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Alexander Lubotzky

Publications and source records attributed to Alexander Lubotzky.

83 records · Page 5Linked to original sources
arXiv2011

Expander Graphs in Pure and Applied Mathematics

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.

math.CO↗
arXiv2011

The Finitary Andrews-Curtis Conjecture

The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent importance for computational group theory. It also resolves a question asked in [5] and shows that a computation in finite groups cannot lead to a counterexample to the classical conjecture, as suggested in [5].

math.GR↗
arXiv2010

Beauville surfaces and finite simple groups

A Beauville surface is a rigid complex surface of the form (C1 x C2)/G, where C1 and C2 are non-singular, projective, higher genus curves, and G is a finite group acting freely on the product. Bauer, Catanese, and Grunewald conjectured that every finite simple group G, with the exception of A5, gives rise to such a surface. We prove that this is so for almost all finite simple groups (i.e., with at most finitely many exceptions). The proof makes use of the structure theory of finite simple groups, probability theory, and character estimates.

math.GR↗
arXiv2009

Lattices with and lattices without spectral gap

The following two results are shown. 1) Let $G$ be the $k$-rational points of a simple algebraic group over a local field $k$ and let $H$ be a lattice in $G.$ Then the regular representation of $G$ on $L^2(G/H)$ has a spectral gap (that is, there are no almost invariant unit vectors in the subspace of functions in $L^2(G/H)$ with zero mean). 2) There exist locally compact simple groups $G$ and lattices $H$ for which $L^2(G/H)$ has no spectral gap. This answers in the negative a question asked by Margulis. In fact, $G$ can be taken to be the group of orientation preserving automorphisms of a $k$-regular tree for $k>2.$

math.DS↗
arXiv2006

Representation Growth for Linear Groups

We investigate the rate of growth of the function of n which counts the number of complex irreducible representations of a fixed group of degree less than or equal to n. The emphasis is on linear groups, especially compact real and p-adic Lie groups arithmetic groups.

math.GR↗
arXiv2006

Linear Representations of the Automorphism Group of a Free Group

Let $F_n$ be the free group on $n\ge 2$ elements and $\A(F_n)$ its group of automorphisms. In this paper we present a rich collection of linear representations of $\A(F_n)$ arising through the action of finite index subgroups of it on relation modules of finite quotient groups of $F_n$. We show (under certain conditions) that the images of our representations are arithmetic groups.

math.GR↗
arXiv2005

Finite Simple Groups as Expanders

We prove that there exist $k\in N$ and $0<ε\in R$ such that every non-abelian finite simple group $G$, which is not a Suzuki group, has a set of $k$ generators for which the Cayley graph $\Cay(G; S)$ is an $ε$-expander.

math.GR↗
arXiv2005

Division Algebras and Non-Commensurable Isospectral Manifolds

A. Reid showed that if $Γ_1$ and $Γ_2$ are arithmetic lattices in $G = \operatorname{PGL}_2(\mathbb R)$ or in $\operatorname{PGL}_2(\mathbb C)$ which give rise to isospectral manifolds, then $Γ_1$ and $Γ_2$ are commensurable (after conjugation). We show that for $d \geq 3$ and ${\mathcal S} = \operatorname{PGL}_d(\mathbb R) / \operatorname{PGO}_d(\mathbb R)$, or ${\mathcal S} = \operatorname{PGL}_d(\mathbb C) / \operatorname{PU}_d(\mathbb C)$, the situation is quite different: there are arbitrarily large finite families of isospectral non-commensurable compact manifolds covered by $\mathcal S$. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants.

math.SP↗
arXiv2005

Isospectral Cayley graphs of some finite simple groups

We apply spectral analysis of quotients of the Bruhat-Tits buildings of type $\tilde{A}_{d-1}$ to construct isospectral non-isomorphic Cayley graphs of the finite simple groups $\operatorname{PSL}_d({\mathbb F}_q)$ for every $d \geq 5$ ($d \neq 6$) and prime power $q > 2$.

math.CO↗