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

. The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating k-tuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an extensive review of both positive and negative theoretical results in the analysis of the algorithm. Introduction In the past few decades the study of groups by means of computations has become a wonderful success story. The whole new field, Computational Group Theory, was developed out of needs to discover and prove new results on finite groups. More recently, the probabilistic method became an important tool for creating faster and better algorithms. A number of applications were developed which assume a fast access to (nearly) uniform group elements. This led to a development of the so called "product replacement algorithm", which is a commonly used heuristic to generate random group elemen...

The product replacement algorithm is a heuristic designed to generate random group elements. The idea is to run a random walk on generating k-tuples of the group, and then output a random component. The algorithm was designed by Leedham-Green and Soicher ([31]), and further investigated in [12]. It was found to have an outstanding performance, much better than the the previously known algorithms (see [12, 22, 26]). The algorithm is now included in two major group algebra packages GAP [42] and MAGMA [10]. In spite of the many serious attempts and partial results, (see [6, 14, 15, 21, 22, 32, 39, 40]), the analysis of the algorithm remains difficult at best. For small values of k even graph connectivity becomes a serious obstacle (see [19, 37, 39, 40]). The most general results are due to Diaconis and Saloff--Coste [22], who used a state of the art analytic technique to obtain polynomial bounds in special cases, and (sub)-exponential bounds in general case. The main result of this pape...

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. We establish a connection between the expansion coefficient of the product replacement graph \Gamma k (G) and the minimal expansion coefficient of a Cayley graph of G with k generators. In particular, we show that the product replacement graphs \Gamma k \Gamma PSL(2; p) \Delta form an expander family, under assumption that all Cayley graphs of PSL(2; p), with at most k generators are expanders. This gives a new explanation of the outstanding performance of the product replacement algorithm and supports the speculation that all product replacement graphs are expanders [LP,P3].

. Let G be a finite group, and let ' k (G) be the probability that k random group elements generate G. Denote by #(G) the smallest k such that ' k (G) ? 1=e. In this paper we analyze quantity #(G) for different classes of groups. We prove that #(G) --(G) + 1 when G is nilpotent and --(G) is the minimal number of generators of G. When G is solvable we show that #(G) 3:25 --(G) + 10 7 . We also show that #(G) ! C log log jGj, where G is a direct product of simple nonabelian groups, and C is a universal constant. The work is motivatedby the applications to the "product replacement algorithm" (see [CLMNO,P4]). This algorithm is an important recent innovation, designed to efficiently generate (nearly) uniform random group elements. Recent work by Babai and the author [BaP] showed that the output of the algorithm must have a strong bias in certain cases. The precise probabilistic estimates we obtain here, combined with a note [P3], give positive result, proving that no bias exists for...

. We prove that the product replacement graph on generating 3-tuples of An is connected for n 11. We employ an efficient heuristic based on [P1] which works significantly faster than brute force. The heuristic works for any group. Our tests were confined to An due to the interest in Wiegold's Conjecture, usually stated in terms of T -systems (see [P2]). Our results confirm Wiegold's Conjecture in some special cases and are related to the recent conjecture of Diaconis and Graham [DG]. The work was motivated by the study of the product replacement algorithm (see [CLMNO,P2]). Introduction Let G be a finite group, and let N k (G) be the set of generating k-tuples (g) = (g 1 ; : : : ; g k ), where hg 1 ; : : : ; g k i = G. Define moves on N k (G) as follows: R \Sigma i;j : (g 1 ; : : : ; g i ; : : : ; g k ) ! (g 1 ; : : : ; g i \Delta g \Sigma1 j ; : : : ; g k ) L \Sigma i;j : (g 1 ; : : : ; g i ; : : : ; g k ) ! (g 1 ; : : : ; g \Sigma1 j \Delta g i ; : : : ; g k ) ) ...