Product Growth and Mixing in Finite Groups L´aszl´oBabai * Nikolay Nikolov † L´aszl´oPyber ‡ Abstract We prove the following inequality on the convolution of distributions over a finite group G: (0.1) ‖ X ∗ Y − U ‖≤  n/m‖ X − U ‖‖ Y − U ‖, where X, Y are probability distributions over G, the ∗ denotes convolution, U the uniform distribution over G, and ‖ . ‖ the ℓ 2 -norm; n is the order of G, and m denotes the minimum dimension of nontrivial real representations of G. This inequality can be viewed as a new expansion property of a large class of groups, including all Lie-type simple groups of bounded rank, all of which satisfy m > cn β (where c> 0 is an absolute constant and β > 0 depends on the rank bound only). Best among them are the groups G = SL 2 (q) (2 × 2 matrices with determinant 1 over F q ) where m ∼ n 1/3 /2. We derive applications of the convolution inequal- ity (0.1) to a variety of areas, ranging from stochastic processes to additive combinatorics to group theory. An immediate consequence is a product growth inequality for subsets of G: if A, B ⊆ G then |AB| > n/(1 + Δ) where Δ = n 2 /(m|A||B|). On the one hand, this corollary strengthens a recent result of Gowers which served as the inspiration to the present work; on the other hand, it gives a strong (and best possible) affirmative answer to a problem regarding the product growth of subsets of SL 2 (q) recently posed by Venkatesh and Green at a conference in the newly flourishing area of “additive combinatorics.” Another corollary to the main inequality shows that for groups with large m, mixing in the strongest sense (ℓ ∞ -norm) occurs more rapidly than expected; we prove that if X,Y,Z are distributions over G then (0.2) ‖ X ∗ Y ∗ Z − U ‖ ∞ <  n m ‖ X ‖‖ Y ‖‖ Z ‖. * University of Chicago. Email: laci@cs.uchicago.edu. Part of this work was done at the R´ enyi Mathematical Institute, Budapest, and at the Centre Interfacultaire Bernoulli, Lausanne. † Imperial College London. Email: n.nikolov@imperial.ac.uk. ‡ R´ enyi Mathematical Institute, Budapest. Email: pyber@renyi.hu. This generalizes a result of Gowers. By easy induction, our main inequality generalizes to the convolution of multiple terms and thereby results in rapid mixing estimates for time-inhomogeneous Cay- ley walks on G. It also gives estimates for the size of the product of several subsets, resulting in diameter es- timates for Cayley graphs and tying in with the broad subject of “bounded generation” in group theory. An illustration of the connection to diameters: for G = SL 2 (q) it follows that if A ⊆ G and |A|≥ n 2/3+ǫ then A t = G where t = O(1/ǫ); we also show that the elements of G are represented nearly uniformly as words of length t over A. The connection to “bounded generation” is illus- trated by one of the main applications of our results: every finite simple group of Lie type of characteristic p is the product of 5 Sylow p-subgroups. – Results of this type are among the ingredients of the recent breakthrough result that all finite simple groups have bounded degree expander Cayley graphs [KLN]; our results improve and greatly simplify these ingredients. The results and techniques used in this paper were inspired by a link between quasirandomness and group representation theory recently found by Gowers [Go]. 1 Introduction 1.1 Expansion of finite groups. Expansion prop- erties of graphs have been of great interest to the theory of computing for decades, and every new technique in- troduced in the area promises new applications. Many of the best expanders are linked to groups and specifi- cally to the group SL 2 (q) of 2 × 2 matrices with deter- minant 1 over the field F q (q a prime power). In partic- ular, the Ramanujan graphs constructed by Lubotzky, Phillips, Sarnak [LuPS] and by Margulis [Ma] are Cay- ley graphs of this group. A belief in a certain degree of expansion of finite simple groups is expressed in the conjecture [BaSe] that all Cayley graphs of such groups have polylogarithmic diameter (and therefore even for the worst generators, the random walks con- verge rapidly). In a recent breakthrough, Helfgott [He] confirmed this conjecture for the groups SL 2 (p)(p prime), by proving a powerful product growth estimate