Systems of elements preserving measure on varieties of groups

E. I. Timoshenko

Sbornik. Mathematics, Tome 204 (2013) no. 12, pp. 1811-1818

Résumé

It is proved that for any $l$, $1\leqslant l\leqslant r$, a system of elements $ \{v_1,\dots,v_l\}$ of a free metabelian group $S$ of rank $r\geqslant2$ is primitive if and only if it preserves measure on the variety of metabelian groups $\mathfrak A^2$. From this we obtain the result that a system of elements $\{v_1,\dots,v_l\}$ is primitive in the group $S$ if and only if it is primitive in its profinite completion $\widehat{S}$. Furthermore, it is proved that there exist a variety $\mathfrak M$ and a nonprimitive element $v \in F_r(\mathfrak M)$ such that $v$ preserves measure on $\mathfrak M$. Bibliography: 13 titles.

Détail

Keywords: variety of groups, metabelian group, primitive system of elements, measure-preserving system of elements.
Mots-clés : soluble group