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Navigating in the Cayley Graphs of SLN(Z) and SLN(Fp)

Tim R. Riley

Geometriae Dedicata, 113(1), pages 215-229, 2005.

We give a non-deterministic algorithm that expresses elements of SLN(Z), for N>2, as words in a finite set of generators, with the length of these words at most a constant times the word metric. We show that the non-deterministic time-complexity of the subtractive version of Euclid's algorithm for finding the greatest common divisor of N>2 integers a1, ..., aN is at most a constant times N log max {|a1|, ..., |aN|}. This leads to an elementary proof that for N>2 the word metric in SLN(Z) is biLipschitz equivalent to the logarithm of the matrix norm — an instance of a theorem of Mozes, Lubotzky and Raghunathan. And we show constructively that there exists K>0 such that for all N>2 and primes p, the diameter of the Cayley graph of SLN(Fp) with respect to the generating set {eij | i \neq j } is at most K N2 log p .