arXiv subjects
Publications and source records attributed to Alexander Lubotzky.
We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random walks on the mapping class group of a closed orientable surface.
We show that the high dimensional expansion property as defined by Gromov, Linial and Meshulam, for simplicial complexes is a form of testability. Namely, a simplicial complex is a high dimensional expander iff a suitable property is testable. Using this connection, we derive several testability results.
Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are LDPC codes with linear rate and distance $n^ε$. Their rate is evaluated via Euler characteristic arguments and their distance using $\mathbb{Z}_2$-systolic geometry. This construction answers a queston of Zémor, who asked whether homological codes with such parameters could exist at all.
Let X be a 2-dimensional simplicial complex. The degree of an edge e is the number of 2-faces of X containing e. The complex X is an ε-expander if the coboundary d_1(ϕ) of every Z_2-valued 1-cochain ϕ\in C^1(X;Z_2) satisfies |support(d_1(ϕ))| \geq ε|\supp(ϕ+d_0(ψ))| for some 0-cochain ψ. Using a new model of random 2-complexes we show the existence of an infinite family of 2-dimensional ε-expanders with maximum edge degree d, for some fixed ε>0 and d.
Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last three decades with many applications in computer science, combinatorics and even pure mathematics. In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes.
Let $2 \leq a \leq b \leq c \in \mathbb{N}$ with $μ=1/a+1/b+1/c<1$ and let $T=T_{a,b,c}=< x,y,z: x^a=y^b=z^c=xyz=1>$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$? (Classically, for $(a,b,c)=(2,3,7)$ and more recently also for general $(a,b,c)$.) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove a conjecture of Marion [21] showing that various finite simple groups are not quotients of $T$, as well as positive results showing that many finite simple groups are quotients of $T$.
This paper is a continuation of our first paper [10] in which we showed how deformation theory of representation varieties can be used to study finite simple quotients of triangle groups. While in Part I, we mainly used deformations of the principal homomorphism from ${\rm SO}(3,\R)$, in this part we use ${\rm PGL}_2(\R)$ as well as deformations of representations which are very different from the principal homomorphism.
In this note we show that various (geometric/homological) finiteness properties are not profinite properties. For example for every $1 \le k, \ell \le \bbn$, there exist two finitely generated residually finite groups $\Ga_1$ and $\Ga_2$ with isomorphic profinite completions, such that $\Ga_1$ is strictly of type $F_k$ and $\Ga_2$ of type $F_\ell$.
In response to questions by Kassabov, Nikolov and Shalev, we show that a given subset $A$ of a finite simple group $G$ is the image of some word map $w : G\times G\to G $ if and only if (i) $A$ contains the identity and (ii) $A$ is invariant under $\Aut (G)$.
Let F be a finitely generated field of characteristic zero and Γ<GL_n(F) a finitely generated subgroup. For an element g in Γ, let Gal(F(g)/ F) be the Galois group of the splitting field of the characteristic polynomial of g over F. We show that the structure of Gal(F(g)/ F) has a typical behaviour depending on F, and on the geometry of the Zariski closure of Γ(but not on Γ).
Several different areas of group theory, topology and geometry have led to the study of the action of Aut(Fn) | the automorphism group of the free group on n generators | on Hom(Fn;G) when G is either finite,compact or simple Lie group. In this survey, we describe these topics and results, with special emphasis on some similarities and with an effort to give a somewhat uniform treatment. This perspective sometimes suggests new questions, conjectures and methods borrowed from one area to another.
We present a symmetric LDPC code with constant rate and constant distance (i.e. good LDPC code) that its constraint space is generated by the orbit of one constant weight constraint under a group action. Our construction provides the first symmetric LDPC good codes. This solves the main open problem raised by Kaufman and Wigderson in [4].
Let G be a finitely generated group and M_n(G) the number of its normal subgroup subgroups of index at most n. For linear groups G we show that M_n(G) can grow polynomially in n only if the semisimple part of the Zariski closure of G has simple components only of type G2, F4 or E8 (and in this case indeed this can happened!)
We prove that the set of non-pseudo-Anosov elements in the Torelli group is exponentially small.
A general sieve method for groups is formulated. It enables one to "measure" subsets of a finitely generated group. As an application we show that if $Γ$ is a finitely generated non virtually-solvable linear group of characteristic zero then the set of proper powers in $Γ$ is exponentially small. This is a far reaching strengthening of the main result of \cite{HKLS}.
Let $π:\aut(F_n)\rightarrow \aut(\Z^n)$ be the epimorphism induced by the isomorphism $\Z^n \cong F_n/F_n'$ and define $\mathcal{T}_n:=\kerπ$. We prove that the subset of $\mathcal{T}_n$ consists of all non-iwip and all non-hyperbolic elements is exponentially small.
We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GL_d have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.
In "Non arithmetic super rigid groups: counter examples to Platonov's conjecture" Bass and Lubotzky gave a counter example to Platonov's conjecture by presenting an example of a linear group with super-rigidity which is not an arithmetic lattice. In this note, a much richer class of such groups is presented with a somewhat simpler proof.