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Over each nontrivial finite group $G$, there exists a finite system of equations having no solutions in larger finite groups but having a solution in a periodic group containing $G$. We prove several similar facts about amenable, orderable, locally indicable, solvable, nilpotent, and other classes of groups. As a byproduct, we also show that any amalgam of two countable periodic groups with finite intersection embeds into a periodic group, thereby answering a 1960 question of B. Neumann in the countable case.

We construct two CW-complexes $K$ and $L$ admitting a common, but not finite common, covering, where $K$ is homeomorphic to a complex with a single 2-cell., Comment: 6 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V3: details are added, an error and misprints are corrected --- thanks to an anonymous referee

Answering a question of A. V. Vasil'ev, we show that each finite symmetric (or alternating) group $H$ is a retract of any group containing $H$ as a verbally closed subgroup., Comment: 4 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: An argument is simplified (thanks to A. Yu. Olshanskii)

How complex two finite 2-complexes must be to admit a common, but not finite common, covering? We obtain an almost answer: the minimum possible number of triangles in a pseudo-simplicial triangulation of each complex is 3, 4, or 5., Comment: 5 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V3: An error (pointed out by an anonymous referee) is corrected

The assertion in the title implies that many interesting groups (e.g., all non-abelian braid groups or ${\bf SL}_{100}(\mathbb Z)$) are not strongly verbally closed, i.e., they embed into some finitely generated groups as verbally closed subgroups, which are not retracts., Comment: 6 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V4: minor misprint corrections

It is known that, if removing some $n$ edges from a graph $\Gamma$ destroys all subgraphs isomorphic to a given finite graph $K$, then all subgraphs isomorphic to $K$ can be destroyed by removing at most $|E(K)|\cdot n$ edges, which form a set invariant with respect to all automorphisms of $\Gamma$. We construct the first examples of (connected) graphs $K$ for which this estimate is not sharp. Our arguments are based on a ``weighted analogue'' of an earlier known estimate for the cost of symmetry., Comment: 5 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: minor corrections

Not any nonsingular equation over a metabelian group has solution in a larger metabelian group. However, any nonsingular equation over a solvable group with a subnormal series with abelian torsion-free quotients has a solution in a larger group with a similar subnormal series of the same length (and an analogous fact is valid for systems of equations)., Comment: 7 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V3: misprints corrected

The Universal Morphology (UniMorph) project is a collaborative effort providing broad-coverage instantiated normalized morphological inflection tables for hundreds of diverse world languages. The project comprises two major thrusts: a language-independent feature schema for rich morphological annotation and a type-level resource of annotated data in diverse languages realizing that schema. This paper presents the expansions and improvements made on several fronts over the last couple of years (since McCarthy et al. (2020)). Collaborative efforts by numerous linguists have added 67 new languages, including 30 endangered languages. We have implemented several improvements to the extraction pipeline to tackle some issues, e.g. missing gender and macron information. We have also amended the schema to use a hierarchical structure that is needed for morphological phenomena like multiple-argument agreement and case stacking, while adding some missing morphological features to make the schema more inclusive. In light of the last UniMorph release, we also augmented the database with morpheme segmentation for 16 languages. Lastly, this new release makes a push towards inclusion of derivational morphology in UniMorph by enriching the data and annotation schema with instances representing derivational processes from MorphyNet., Comment: LREC 2022; The first two authors made equal contributions

The main result includes as special cases on the one hand, the Gerstenhaber--Rothaus theorem (1962) and its generalisation due to Nitsche and Thom (2022) and, on the other hand, the Brodskii--Howie--Short theorem (1980--1984) generalising Magnus's Freiheitssatz (1930)., Comment: 6 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm V4: minor misprint corrections

We show, in particular, that, if a finite group $H$ is a retract of any finite group containing $H$ as a verbally closed subgroup, then the centre of $H$ is a direct factor of $H$., Comment: 13 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: Results concerning strong retracts are improved slightly. V3: minor errors in the last section are corrected. V4: misprints are corrected

How many two-cells must two finite CW-complexes have to admit a common, but not finite common, covering? Leighton's theorem says that both complexes must have two-cells. We construct an almost (?) minimal example with two two-cells in each complex., Comment: 5 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V.4: Some details were added

The Friedman--Mineyev theorem, earlier known as the (strengthened) Hanna Neumann conjecture, gives a sharp estimate for the rank of the intersection of two subgroups in a free group. We obtain an analogue of this inequality for any two subgroups in a virtually free group (or, more generally, in a group containing a free product of left-orderable groups as a finite-index subgroup)., Comment: 6 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V4: misprints corrected, grants updated

While language identification is a fundamental speech and language processing task, for many languages and language families it remains a challenging task. For many low-resource and endangered languages this is in part due to resource availability: where larger datasets exist, they may be single-speaker or have different domains than desired application scenarios, demanding a need for domain and speaker-invariant language identification systems. This year's shared task on robust spoken language identification sought to investigate just this scenario: systems were to be trained on largely single-speaker speech from one domain, but evaluated on data in other domains recorded from speakers under different recording circumstances, mimicking realistic low-resource scenarios. We see that domain and speaker mismatch proves very challenging for current methods which can perform above 95% accuracy in-domain, which domain adaptation can address to some degree, but that these conditions merit further investigation to make spoken language identification accessible in many scenarios., Comment: The first three authors contributed equally

It is known that the number of homomorphisms from a group $F$ to a group $G$ is divisible by the greatest common divisor of the order of $G$ and the exponent of $F/[F,F]$. We investigate the number of homomorphisms satisfying some natural conditions such as injectivity or surjectivity. The simplest nontrivial corollary of our results is the following fact: {\it in any finite group, the number of generating pairs $(x,y)$ such that $x^3=1=y^5$, is a multiple of the greatest common divisor of 15 and the order of the group $[G,G]\cdot\{g^{15}\;|\;g\in G\}$., Comment: 5 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: minor corrections. arXiv admin note: text overlap with arXiv:1806.08870

Given groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered., Comment: 11 pages, 4 figures. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: misprints corrected. V3: several corrections in the proofs --- the results remain unchanged. V4: A misprint is corrected

According to Mazhuga's theorem, the fundamental group $H$ of any connected surface, possibly except for the Klein bottle, is a retract of each finitely generated group containing $H$ as a verbally closed subgroup. We prove that the Klein bottle group is indeed an exception but has a very close property., Comment: 4 pages. V.2: A misprint corrected, thanks to the referee. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm

A broad goal in natural language processing (NLP) is to develop a system that has the capacity to process any natural language. Most systems, however, are developed using data from just one language such as English. The SIGMORPHON 2020 shared task on morphological reinflection aims to investigate systems' ability to generalize across typologically distinct languages, many of which are low resource. Systems were developed using data from 45 languages and just 5 language families, fine-tuned with data from an additional 45 languages and 10 language families (13 in total), and evaluated on all 90 languages. A total of 22 systems (19 neural) from 10 teams were submitted to the task. All four winning systems were neural (two monolingual transformers and two massively multilingual RNN-based models with gated attention). Most teams demonstrate utility of data hallucination and augmentation, ensembles, and multilingual training for low-resource languages. Non-neural learners and manually designed grammars showed competitive and even superior performance on some languages (such as Ingrian, Tajik, Tagalog, Zarma, Lingala), especially with very limited data. Some language families (Afro-Asiatic, Niger-Congo, Turkic) were relatively easy for most systems and achieved over 90% mean accuracy while others were more challenging., Comment: 39 pages, SIGMORPHON

The paper describes the results of the first shared task on morphological analysis for the languages of Russia, namely, Evenki, Karelian, Selkup, and Veps. For the languages in question, only small-sized corpora are available. The tasks include morphological analysis, word form generation and morpheme segmentation. Four teams participated in the shared task. Most of them use machine-learning approaches, outperforming the existing rule-based ones. The article describes the datasets prepared for the shared tasks and contains analysis of the participants' solutions. Language corpora having different formats were transformed into CONLL-U format. The universal format makes the datasets comparable to other language corpura and facilitates using them in other NLP tasks., Comment: 16 pages, 4 tables, 2 figures, published in the conference proceeding

Suppose that one can destroy all 100-gons in a graph by removing 2019 edges. How many edges must be removed to destroy all 100-gons in such a way that the set of removed edges is invariant with respect to all automorphisms the initial graph? This paper contains solutions to such kind of problems. Several open questions are raised., Comment: 11 pages, 9 figures. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V3: some misprints corrected and details added

This is a collection of teaching materials used in several Russian universities, schools, and mathematical circles. Most problems are chosen in such a way that in the course of the solution and discussion a reader learns important mathematical ideas and theories. The materials can be used by pupils and students for self-study, and by teachers. This is an abridged pre-copyedit version of the published book submitted with the permission of the publisher. Each included individual material is self-contained and ready-for-use. Solutions to problems are not included intentionally. This collection consolidates updates of several arXiv submissions, e.g., arXiv:1305.2598., Comment: Edited by A. Skopenkov, M. Skopenkov, A. Zaslavsky. In Russian. Optimized for printing on A5 paper. Moscow Center for Continuous Mathematical Education, 2018, 592pp (in Russian)

This note contains a (short) proof of the following generalisation of the Friedman--Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $rank(A\cap B)-1\leqslant n\cdot(rank(A)-1)\cdot(rank(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result., Comment: 3 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V.2: minor corrections and refinements

The Gordon--Rodriguez-Villegas theorem says that, in a finite group, the number of solutions to a system of coefficient-free equations is divisible by the order of the group if the rank of the matrix composed of the exponent sums of $j$-th unknown in $i$-th equation is less than the number unknowns. We obtain analogues of this and similar facts for algebraic groups. In particular, our results imply that the dimension of each irreducible component of the variety of homomorphisms from a finitely generated group with infinite abelianisation into an algebraic group $G$ is at least $\dim G$., Comment: 6 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V.2: An open question added, misprints corrected. V.3: misprints corrected

Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation $x^n=1$ in a group, the Solomon theorem (1969) on the number of solutions in a group to a system of equations having fewer equations than unknowns, and the Iwasaki theorem (1985) on roots of subgroups. There are other curious corollaries on groups and rings., Comment: 9 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: misprints corrected, an acknowledgement added. V3: misprints corrected

Any virtually free group $H$ containing no non-trivial finite normal subgroup (e.g., the infinite dihedral group) is a retract of any finitely generated group containing $H$ as a verbally closed subgroup., Comment: 6 pages. V2: minor corrections. V3: misprint corrections. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm

A theorem of Myasnikov and Roman'kov says that any verbally closed subgroup of a finitely generated free group is a retract. We prove that all free (and many virtually free) verbally closed subgroups are retracts in any finitely generated group., Comment: 5 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V.5: An error in an example is corrected (Section 2); thanks to Mikhail Mikheenko

It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation $[ x_1, y_1] \ldots [ x_k, y_k] = z^n$, where $n \ge 2k$, in a free product $\mathcal{F}$ of groups without nontrivial elements of order $\le n$ implies that $z$ is conjugate to an element of a free factor of $\mathcal{F}$. If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct., Comment: 14 pages, 5 figures. V3: minor corrections