arXiv subjects
Publications and source records attributed to Olga Kharlampovich.
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.
We prove that the first-order theory of nonabelian free groups does not admit a model companion.
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.
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.
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.
We show that a first-order sentence is almost surely true in a random group of density d<1/2 if and only if it is true in a non-abelian free group.
We describe groups elementarily equivalent to a free metabelian group with n generators. We also explore an exponentiation that naturally occurs in metabelian groups.
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 $G$.
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.
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.
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.
We prove that a random group, in Gromov's density model with $d < 1/16$ satisfies with overwhelming probability a universal-existential first-order sentence $σ$ (in the language of groups) if and only if $σ$ is true in a nonabelian free group.
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.
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.
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).
We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not $\forall$-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated elementary free groups, and non-abelian limit groups form special kinds of Fraïssé classes in which embeddings must preserve $\forall$-formulas. We also provide interesting examples of countable non-finitely generated elementary free groups.
We show that, given a finitely generated group $G$ as the coordinate group of a finite system of equations over a torsion-free hyperbolic group $Γ$, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from $G$ to $Γ$ as compositions of factorizations through $Γ$-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of $Γ$-limit groups as iterated generalized doubles over $Γ$.
The aim of this short note is to provide a proof of the decidability of the generalized membership problem for relatively quasi-convex subgroups of finitely presented relatively hyperbolic groups, under some reasonably mild conditions on the peripheral structure of these groups. These hypotheses are satisfied, in particular, by toral relatively hyperbolic groups.