Abstract
Given groups
$A$
and
$B$
, what is the minimal commutator length of the 2020th (for instance) power of an element
$g\in A*B$
not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on
$A$
and
$B$
). Other similar problems are also considered.
We study congruences on the partial automorphism monoid of a finite rank free group action. We determine a decomposition of a congruence on this monoid into a Rees congruence, a congruence on a Brandt semigroup and an idempotent separating congruence. The constituent parts are further described in terms of subgroups of direct and semidirect products of groups. We utilize this description to demonstrate how the number of congruences on the partial automorphism monoid depends on the group and the rank of the action.
AbstractTwo subgroups 𝐴 and 𝐵 of a group 𝐺 are said to be 𝒩-connected if, for all 𝑎 in 𝐴 and 𝑏 in 𝐵, the subgroup generated by 𝑎 and 𝑏 is a nilpotent group.
In this paper, we study the structure of a group 𝐺 assuming that G=AB and 𝐴 and 𝐵 are 𝒩-connected subgroups satisfying Max or Min.
We prove that generalized free products of certain abelian subgroup separable groups are abelian subgroup separable. Applying this, we show that tree products of polycyclic-by-finite groups, amalgamating central subgroups or retracts are abelian subgroup separable.