Searcharxiv⌕ Search

arXiv subjects

Alexander Olshanskii

Publications and source records attributed to Alexander Olshanskii.

At least 19 recordsLinked to original sources
arXiv2026

Finite groups and rings generating varieties of rapid growth

Let $A$ be a finite universal algebra. Then the orders of the $n$-generated free algebras $F_n$ in the variety (equational class) generated by $A$ satisfy G. Birkhoff's inequality: $|F_n|\le |A|^{|A|^n}$ for $n=1,2,\dots$ It follows that $\limsup_{n\to\infty}\sqrt[n]{\log |F_n|}\le |A|$. When $A$ is a finite group or a finite nonassociative algebra, we obtain a criterion for equality in this estimate; equivalently, a criterion for maximal growth of the sequence $\{|F_n|\}_{n=1}^{\infty}$.

math.GR↗
arXiv2026

Locally finite varieties of nonassociative algebras

We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such as: nilpotence, solvability, simplicity, freeness, projectivity, and injectivity. We are also interested in the numerical estimates of the ratio of the number of algebras with various classical properties to the total number of all algebras of a fixed dimension $n$. Among these properties are having no proper nontrivial subalgebras or no nontrivial automorphisms, etc.

math.RA↗
arXiv2025

Linear average-case complexity of algorithmic problems in groups

The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case time complexity of the word problem in several classes of groups and show that it is often the case that the average-case complexity is linear with respect to the length of an input word. The classes of groups that we consider include groups of matrices over rationals (in particular, polycyclic groups), some classes of solvable groups, as well as free products. Along the way, we improve several bounds for the worst-case complexity of the word problem in groups of matrices, in particular in nilpotent groups. For free products, we also address the average-case complexity of the subgroup membership problem and show that it is often linear, too. Finally, we discuss complexity of the identity problem that has not been considered before.

math.GR↗
arXiv2024

Remembering Mark Sapir

This memorial article for Mark Sapir provides a brief overview of his life and career. Among his many contributions we highlight two of his most celebrated achievements: his groundbreaking solutions to Burnside-type problems for semigroups and his innovative construction of S-machines. Additionally, reflections from his colleagues and friends offer a heartfelt tribute, blending professional insights with personal memories.

math.HO↗
arXiv2024

On generic properties of nilpotent algebras

We study general nilpotent algebras. The results obtained are new even for the classical algebras, such as associative or Lie algebras. We single out certain generic properties of finite-dimensional algebras, mostly over infinite fields. The notion of being generic in the class of $n$-generated algebras of an arbitrary primitive class of $c$-nilpotent algebras appears naturally in the following way. On the set of the isomorphism classes of such algebras one can introduce the structure of an algebraic variety. As a result, the subsets are endowed with the dimensions as algebraic varieties. A subset $Y$ of a set $X$ of lesser dimension can be viewed as negligible in $X$. For example, if $n\gg c$, we determine that an automorphism group of a generic algebra $P$ consists of the automorphisms, which are scalar modulo $P^2$. Generic ideals are in $I(P)$, the annihilator of $P$. In the case of classical nilpotent algebras as above, the generic algebras are graded by the degrees with respect to some generating sets.

math.RA↗
arXiv2023

Steep uncountable groups

We produce a simple group $G$ of cardinality $\aleph_1$ which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset $Y \subseteq G$ there exists a natural number $n_Y$ for which every element of $G$ may be expressed as a product of length at most $n_Y$ of elements in $Y^{\pm 1}$. In particular this group is Jónsson (every proper subgroup is of strictly smaller cardinality) and strongly bounded (every abstract action on a metric space has bounded orbits); this is the first example of an uncountable group having both of these properties which is constructed without using the continuum hypothesis. The group $G$ can also be made so that all subgroups are simple and all nontrivial subgroups are malnormal in $G$.

math.GR↗
arXiv2022

Nilpotent algebras, implicit function theorem, and polynomial quasigroups

We study finite-dimensional nonassociative algebras. We prove the implicit function theorem for such algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.

math.RA↗
arXiv2019

Subgroups of groups finitely presented in Burnside varieties

For all sufficiently large odd integers $n$, the following version of Higman's embedding theorem is proved in the variety ${\cal B}_n$ of all groups satisfying the identity $x^n=1$. A finitely generated group $G$ from ${\cal B}_n$ has a presentation $G=\langle A\mid R\rangle$ with a finite set of generators $A$ and a recursively enumerable set $R$ of defining relations if and only if it is a subgroup of a group $H$ finitely presented in the variety ${\cal B}_n$. It follows that there is a 'universal' $2$-generated finitely presented in ${\cal B}_n$ group containing isomorphic copies of all finitely presented in ${\cal B}_n$ groups as subgroups.

math.GR↗
arXiv2014

On identities in the products of group varieties

Let ${\cal B}_n$ be the variety of groups satisfying the law $x^n=1$. It is proved that for every sufficiently large prime $p$, say $p>10^{10}$, the product ${\cal B}_p{\cal B}_p$ cannot be defined by a finite set of identities. This solves the problem formulated by C.K. Gupta and A.N. Krasilnikov in 2003. We also find the axiomatic and the basis ranks of the variety ${\cal B}_p{\cal B}_p$. For this goal, we improve the estimate for the basis rank of the product of group varieties obtained by G. Baumslag, B.H. Neumann, H. Neumann and P.M. Neumann long ago.

math.GR↗
arXiv2014

Subnormal subgroups in free groups, their growth and cogrowth

In this paper, the author (1) compares subnormal closures of finite sets in free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the limit of the n-th roots of g(n), where g(n) is the growth function of a subgroup H with respect to a finite free basis of F, exists for any subgroup H of the free group F; (3) gives sharp estimates from below for the e.g.r. of subnormal subgroups in free groups; and (4) finds cogrowth for the subnormal closures of free generators in F.

math.GR↗
arXiv2013

Growth of subalgebras and subideals in free Lie algebras

We investigate subalgebras in free Lie algebras, the main tool being relative growth and cogrowth functions. Our study reveals drastic differences in the behavior of proper finitely generated subalgebras and nonzero subideals. For instance, the \textit{growth} of a proper finitely generated subalgebra $H$ of a free Lie algebra $L$, with respect to any fixed free basis $X$, is exponentially small compared to the growth of the whole of $L$. Quite opposite, the \textit{cogrowth} of any nonzero subideal $S$ is exponentially small compared to the growth of $L$.

math.RA↗
arXiv2011

Space functions and complexity of the word problem in semigroups

We introduce the space function $s(n)$ of a finitely presented semigroup $S = .$ To define $s(n)$ we consider pairs of words $w,w'$ over $A$ of length at most $n$ equal in $S$ and use relations from $R$ for the transformations $w=w_0\to...\to w_t= w'$; $s(n)$ bounds from above the tape space (or computer memory) sufficient to implement all such transitions $w\to...\to w'.$ One of the results obtained is the following criterion: A finitely generated semigroup $S$ has decidable word problem of polynomial space complexity if and only if $S$ is a subsemigroup of a finitely presented semigroup $H$ with polynomial space function.

math.GR↗
arXiv2011

Filtrations and Distortion in Infinite-Dimensional Algebras

A tame filtration of an algebra is defined by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another. A geometric analogue is the distortion of the (Riemannian) metric of a (Lie) subgroup when compared to the metric induced from the ambient (Lie) group. The distortion of a subalgebra in an algebra also reflects the degree of complexity of the membership problem for the elements of this algebra in this subalgebra. One of our goals here is to investigate, mostly in the case of associative or Lie algebras, if a tame filtration of an algebra can be induced from the degree filtration of a larger algebra.

math.RA↗
arXiv2011

Subgroup Distortion in Wreath Products of Cyclic Groups

We study the effects of subgroup distortion in the wreath products A wr Z, where A is finitely generated abelian. We show that every finitely generated subgroup of A wr Z has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of A wr Z having distortion function equivalent to the given polynomial. Also a formula for the length of elements in arbitrary wreath product H wr G easily shows that the group Z_2 wr Z^2 has distorted subgroups, while the lamplighter group Z_2 wr Z has no distorted (finitely generated) subgroups.

math.GR↗
arXiv2010

Space functions of groups

We consider space functions $s(n)$ of finitely presented groups $G =< A\mid R> .$ (These functions have a natural geometric analog.) To define $s(n)$ we start with a word $w$ over $A$ of length at most $n$ equal to 1 in $G$ and use relations from $R$ for elementary transformations to obtain the empty word; $s(n)$ bounds from above the tape space (or computer memory) one needs to transform any word of length at most $n$ vanishing in $G$ to the empty word. One of the main obtained results is the following criterion: A finitely generated group $H$ has decidable word problem of polynomial space complexity if and only if $H$ is a subgroup of a finitely presented group $G$ with a polynomial space function.

math.GR↗
arXiv2010

Space functions of groups

We study the interrelation of space functions of groups and the space complexity of the algorithmic word problem in groups.

math.GR↗
arXiv2009

Actions of Maximal Growth

We study acts and modules of maximal growth over finitely generated free monoids and free associative algebras as well as free groups and free group algebras. The maximality of the growth implies some other specific properties of these acts and modules that makes them close to the free ones; at the same time, we show that being a strong "infiniteness" condition, the maximality of the growth can still be combined with various finiteness conditions, which would normally make finitely generated acts finite and finitely generated modules finite-dimensional.

math.GR↗