In the wikipedia article for Kazhdan's Property (T), there's an intriguing application:

Similarly, groups with property (T) can be used to construct finite sets of invertible matrices which can efficiently approximate any given invertible matrix, in the sense that every matrix can be approximated, to a high degree of accuracy, by a finite product of matrices in the list or their inverses, so that the number of matrices needed is proportional to the number of significant digits in the approximation.

Unfortunately, there's no citation for this and googling invertible matrices + property (T) doesn't give much, nor have I found it in the book by Bekka, de la Harpe, and Valette. Does anyone have a reference for the above quote?

  • 2
    I think the idea is that if we have a dense subgroup in $G=SO(n)$ with Property T generated by a finite subset $S$, the spectral gap ensures a fast convergence of the $n$-fold convolution $(1_S)^{\ast n}$ to the constant $1_G$, in a suitable sense. Possibly a good reference is Lubotzky's 1994 Birkhäuser book "Discrete groups, expanding graphs, and invariant measures". – YCor Jan 22 at 17:02
  • PS 1) I meant convergence in $L^2$-operator norm. 2) This is surveyed in Sevennec's article (in French) umpa.ens-lyon.fr/sevennec/txt-lebesgue.ps – YCor Jan 22 at 17:51
  • @ycor might be possible to make it concrete an engineer understands? – Anon. Jan 22 at 18:25
  • @ao. hopefully yes but maybe not in 4 lines... I remember nice slides in Sevennec's talk (in Lebesgue measure's 100th birthday, in 2001) with pictures of well and bad distributed points on the sphere, unfortunately they're no in the linked pdf. – YCor Jan 22 at 18:58
  • @YCor what is the closest you have explicitly? – Anon. Jan 22 at 19:00

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