Let f be the smallest function satisfying the following:

Suppose given two matrices A and B in SL_3(Z), with all entries at most N. If there is a word w(A,B,A^{-1},B^{-1}) which vanishes in SL_3(Z), then there is a word w'(A,B,A^{-1},B^{-1}) of length at most f(N) which vanishes in SL_3(Z).

What are the asymptotics of f(N)?

The reason for the title is that, if SL_3(Z) is replaced by Z^n, this is Siegel’s lemma: if two (or, for that matter, k) vectors in [-N..N]^n are linearly dependent, then there is a linear dependency whose height is polynomial in N. (Here k and n are constants and N is growing.)

I don’t have any particular need to know this — the question came up in conversation at the very stimulating MSRI Thin Groups workshop just concluded. Sarnak’s notes are an excellent guide to the topics discussed there.

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