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It seems pretty clear a priori that Fourier analysis on a discrete cocompact group of motions of $\mathbb{E}^n$ should be quite tractable (since, by Bieberbach, such groups are almost $\mathbb{Z}^n,$ but I have not seen a lucid exposition of such, so any pointers or words of wisdom would be welcome.

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I don't too much about this area, but I think the jump is not quite as small as one might first hope. That said, one place to start with some aspects of their Fourier analysis might be with a description of their $C^*$-algebras, for which this paper of K. F. Taylor could be worth a look. – Yemon Choi Jul 2 '12 at 4:40
The finite dimensional reps are given here: – Marc Palm Jul 2 '12 at 12:11
I will check these out, thanks to both @Yemon and @Mrc... – Igor Rivin Jul 2 '12 at 21:36

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