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I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the (left invariant) riemannian metric, and the volume form of the considered group. Are they unique?

Could you suggest an example based reference on Lie groups?

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Of course any left invariant metric with respect the group of orientation preserving euclidean isometries (I believe that it what you call SE(n) ) is a multiple of the standard Euclidean metric and any invariant volume form is a multiple of the standard volume form.

I do not really know what reference to suggest you basing on your question. Actually, your question belongs IMHO rather to the elementary geometry than to the Lie group theory and if I may I suggest you ``Geometry I, II'' of M. Berger.

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  • $\begingroup$ I'm sorry. Your answer doesn't say too much to me. $\endgroup$
    – Melquíades Ochoa
    Commented Apr 25, 2014 at 21:46
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    $\begingroup$ Unfortunately, your comment does not say too much to me either. Where the desired volume form and left invariant metric lives? On $R^n$ or on E(n) resp. SE(n)? In the second case, the volume form is unique, up to scaling. But then I change my answer for the left invariant metric: fixing a positively definite quadratic form at the Lie algebra gives a left invarinat metric. $\endgroup$
    – Vladimir S Matveev
    Commented Apr 26, 2014 at 10:32

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