arXiv subjects
Publications and source records attributed to Artem Semidetnov.
Let $F$ be the fraction field of a discrete valuation ring. We give an explicit matrix in $\mathrm{SL}_2(F[X^{\pm1},Y^{\pm1}])\setminus \mathrm{E}_2(F[X^{\pm1},Y^{\pm1}])$. The proof of non-elementarity is based on techniques developed by Peter Abramenko in his preprint arXiv:0808.1095v1.
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.
We show that, for the words $r=\mathtt{aabABabABBAbaabABBAb}$ and $r'=\mathtt{aabABabABabABBAbaBAb}$, both in the commutator subgroup of the free group on $\mathtt{a}$ and $\mathtt{b}$, the one-relator groups $\langle \mathtt{a,b}\mid r=1\rangle$ and $\langle \mathtt{a,b}\mid r'=1\rangle$ are isomorphic, whereas $r$ and $r'$ have distinct stable commutator lengths. This provides a negative answer to a question posed by Heuer and Löh.
We introduce and study structured enhancement of the notion of a crossed simplicial group, which we call an operadic crossed simplicial group. We show that with each operadic crossed simplicial group one can associate a certain operad in groupoids. We demonstrate that symmetric and braid crossed simplicial groups can be made into operadic crossed simplicial groups in a natural way. For these two examples, we show that our construction of the associated operad recovers the $E_\infty$-operad and the $E_2$-operad respectively. We demonstrate the utility of this framework through two main applications: a generalized bar construction that specializes to Fiedorowicz's symmetric and braided bar constructions, and an identification of the associated group-completed monads with Baratt-Priddy-Quillen type spaces.
In this article, we study geometric properties of nilpotent groups. We find a geometric criterion for the word problem for the finitely generated free nilpotent groups. By geometric criterion, we mean a way to determine whether two words represent the same element in a free nilpotent group of rank $r$ and class $k$ by analyzing their behavior on the Cayley graph of the free nilpotent group of rank $r$ and class $k-1$.