline sheet

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.

  • $\begingroup$ In your setup is $\rho(\Gamma)$ a discrete subgroup of $E(d)$? $\endgroup$ – Igor Belegradek Dec 14 '15 at 22:25
  • $\begingroup$ It's natural to consider the space of homomorphisms as it can be viewed as set of real points of some algebraic variety (of commuting $m$-tuples in $E(d)$); then study its subset (probably open in the real topology) of injective discrete representations. $\endgroup$ – YCor Dec 14 '15 at 22:35
  • $\begingroup$ Also you probably want $\rho$ to be injective. Assuming this there aren't many choices for discrete injective homomorphisms $\mathbb Z^2\to E(3)$. The group $\rho(\Gamma)$ will have to stabilize a $2$-plane where it acts cocompactly in a familiar way (tiling with parallelograms), and then $\rho(\Gamma)$ also acts isometrically in the normal direction. The normal direction is one dimensional so the action is simply a homomorphism $\mathbb Z^2\to O(1)\cong\mathbb Z_2$. For general $m$, $d$ the picture is similar, except that homomorphisms $\mathbb Z^m\to O(d-m)$ could be more complicated. $\endgroup$ – Igor Belegradek Dec 14 '15 at 22:40
  • $\begingroup$ @IgorBelegradek Actually, thinking more carefully about the pictures it seems that requiring $\rho(\Gamma)$ to be discrete and $\rho$ to be injective may be too strong (so I've deleted my earlier comment). If we require that then it seems to rule out the deformation to "cylinders" that I want to capture. $\endgroup$ – j.c. Dec 14 '15 at 22:43
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    $\begingroup$ Concerning local deformations, an important reference is Weil's 1964 paper (jstor.org/stable/1970495, unfortunately in restricted access). $\endgroup$ – YCor Dec 14 '15 at 23:44

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