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Alireza Salehi Golsefidy

Publications and source records attributed to Alireza Salehi Golsefidy.

16 recordsLinked to original sources
arXiv2020

Locally Random Groups

In this work, we will introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we will develop a Littlewood-Paley decomposition and explore its connection to the existence of the spectral gap for random walks. Moreover, under the dimension condition alone, we will prove a multi-scale entropy gain result `a la Bourgain-Gamburd and Tao.

math.GR↗
arXiv2020

Towards super-approximation in positive characteristic

In this note we show that the family of Cayley graphs of a finitely generated subgroup of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$ modulo some admissible square-free polynomials is a family of expanders under certain algebraic conditions. Here is a more precise formulation of our main result. For a positive integer $c_0$, we say a square-free polynomial is $c_0$-admissible if degree of irreducible factors of $f$ are distinct integers with prime factors at least $c_0$. Suppose $Ω$ is a finite symmetric subset of ${\rm GL}_{n_0}(\mathbb{F}_p(t))$, where $p$ is a prime more than $5$. Let $Γ$ be the group generated by $Ω$. Suppose the Zariski-closure of $Γ$ is connected, simply-connected, and absolutely almost simple; further assume that the field generated by the traces of ${\rm Ad}(Γ)$ is $\mathbb{F}_p(t)$. Then for some positive integer $c_0$ the family of Cayley graphs ${\rm Cay}(π_{f(x)}(Γ),π_{f(x)}(Ω))$ as $f$ ranges in the set of $c_0$-admissible polynomials is a family of expanders, where $π_{f(t)}$ is the quotient map for the congruence modulo $f(t)$.

math.GR↗
arXiv2018

Sum-product phenomena: p-adic case

The sum-product phenomena over a finite extension K of $\mathbb{Q}_p$ is explored. The main feature of the results is the fact that the implied constants are independent of $p$.

math.CO↗
arXiv2018

Super-approximation, II: the p-adic and bounded power of square-free integers cases

Let $Ω$ be a finite symmetric subset of GL$_n(\mathbb{Z}[1/q_0])$, and $Γ:=\langle Ω\rangle$. Then the family of Cayley graphs $\{{\rm Cay}(π_m(Γ),π_m(Ω))\}_m$ is a family of expanders as $m$ ranges over fixed powers of square-free integers and powers of primes that are coprime to $q_0$ if and only if the connected component of the Zariski-closure of $Γ$ is perfect. Some of the immediate applications, e.g. orbit equivalence rigidity, {\em largeness} of certain $\ell$-adic Galois representations, are also discussed.

math.GR↗
arXiv2018

Inducing Super-Approximation

Let $Γ_2\subseteq Γ_1$ be finitely generated subgroups of ${\rm GL}_{n_0}(\mathbb{Z}[1/q_0])$. For $i=1$ or $2$, let $\mathbb{G}_i$ be the Zariski-closure of $Γ_i$ in $({\rm GL}_{n_0})_{\mathbb{Q}}$, $\mathbb{G}_i^{\circ}$ be the Zariski-connected component of $\mathbb{G}_i$, and let $G_i$ be the closure of $Γ_i$ in $\prod_{p\nmid q_0}{\rm GL}_{n_0}(\mathbb{Z}_p)$. In this article we prove that, if $\mathbb{G}_1^{\circ}$ is the smallest closed normal subgroup of $\mathbb{G}_1^{\circ}$ which contains $\mathbb{G}_2^{\circ}$ and $Γ_2\curvearrowright G_2$ has spectral gap, then $Γ_1\curvearrowright G_1$ has spectral gap.

math.GR↗
arXiv2018

Super-approximation, I: p-adic semisimple case

Let $k$ be a number field, $Ω$ be a finite symmetric subset of $\mathbb{GL}_{n_0}(k)$, and $Γ=\langle Ω\rangle$. Let \[ C(Γ):=\{\mathfrak{p}\in V_f(k)|\hspace{1mm} Γ\text{is a bounded subgroup of} \mathbb{GL}_{n_0}(k_{\mathfrak{p}})\}, \] and $Γ_{\mathfrak{p}}$ be the closure of $Γ$ in $\mathbb{GL}_{n_0}(k_{\mathfrak{p}})$. Assuming that the Zariski-closure of $Γ$ is semisimple, we prove that the family of left translation actions $\{Γ\curvearrowright Γ_{\mathfrak{p}}\}_{\mathfrak{p}\in C(Γ)}$ has {\em uniform spectral gap}. As a corollary we get that the left translation action $Γ\curvearrowright G$ has {\em local spectral gap} if $Γ$ is a countable dense subgroup of a semisimple $p$-adic analytic group $G$ and Ad$(Γ)$ consists of matrices with algebraic entries in some $\mathbb{Q}_p$-basis of Lie$(G)$. This can be viewed as a (stronger) $p$-adic version of \cite[Theorem A]{BISG}, which enables us to give applications to the Banach-Ruziewicz problem and orbit equivalence rigidity.

math.GR↗
arXiv2016

Local spectral gap in simple Lie groups and applications

We introduce a novel notion of {\it local spectral gap} for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $Γ\curvearrowright G$, whenever $Γ$ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group $G$. This extends to the non-compact setting recent works of Bourgain and Gamburd \cite{BG06,BG10}, and Benoist and de Saxcé \cite{BdS14}. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on $G$. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique $Γ$-invariant finitely additive measure defined on all bounded measurable subsets of $G$.

math.GR↗
arXiv2013

Translate of horospheres and counting problems

Let G be a semisimple Lie group without compact factors, Γbe an irreducible lattice in G. In the first part of the article we give the necessary and sufficient condition under which a sequence of translates of probability "horospherical measures" is convergent. And the limiting measure is also determined when it is convergent (see Theorems 1 and 2 for the precise statements). In the second part, two applications are presented. The first one is of geometric nature and the second one gives an alternative way to count the number of rational points on a flag variety.

math.DS↗
arXiv2012

Expansion in perfect groups

Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an integer q, denote by Ga_q the subgroup of Ga consisting of the elements that project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q with respect to the generating set S form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of Ga is perfect.

math.GR↗
arXiv2011

Affine Sieve

We establish the main saturation conjecture in [BGS10] connected with executing a Brun sieve in the setting of an orbit of a group of affine linear transformations. This is carried out under the condition that the Zariski closure of the group is Levi-semisimple. It is likely that this condition is also necessary for such saturation to hold.

math.NT↗
arXiv2011

S-Arithmetic Khintchine-Type Theorem

In this article we prove a convergence S-arithmetic Khintchine-type theorem for product of non-degenerate v-adic manifolds, where one of them is the Archimedian place.

math.NT↗
arXiv2011

Counting lattices in simple Lie groups: the positive characteristic case

In this article we prove a conjecture of A. Lubotzky: let G = G_0(K), where K is a local field of characteristic p>5, G_0 is a simply connected, absolutely almost simple K-group of K-rank at least 2. We give the rate of growth of r_x(G) :={H< G| H a lattice in G; vol(G/H)<x}/~, where H_1\simH_2 if and only if there is an abstract automorphism t of G such that H_2 =t(H_1). We also study the rate of subgroup growth s_x(H) of any lattice H in G. As a result we show that these two functions have the same rate of growth which proves Lubotzky's conjecture. Along the way, we also study the rate of growth of the number of equivalence classes of maximal lattices in G with covolume at most x.

math.GR↗