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Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
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Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
The sumproduct phenomenon in arbitrary rings
 Cont. to Disc. Math
"... Abstract. The sumproduct phenomenon predicts that a finite set A in a ring R should have either a large sumset A + A or large product set A · A unless it is in some sense “close ” to a finite subring of R. This phenomenon has been analysed intensively for various specific rings, notably the reals R ..."
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Abstract. The sumproduct phenomenon predicts that a finite set A in a ring R should have either a large sumset A + A or large product set A · A unless it is in some sense “close ” to a finite subring of R. This phenomenon has been analysed intensively for various specific rings, notably the reals R and cyclic groups Z/qZ. In this paper we consider the problem in arbitrary rings R, which need not be commutative or contain a multiplicative identity. We obtain rigorous formulations of the sumproduct phenomenon in such rings in the case when A encounters few zerodivisors of R. As applications we recover (and generalise) several sumproduct theorems already in the literature. 1.
STATIONARY MEASURES AND EQUIDISTRIBUTION FOR ORBITS OF NONABELIAN SEMIGROUPS ON THE TORUS
"... Abstract. Let ν be a probability measure on SLd(Z) satisfying the moment condition Eν(‖g ‖ ɛ) < ∞ for some ɛ. We show that if the group generated by the support of ν is large enough, in particular if this group is Zariski dense in SLd, for any irrational x ∈ T d the probability measures ν ∗n ∗ δx te ..."
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Abstract. Let ν be a probability measure on SLd(Z) satisfying the moment condition Eν(‖g ‖ ɛ) < ∞ for some ɛ. We show that if the group generated by the support of ν is large enough, in particular if this group is Zariski dense in SLd, for any irrational x ∈ T d the probability measures ν ∗n ∗ δx tend to the uniform measure on T d. If in addition x is Diophantine generic, we show this convergence is exponentially fast. 1. Introduction and Statement
Strong spectral gaps for compact quotients of products of PSL(2
 R). J. Eur. Math. Soc
"... Abstract. The existence of a strong spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the RamanujanSelberg Conjectures ..."
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Abstract. The existence of a strong spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the RamanujanSelberg Conjectures. If G has no compact factors then for general lattices a strong spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the strong spectral gap for an irreducible cocompact lattice Γ in G = PSL(2, R) d for d ≥ 2 which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting. introduction This note is concerned with the strong spectral gap property for an irreducible cocompact lattice Γ in G = PSL(2, R) d, d ≥ 2. Before stating our main result we review in some detail what is known about such spectral gaps more generally. Let G be a noncompact connected semisimple Lie group with finite center and let Γ be a lattice in G. For π an irreducible unitary representation of G on a Hilbert space H, we let p(π) be the infimum of all p such that there is a dense set of vectors v ∈ H with 〈π(g)v, v 〉 in L p (G). Thus if π is finite dimensional p(π) = ∞, while π is tempered if and only if p(π) = 2. In general p(π) can be computed from the Langlands parameters of π and for many purposes it is a suitable measure of the nontemperedness of π (if p(π)> 2). The regular representation, f(x) ↦ → f(xg), of G on L 2 (Γ\G) is unitary and if Γ\G is compact it decomposes into a discrete direct sum of irreducibles while if Γ\G is noncompact the decomposition involves also continuous integrals via Eisenstein series. In any case, let
SPECTRAL GAP FOR PRODUCTS OF PSL(2, R)
"... Abstract. The existence of a spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the RamanujanSelberg Conjectures. If G ..."
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Cited by 2 (0 self)
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Abstract. The existence of a spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the RamanujanSelberg Conjectures. If G has no compact factors then for general lattices a spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible cocompact lattice Γ in G = PSL(2, R) d for d ≥ 2 which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting. introduction This note is concerned with the strong spectral gap property for an irreducible cocompact lattice Γ in G = PSL(2, R) d, d ≥ 2. Before stating our main result we review in some detail what is known about such spectral gaps more generally. Let G be a noncompact connected semisimple Lie group with finite center and let Γ be a lattice in G. For π an irreducible unitary representation of G on a Hilbert space H, we let p(π) be the infimum of all p such that there is a dense set of vectors v ∈ H with 〈π(g)v, v 〉 in Lp (G). Thus if π is finite dimensional p(π) =∞, while π is tempered if and only if p(π) = 2. In general p(π) can be computed from the Langlands parameters of π and for many purposes it is a suitable measure of the nontemperedness of π (if p(π)> 2). The regular representation, f(x) ↦ → f(xg), of G on L2 (Γ\G) is unitary and if Γ\G is compact it decomposes into a discrete direct sum of irreducibles while if Γ\G is noncompact the decomposition involves also continuous integrals via Eisenstein series. In any case, let E denote the exceptional exponent set defined by
Monotone expanders constructions and applications
"... The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree ..."
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The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree dimension expanders in finite fields, resolving a question of [BISW04]. 2. O(1)page and O(1)pushdown expanders, resolving a question of [GKS86], and leading to tight lower bounds on simulation time for certain Turing Machines. Bourgain [Bou09] gave recently an ingenious construction of such constant degree monotone expanders. The first application (1) above follows from a reduction in [DS08]. We give a short exposition of both construction and reduction. The new contributions of this paper are simple. First, we explain the observation leading to the second application (2) above, and some of its consequences. Second, we observe that a variant of the zigzag graph product preserves monotonicity, and use it to give a simple alternative construction of monotone expanders, with nearconstant degree. 1
Notes prepared for the Colloquium Lectures at the
"... 1.1 The basic definition.......................... 5 1.2 Eigenvalues, random walks and Ramanujan graphs......... 6 1.3 Cayley graphs and representation theory............... 8 ..."
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1.1 The basic definition.......................... 5 1.2 Eigenvalues, random walks and Ramanujan graphs......... 6 1.3 Cayley graphs and representation theory............... 8
Random Walks in Compact Groups
 DOCUMENTA MATH.
, 2013
"... Let X1,X2,... be independent identically distributed random elements of a compact group G. We discuss the speed of convergence of the law of the product Xl···X1 to the Haar measure. We give polylog estimates for certain finite groups and for compact semisimple Lie groups. We improve earlier resul ..."
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Let X1,X2,... be independent identically distributed random elements of a compact group G. We discuss the speed of convergence of the law of the product Xl···X1 to the Haar measure. We give polylog estimates for certain finite groups and for compact semisimple Lie groups. We improve earlier results of Solovay, Kitaev, Gamburd, Shahshahani and Dinai.