arXiv subjects
Publications and source records attributed to Vasily Ionin.
We give solutions to several problems in combinatorial group theory recorded as open in the literature. Our main source is the Kourovka Notebook, although we also consider questions from elsewhere. All solutions were found with the assistance of large language models and checked by the authors.
For every base field, we construct a finitely generated parafree augmented associative algebra with countably infinite-dimensional second homology. This disproves the analogue of the Parafree Conjecture for associative algebras and answers a question of Ivanov and Lopatkin. Our example is the monoid algebra of a finitely generated but not finitely presented submonoid of a free monoid.
We isolate a tractable class of HNN-extensions of a free group, namely, multiple HNN-extensions by basis-conjugating embeddings. For this class, we construct a normal form and establish a practical version of the ping-pong lemma that provides verifiable sufficient conditions for a set of elements to generate a free subgroup. We then apply these results to the pure braid group $P_{n+1}$, exploiting its well-known decomposition as a semidirect product of free groups. Our approach yields new families of free subgroups within the first two factors $F_n \rtimes F_{n-1}$ of this decomposition.
There is a general phenomenon in algebra that numerous functors of homological significance admit characterization as derived limits of elementary functors defined over categories of free extensions. We demonstrate that upon restriction to appropriate subcategories of the category of groups, one may express analogously more interesting functors, including homology groups with cyclic coefficients. Moreover, we are laying the foundations of the so-called $\mathbf{fr}_\infty$-language, extending the $\mathbf{fr}$-language of Roman Mikhailov and Sergei O. Ivanov. This language is constructed by augmenting the $\mathbf{fr}$-language through the introduction of an infinite family of letters $\mathbf{fr}$ corresponding to the lower central series $γ_m(R)$ of the group of relations and leads to some neat computations.
We give a new elementary proof of the theorem that a natural map from Milnor's construction $F[S^1]$ to the simplicial group $\mathrm{AP}$ of pure braids is injective. Our approach is group-theoretic and does not rely on Lie algebras.
We examine the Moore complex of the Delta-group structure related to the pure braid groups and introduced by Berrick, Cohen, Wong, and Wu. We prove that the cycle and the boundary groups are invariant under all automorphisms of the pure braid groups, and thereby, we extend the results of Li and Wu on the reflection automorphism. We conclude that there is an induced action of all automorphisms of the pure braid groups on the homotopy groups of the two-sphere. Besides, we compute this action for a small number of strands.