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Aner Shalev

Publications and source records attributed to Aner Shalev.

At least 37 records · Page 2Linked to original sources
arXiv2018

The length and depth of algebraic groups

Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group $G$ in terms of the dimension of its unipotent radical $R_u(G)$ and the dimension of a Borel subgroup $B$ of the reductive quotient $G/R_u(G)$. In particular, a simple algebraic group of rank $r$ has length $\dim B + r$, which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group $G$ exceeds $\frac{1}{2} \dim G$. We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most $6$ (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group $G$, the dimension of $G/R(G)$ is bounded above in terms of the chain difference of $G$.

math.GR↗
arXiv2018

Probabilistically nilpotent groups

We show that, for a finitely generated residually finite group $Γ$, the word $[x_1, \ldots, x_k]$ is a probabilistic identity of $Γ$ if and only if $Γ$ is virtually nilpotent of class less than $k$. Related results, generalizations and problems are also discussed.

math.GR↗
arXiv2017

The depth of a finite simple group

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups. We determine the simple groups of minimal depth, and show, somewhat surprisingly, that alternating groups have bounded depth. We also establish general upper bounds on the depth of simple groups of Lie type, and study the relation between the depth and the much studied notion of the length of simple groups. The proofs of our main theorems depend (among other tools) on a deep number-theoretic result, namely, Helfgott's recent solution of the ternary Goldbach conjecture.

math.GR↗
arXiv2017

Character bounds for finite groups of Lie type

We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $|χ(g)| \le χ(1)^{α_g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular we deduce that, if $G$ is a classical group in dimension $n$, then, under some conditions on $G$ and $g \in G$, the mixing time of the random walk on $G$ with the conjugacy class of $g$ as a generating set is (up to a small multiplicative constant) $n/s$, where $s$ is the support of $g$.

math.GR↗
arXiv2017

Words, Hausdorff dimension and randomly free groups

We study fibers of word maps in finite, profinite, and residually finite groups. Our main result is that, for any word w in the free group on d generators, there exists $ε> 0$ such that if G is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $w\colon G^d \to G$ have Hausdorff dimension at most $d-ε$. We conclude that the profinite completion of a group G as above satisfies no probabilistic identity. It is therefore randomly free; namely, for any d > 0, the probability that d randomly chosen elements freely generate a free subgroup of G is 1. This solves an open problem of Dixon, Pyber, Seress, and Shalev. Additional applications and related results are also established. For example, combining our results with recent results of Bors, we conclude that a profinite group in which the set of elements of finite odd order has positive measure has an open prosolvable subgroup. This may be regarded as a probabilistic version of the Feit-Thompson theorem.

math.GR↗
arXiv2016

Generation of second maximal subgroups and the existence of special primes

Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by 3 elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper we consider subgroups at the next level of the subgroup lattice - the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of $G$ is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes $r$ for which there is a prime power $q$ such that $(q^r-1)/(q-1)$ is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.

math.GR↗
arXiv2016

Simple groups, interleaved products, complexity and conjectures of Gowers and Viola

We study the distribution of products of conjugacy classes in finite simple groups, obtaining various effective uniformity results, which give rise to an approximation to a conjecture of Thompson. Our results, combined with work of Gowers and Viola, also lead to the solution of recent conjectures they posed on interleaved products and related complexity lower bounds, extending their work on the groups SL$(2,q)$ to all (nonabelian) finite simple groups. In particular it follows that, if $G$ is a finite simple group, and $A, B \subseteq G^t$ for $t \ge 2$ are subsets of fixed positive densities, then, as $a = (a_1, \ldots , a_t) \in A$ and $b = (b_1, \ldots , b_t) \in B$ are chosen uniformly, the interleaved product $a \bullet b := a_1b_1 \cdots a_tb_t$ is almost uniform on $G$ (with quantitative estimates) with respect to the $\ell_{\infty}$-norm. It also follows that the communication complexity of an old decision problem related to interleaved products of $a, b \in G^t$ is at least $Ω(t \log |G|)$ when $G$ is a finite simple group of Lie type of bounded rank, and at least $Ω(t \log \log |G|)$ when $G$ is any finite simple group. Both these bounds are best possible.

math.GR↗
arXiv2015

A probabilistic Tits alternative and probabilistic identities

We introduce the notion of a probabilistic identity of a residually finite group. We prove that a finitely generated linear group satisfies a probabilistic identity if and only if it is virtually solvable. As an application, we prove a probabilistic variant of the Tits alternative: let G be the profinite completion of a finitely generated linear group. Then either G is virtually solvable, or for any positive integer n, with probability one, n independent, uniformly distributed elements of G freely generate a free subgroup of G of rank n.

math.GR↗
arXiv2015

Surjective word maps and Burnsides $p^aq^b$ theorem

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map sending (x,y) to the product of the Nth powers of x and y is surjective on every finite non-abelian simple group; If is an odd integer, then the word map sending the triple (x,y,z) to the product of the Nth powers is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit-Thompson. We also prove asymptotic results about the surjectivity of the word map sending (x,y) to the product of the Nth powers of x and y that depend on the number of prime factors of the integer N.

math.GR↗
arXiv2014

Invariable Generation of Infinite Groups

A subset S of a group G invariably generates G if G = for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a finitely generated linear group is invariably generated by some finite set of elements if and only if it is virtually solvable. We also show that the profinite completion of an arithmetic group having the congruence subgroup property is invariably generated by a finite set of elements.

math.GR↗
arXiv2014

The Waring problem for Lie groups and Chevalley groups

The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given non-trivial word w. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed non-trivial word w and for a classical connected real compact Lie group G of sufficiently large rank we have w(G)^2=G, namely every element of G is a product of 2 values of w. We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over R or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares.

math.GR↗
arXiv2011

Waring Problem for Finite Quasisimple Groups

The classical Waring problem deals with expressing every natural number as a sum of g(k) kth powers. Similar problems for finite simple groups have been studied recently, and in this paper we study them for finite quasisimple groups G. We show that for a fixed non-trivial group word w and large enough G we have w(G)^3=G, namely every element of G is a product of 3 values of w. For various families of finite quasisimple groups, including covers of alternating groups, we obtain a stronger result, namely w(G)^2=G. However, in contrast with the case of simple groups, we show that w(G)^2 need not equal G for all large G. If k>2 then x^k y^k fails to be surjective for infinitely many finite quasisimple groups. The case k=2 turns out to be exceptional. Indeed, our last result shows that every element of a finite quasisimple group is a product of two squares. This can be regarded as a non-commutative analogue of Lagrange's four squares theorem.

math.GR↗
arXiv2011

Product decompositions in finite simple groups

We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of Lie type of bounded rank. Some of our arguments apply recent advances in the theory of growth in finite simple groups of Lie type, and provide a variety of new product decompositions of these groups.

math.GR↗
arXiv2010

Counting arithmetic lattices and surfaces

We give estimates on the number $AL_H(x)$ of arithmetic lattices $Γ$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $Γ$ as a function of its covolume), number theoretic (bounding the number of maximal such $Γ$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $Γ$).

math.GR↗
arXiv2007

Commutator maps, measure preservation, and T-systems

Let G be a finite simple group. We show that the commutator map $a : G \times G \to G$ is almost equidistributed as the order of G goes to infinity. This somewhat surprising result has many applications. It shows that for a subset X of G we have $a^{-1}(X)/|G|^2 = |X|/|G| + o(1)$, namely $a$ is almost measure preserving. From this we deduce that almost all elements $g \in G$ can be expressed as commutators $g = [x,y]$ where x,y generate G. This enables us to solve some open problems regarding T-systems and the Product Replacement Algorithm (PRA) graph. We show that the number of T-systems in G with two generators tends to infinity as the order of G goes to infinity. This settles a conjecture of Guralnick and Pak. A similar result follows for the number of connected components of the PRA graph of G with two generators. Some of our results apply for more general finite groups, and more general word maps. Our methods are based on representation theory, combining classical character theory with recent results on character degrees and values in finite simple groups. In particular the so called Witten zeta function plays a key role in the proofs.

math.GR↗