A Non - spectral Estimate of the Cayley Diameter of SL 2 ( F p )

It is well known that SL2(Fp) is generated by ( 1 1 0 1 ) and ( 1 0 1 1 ) . It is a much deeper theorem [4] that the Cayley diameter of this group with respect to these generators is O(log p). The proof depends on uniformly bounding the eigenvalues of the Laplacian on L0(X(p)) away from zero. This argument is, of course, very far from constructive. (For certain other sets of generators, A. Lubotzky, R. Phillips, and P. Sarnak gave a non-spectral, but still non-constructive, argument for the logarithmic growth of the Cayley diameter of SL2(Fp) [5].) Lubotzky asked [4] whether one can efficiently find short word representations of general elements of SL2(Fp). In this note we give an elementary, constructive proof that every element has a word representation of length O(log p log log p). The argument gives a probabilistic algorithm with average running time polynomial in log p for finding a word representing a given element. More precisely, we prove