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Olga Kharlampovich

Publications and source records attributed to Olga Kharlampovich.

At least 19 recordsLinked to original sources
arXiv2026

Aleshin graphs are expanders

We prove that the Aleshin level graphs are two-sided expanders. We establish linear diameter bounds for Belaterra and Aleshin graph families. At level $n$, the diameters of the Bellaterra and undirected Aleshin graphs are a $|V_n|=2^n$. This improves the quadratic diameter bounds of Malyshev and Pak to the correct order of growth. For the Aleshin graphs we obtain a normalized spectral gap of at least $1-2\sqrt2/3$ and an edge-expansion constant of at least $3-2\sqrt2$, thereby establishing a Malyshev--Pak's Conjecture with explicit constants. An exact calculation at level ten also proves that the undirected Aleshin graphs are not Ramanujan for any $n\geq10$.

math.CO↗
arXiv2026

Model Theory and Algorithms for One-Relator Groups

We study the elementary theories and algorithmic properties of one-relator groups. We give a first-order sentence characterizing Baumslag-Solitar-freeness in this class. For fixed $|m|>1$, we regularly interpret $\BS(1,m)$ as a definable subgroup in every one-relator group containing it. We uniformly, regularly and injectively interpret integer arithmetic in countable groups of cohomological dimension at most two. We obtain one uniform interpretation for all one-relator groups containing a non-unimodular Baumslag--Solitar subgroup. Every fixed finitely presented group is regularly interpretable in these groups, and their elementary theories are undecidable. We also describe the GBS overgroups of a non-unimodular Baumslag--Solitar subgroup. For the unimodular groups we construct finite-index direct products of free groups and $\Z$, and reduce the elementary theory of $\BS(k,k)$ to that of a free group with a named periodic automorphism. Using the recent Open AI result that BS-free one-relator groups are hyperbolic we obtain a terminating algorithm recognizing this class and obtain consequences. From the virtual compact specialness result of another Open AI paper, we obtain an effective construction of virtual free-by-cyclic presentations for hyperbolic one-relator groups, the free kernel may have infinite rank. For classes of torsion-free one-relator groups in which CSA implies toral relative hyperbolicity, we give a uniform algorithm deciding CSA and, in the affirmative case, the existence of a subgroup isomorphic to $\Z^2$.

math.GR↗
arXiv2026

The Bellaterra graphs are Ramanujan

The Bellaterra automaton gives three involutions of every set of binary words of a fixed length. We identify their Schreier graph at level $n$ with a cubic arithmetic graph over $\Q(\sqrt{17})$, and prove that every nonconstant adjacency eigenvalue satisfies the Ramanujan bound. In particular, this proves Malyshev-Pak's Conjecture that the Bellaterra graphs form a family of two-sided expanders.

math.CO↗
arXiv2026

Uniform small cancellation in torsion-free hyperbolic groups

We establish a uniform small cancellation theorem for non-elementary torsion-free hyperbolic groups: given any prescribed small cancellation constant, finitely many fixed words suffice to produce a small cancellation element from every pair of noncommuting elements. As an application, we prove that the first-order theory of every non-elementary torsion-free hyperbolic group does not admit a model companion.

math.GR↗
arXiv2026

Equations in Products of Free Groups and 3-Manifold Groups II: Olshanskii Epimorphisms

In 1989 Olshanskii introduced a three-parameter family of coordinate-surjective homomorphisms from the genus-two surface group to a direct product of two rank-two free groups. When the common quotient of the two coordinate images is finite of order $n$, restriction to the corresponding regular cover produces an epimorphism \[ π_1(S_{n+1})\longrightarrow F_{n+1}\times F_{n+1}. \] We call these maps \emph{Olshanskii epimorphisms}. They form an explicit high-genus test family for the standardness problem for splitting epimorphisms. We prove that every Olshanskii epimorphism is standard. The genus-two homomorphism determines a Heegaard splitting of a Seifert fibered $3$-manifold over $S^2$ with at most three exceptional fibers. In the finite-quotient cases, classical Seifert theory shows that its universal cover is $S^3$; Waldhausen's theorem then implies that the lifted genus-$(n+1)$ Heegaard splitting is standard. The proof uses neither Perelman's theorem nor the general Poincaré theorem. We also give a constructive treatment of the quaternion case $Q(2,2,2)\cong Q_8$, whose covering surface has genus nine. A maximal tree in the quaternion Schreier graph yields the covering handlebody and explicit Schreier bases. Using geometric longitude--meridian pairs and explicit surface automorphisms supported on the nine one-holed tori, we transform the lifted meridian words into a free basis. A separate fixed-rank Andrews--Curtis certificate reduces the associated balanced presentation. This paper supplies the detailed proof of the result announced in the previous Kharlampovich, Vdovina paper.

math.GR↗
arXiv2026

Equations in Products of Free Groups and 3-Manifold Groups, I

Perelman's proof of the Poincare conjecture shows that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. The fundamental groups of 3-manifolds attract lots of interest from mathematicians of different fields. As it was stated in a famous survey of Allen Hatcher "The classification of 3-manifolds", one would want to know exactly which groups occur as fundamental groups of these manifolds. The Stallings-Jaco-Hempel reformulation of the Poincare conjecture inspired several connections between low-dimensional topology, equations over free groups, and combinatorial group theory. The reformulation reduces the problem to study epimorphisms from the fundamental group of a closed orientable surface onto the direct product of two free groups (they correspond to Heegaard splittings of 3-manifolds and were named splitting homomorphisms). Olshankii (1989) constructed (in non-explicit form) first non-trivial examples of such splitting epimorphisms and verified the standardness of some of them. We construct up to equivalence all the splitting coordinate-surjective homomorphisms (among them, the genuine splitting epimorphisms are exactly those for which our constructed associated group balanced presentation is trivial). We give generators and relations of the corresponding balanced presentation (so all closed orientable 3-manifold groups) that can be studied by algebraic methods. We analyse a big class of such homomorphisms/presentations (including all Olshanskii's epimorphisms) and show that splitting epimorphisms are rare, in this case the corresponding balanced presentation of the trivial group can be reduced to the standard one by Andrews-Curtis transformations and the epimorphisms are standard.

math.GR↗
arXiv2026

On the first-order genus of wreath products and their central extensions

We prove that groups of the form $\mathbb Z^m {\,\rm wr\,} \mathbb Z^n$, where $m,n \in \mathbb N$, are regularly bi-interpretable with $\mathbb Z$ and therefore are first-order rigid: every finitely generated group elementarily equivalent to $\mathbb Z^m {\,\rm wr\,} \mathbb Z^n$ is isomorphic to $\mathbb Z^m {\,\rm wr\,} \mathbb Z^n$. On the other hand, we show that $\mathbb Z^2 {\,\rm wr\,} \mathbb Z$ admits $2^{\aleph_0}$ elementarily equivalent, pairwise non-isomorphic central extensions with finite kernel.

math.GR↗
arXiv2025

The Diophantine problem in iterated wreath products of free abelian groups is undecidable

In this paper we prove that the Diophantine problem in iterated restricted wreath products $G$ of arbitrary non-trivial free abelian groups $A_1,\ldots, A_k$, $k>1$ of finite ranks is undecidable, i.e., there is no algorithm that given a finite system of group equations with coefficients in $G$ decides whether or not the system has a solution in $G$.

math.GR↗
arXiv2025

Quantifying separability in RAAGs via representations

We answer the question asked by Louder, McReynolds and Patel, and prove the following statement. Let L be a RAAG, H a word quasiconvex subgroup of L, then there is a finite dimensional representation of L that separates the subgroup H in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate H in L. This implies the same statement for a virtually special group L and, in particular, a fundamental groups of a hyperbolic 3-manifold.

math.GR↗
arXiv2023

Diophantine Problem in Some Metabelian Groups

In this paper we show that Diophantine problem for quadratic equations in Baumslag-Solitar groups $BS(1,k)$ and in wreath products $A \wr \mathbb{Z}$, where $A$ is a finitely generated abelian group and $\mathbb{Z}$ is an infinite cyclic group, is decidable.

math.GR↗
arXiv2023

Quantifying separability in limit groups via representations

We show that for any finitely generated subgroup $H$ of a limit group $L$ there exists a finite-index subgroup $K$ containing $H$, such that $K$ is a subgroup of a group obtained from $H$ by a series of extensions of centralizers and free products with $\mathbb Z$. If $H$ is non-abelian, the $K$ is fully residually $H$. We also show that for any finitely generated subgroup of a limit group, there is a finite-dimensional representation of the limit group which separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a limit group. This generalizes the results of Louder, McReynolds and Patel. Another corollary is that a hyperbolic limit group satisfies the Geometric Hanna Neumann conjecture.

math.GR↗
arXiv2022

Rich groups, weak second order logic, and applications

In this paper we initiate a study of first-order rich groups, i.e., groups where the first-order logic has the same power as the weak second order logic. Surprisingly, there are quite a lot of finitely generated rich groups, they are somewhere in between hyperbolic and nilpotent groups (these ones are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev's problems in various groups.

math.LO↗
arXiv2021

Regular completions of $\mathbb{Z}^n$-free groups

In the present paper we continue studying regular free group actions on $\mathbb{Z}^n$-trees. We show that every finitely generated $\mathbb{Z}^n$-free group $G$ can be embedded into a finitely generated $\mathbb{Z}^n$-free group $H$ acting regularly on the underlying $\mathbb{Z}^n$-tree (we call $H$ a {\em regular $\mathbb{Z}^n$-completion} of $G$) so that the action of $G$ is preserved. Moreover, if $G$ is effectively represented as a group of $\mathbb{Z}^n$-words then the construction of $H$ is effective and $H$ is also effectively represented as a group of $\mathbb{Z}^n$-words.

math.GR↗
arXiv2021

On countable elementary free groups

We prove that if a countable group is elementarily equivalent to a non-abelian free group and all of its abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).

math.LO↗