The Product Replacement Algorithm and Kazhdan’s Property (t)

A problem of great importance in computational group theory is to generate (nearly) uniformly distributed random elements in a finite groupG. A good example of such an algorithm should start at any given set of generators, use no prior knowledge of the structure of G, and in a polynomial number of group operations return the desired random group elements (see [Bb2]). Then these random elements can be used further to determine the structure of G. In a pioneer paper [Bb1] Babai found such an algorithm which provably generates (nearly) uniformly distributed random elements in O(log |G|) group multiplications, too many for practical applications. A different heuristic, the “product replacement algorithm”, was designed by Leedham-Green and Soicher [LG], and studied by Celler et al. in [CLMNO]. In spite of the fact that very little theoretical justification was known, practical experiments showed excellent performance. So, it quickly became the most popular “practical” algorithm to generate random group elements, and was included in the two most frequently used group algebra packages GAP ([Sc]) and MAGMA ([BCP]). A systematic and quantitative approach was carried out by Diaconis and SaloffCoste [DS1], [DS2] (see also [Bb2], [CG]), but their results did not reveal the mystery of the truly outstanding performance of the algorithm. The aim of this paper is to propose a conceptual explanation based on Kazhdan’s property (T) from representation theory of Lie groups and to improve some of the previous estimates on the running time. The product replacement algorithm works as follows ([CLMNO]): Given a finite group G, let Γk(G) be the set of k-tuples (g) = (g1, . . . , gk) of elements of G such that 〈g1, . . . , gk〉 = G. We call elements of Γk(G) the generating k-tuples. Given a generating k-tuple (g1, . . . , gk), define a move on it in the following way: Choose uniformly a pair (i, j), such that 1 ≤ i 6= j ≤ k, then apply one of the following four operations with equal probability: