The product replacement algorithm and Kazhdan’s property (T)

The “product replacement algorithm” is a commonly used heuristic to generate random group elements in a finite group G G , by running a random walk on generating k k -tuples of G G . While experiments showed outstanding performance, the theoretical explanation remained mysterious. In this paper we propose a new approach to the study of the algorithm, by using Kazhdan’s property (T) from representation theory of Lie groups.