On the collection formulas for positive words

Vladimir M. Leontiev

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 3, pp. 365-377

Résumé

For any formal commutator $R$ of a free group $F$, we constructively prove the existence of a logical formula $\mathcal{E}_R$ with the following properties. First, if we apply the collection process to a positive word $W$ of the group $F$, then the structure of $\mathcal{E}_R$ is determined by $R$, and the logical values of $\mathcal{E}_R$ are determined by $W$ and the arrangement of the collected commutators. Second, if the commutator $R$ was collected during the collection process, then its exponent is equal to the number of elements of the set $D(R)$ that satisfy $\mathcal{E}_R$, where $D(R)$ is determined by $R$. We provide examples of $\mathcal{E}_R$ for some commutators $R$ and, as a consequence, calculate their exponents for different positive words of $F$. In particular, an explicit collection formula is obtained for the word $(a_1 \ldots a_n)^m$, $n,m \geqslant 1$, in a group with the Abelian commutator subgroup. Also, we consider the dependence of the exponent of a commutator on the arrangement of the commutators collected during the collection process.

Détail

Keywords: commutator, collection process, free group.