Speakers

2025-11-03 : Serge Cantat (CNRS, Universite de Rennes). 

Title: The product replacement algorithm 

 Abstract: Choose a compact Lie group G, say the group SO_k,  and an integer n > 2. We will be interested in n-tuples (g_1, …, g_n) of elements in G and in the smallest closed subgroup containing such a tuple. The product replacement algorithm can be considered  as a group action on G^n generated by the following simple moves: 

    (a) permuting the g_i, 

    (b) changing one of them into its inverse, and 

    (c) changing some g_j into g_jg_i with i distinct from j.

Such ``moves’’ do not change the group generated by (g_1, …, g_n). The problem is to  describe the orbits of the product replacement algorithm in G^n. I will describe this problem  and explain some results that hold when n is large. (based on a joint work with C. Dupont  and F. Martin-Baillon).