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Almost all one-relator groups with at least three generators are residually finite

Mark V. Sapir, Iva Špakulová (2011)

Journal of the European Mathematical Society

We prove that with probability tending to 1, a one-relator group with at least three generators and the relator of length n is residually finite, is a virtually residually (finite p -)group for all sufficiently large p , and is coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions.

Amenability of linear-activity automaton groups

Gideon Amir, Omer Angel, Bálint Virág (2013)

Journal of the European Mathematical Society

We prove that every linear-activity automaton group is amenable. The proof is based on showing that a random walk on a specially constructed degree 1 automaton group – the mother group – has asymptotic entropy 0. Our result answers an open question by Nekrashevych in the Kourovka notebook, and gives a partial answer to a question of Sidki.

Cogrowth and spectral gap of generic groups

Yann Ollivier (2005)

Annales de l’institut Fourier

The cogrowth exponent of a group controls the random walk spectrum. We prove that for a generic group (in the density model) this exponent is arbitrarily close to that of a free group. Moreover, this exponent is stable under random quotients of torsion-free hyperbolic groups.

Expansion in finite simple groups of Lie type

Emmanuel Breuillard, Ben J. Green, Robert Guralnick, Terence Tao (2015)

Journal of the European Mathematical Society

We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].

Groupes aléatoires

Étienne Ghys (2002/2003)

Séminaire Bourbaki

Quelles sont les propriétés d’un groupe de présentation finie “tiré au hasard” ? La réponse à cette question dépend bien entendu de la méthode choisie pour le tirage au sort. On peut par exemple fixer n générateurs et choisir p relations aléatoirement parmi les mots de longueur L , puis faire tendre L vers l’infini. On peut aussi choisir un graphe fini, étiqueter aléatoirement ses arêtes par des générateurs, et considérer le groupe engendré par ces générateurs, soumis aux relations lues sur les cycles...

Homomorphisms to constructed from random walks

Anna Erschler, Anders Karlsson (2010)

Annales de l’institut Fourier

We give a construction of homomorphisms from a group into the reals using random walks on the group. The construction is an alternative to an earlier construction that works in more general situations. Applications include an estimate on the drift of random walks on groups of subexponential growth admitting no nontrivial homomorphism to the integers and inequalities between the asymptotic drift and the asymptotic entropy. Some of the entropy estimates obtained have applications independent of the...

Jucys-Murphy element and walks on modified Young graph

Akihito Hora (2006)

Banach Center Publications

Biane found out that irreducible decomposition of some representations of the symmetric group admits concentration at specific isotypic components in an appropriate large n scaling limit. This deepened the result on the limit shape of Young diagrams due to Vershik-Kerov and Logan-Shepp in a wider framework. In particular, it is remarkable that asymptotic behavior of the Littlewood-Richardson coefficients in this regime was characterized in terms of an operation in free probability of Voiculescu....

Probability that an element of a finite group has a square root

M. S. Lucido, M. R. Pournaki (2008)

Colloquium Mathematicae

Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 - ⌊√|G|⌋/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 - 1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G.

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G , then for every subset B of G with | B | > | G | / k 1 / 3 we have B 3 = G . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if k 2 , then G has a proper subgroup...

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