arXiv subjects

Anton Klyachko

Publications and source records attributed to Anton Klyachko.

2 recordsLinked to original sources

arXiv2025

Equationally separable classes of groups

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.

math.GR↗

arXiv2016

New topological methods to solve equations over groups

We show that the equation associated with a group word $w \in G \ast {\mathbf F}_2$ can be solved over a hyperlinear group $G$ if its content - that is its augmentation in ${\mathbf F}_2$ - does not lie in the second term of the lower central series of ${\mathbf F}_2$. Moreover, if $G$ is finite, then a solution can be found in a finite extension of $G$. The method of proof extends techniques developed by Gerstenhaber and Rothaus, and uses computations in $p$-local homotopy theory and cohomology of compact Lie groups.

math.GR↗