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Lev Glebsky

Publications and source records attributed to Lev Glebsky.

At least 19 recordsLinked to original sources
arXiv2023

Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Namely, we show that for (most) high-rank lattices, every finite-dimensional unitary "almost-representation" of $\Gamma$ is a small deformation of a (true) unitary representation. This extends a result of Kazhdan (1983) for amenable groups and of Burger-Ozawa-Thom (2013) for SL(n,Z) (for n>2). Towards this goal, we first build an elaborate cohomological theory capturing the obstruction to such stability, and show that the vanishing of second cohomology implies uniform stability in this setting. This cohomology can be roughly thought of as an asymptotic version of bounded cohomology, and sheds light on a question raised in Monod (2006) about a possible connection between vanishing of second bounded cohomology and Ulam stability.

math.GR↗
arXiv2019

On the number of even roots of permutations

Let $\sigma$ be a permutation on $n$ letters. We say that a permutation $\tau$ is an even (resp. odd) $k$th root of $\sigma$ if $\tau^k=\sigma$ and $\tau$ is an even (resp. odd) permutation. In this article, we obtain generating functions for the number of even and odd $k$th roots of permutations. Our result implies know generating functions of Moser and Wyman and also some generating functions for sequences in The On-line Encyclopedia of Integer Sequences (OEIS).

math.CO↗
arXiv2018

What is a true spectra of a finite Fourier transform

In this paper we deal with a finite abelian group $G$ and the abstract Fourier transform ${\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$. Then, we consider $\tilde{j}\circ {\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$ where $\tilde j:{\mathbb C}^\hat{G}\to {\mathbb C}^G$ is defined by the composition with a bijection $j:G\to \hat{G}$. ($\tilde j$ is a pullback of $j$.) In particular, we show that $(\tilde{j}\circ {\mathcal F})^2$ is a permutation if and only if $j$ is a group isomorphism. Then, we study how the spectra of $\tilde{j}\circ{\mathcal F}$ depends on the isomorphism $j$.

math.GR↗
arXiv2018

Unrestricted wreath products and sofic groups

We show that the unrestricted wreath product of a sofic group by an amenable group is sofic. We use this result to present an alternative proof of the known fact that any group extension with sofic kernel and amenable quotient is again a sofic group. Our approach exploits the famous Kaloujnine-Krasner theorem and extends, with an additional argument, to hyperlinear-by-amenable groups.

math.GR↗
arXiv2017

Stability, cohomology vanishing, and non-approximable groups

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\mathrm{Sym}(n)$ (in the sofic case) or the finite dimensional unitary groups ${\rm U}(n)$ (in the hyperlinear case)? In the case of ${\rm U}(n)$, the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by ${\rm U}(n)$ with respect to the Frobenius norm $\|T\|_{\rm{Frob}}=\sqrt{\sum_{i,j=1}^n|T_{ij}|^2},T=[T_{ij}]_{i,j=1}^n\in\mathrm{M}_n(\mathbb C)$. Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain non-residually finite central extensions of lattices in some simple $p$-adic Lie groups. These groups act on high rank Bruhat-Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

math.GR↗
arXiv2016

p-quotients of the G.Higman group

These notes are based on the mini-course "On the Graham Higman group", given at the Erwin Schr\"odinger Institute in Vienna, January 20, 22, 27 and 29, 2016, as a part of the Measured Group Theory program. The main purpose is to describe p-quotients of the Higman group $H(k)$ for $p|(k-1)$. (One may check that the condition $p|(k-1)$ is necessary for the existence of such quotients.)

math.GR↗
arXiv2015

Approximation of groups, characterizations of sofic groups, and equations over groups

We give new characterizations of sofic groups: -- A group $G$ is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups. -- A group $G$ is sofic if and only if any system of equations solvable in all alternating groups is solvable over $G$. The last characterization allows to express soficity of an existentially closed group by $\forall\exists$-sentences. Keywords: sofic groups, approximations, equations over groups.

math.GR↗
arXiv2012

A proof of Hilbert's Nullstellensatz based on Groebner bases

The aim of this note is to present an easy proof of Hilbert's Nullstellensatz using Groebner basis. I believe, that the proof has some methodical advantage in a course on Groebner bases. Key words: Hilbert's Nullstellensatz, Groebner bases.

math.AC↗
arXiv2011

Balanced $0,1$-words and the Galois group of $(x+1)^n-λx^p$

We study the number of $0,1$-words where the fraction of 0 is "almost" fixed for any initial subword. It turns out that this study use and reveal the structure of the Galois group (the monodromy group) of the polynomials $(x+1)^n-λx^p$. ($p$ is not necessary a prime here.)

math.CO↗
arXiv2010

Cycles in Repeated Exponentiation Modulo $p^n$

Given a number $r$, we consider the dynamical system generated by repeated exponentiations modulo $r$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod r$ and $0 \le f_g(u) \le r-1$. The number of cycles of the defined above dynamical system is considered for $r=p^n$.

math.NT↗
arXiv2009

Short Cycles in Repeated Exponentiation Modulo a Prime

Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system.

math.NT↗
arXiv2008

Sofic groups and profinite topology on free groups

We give a definition of weakly sofic groups (w-sofic groups). Our definition is rather natural extension of the definition of sofic groups where instead of Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non w-sofic groups is equivalent to some conjecture about profinite topology on free groups.

math.GR↗
arXiv2008

On low rank perturbation of matrices

The article is devoted to different aspects of the question "What can be done with a matrix by low rank perturbation?" It is proved that one can change a geometrically simple spectrum drastically by a rank 1 permutation, but the situation is quite different if one restricts oneself to normal matrices. Also, the Jordan normal form of a perturbed matrix is considered. It is proved that with respect to rank as a distance all almost unitary matrices are near unitary.

math.FA↗