ON A CHARACTERIZATION OF FREE PRODUCTS OF GROUPS WITH COMMUTING SUBGROUPS BY THE TREES

In this paper, we prove that when a group $G$ acts on a tree $\Gamma$ such that the quotient graph $G/\Gamma$ is $path\ 2$ and when in $G$, any element of the stabilizer of one of the segments of this path commutes with those of the stabilizer of the other segment, $G$ may be identified with the free product of groups with commuting subgroups and every free product of groups with commuting subgroups is obtained uniquely in this way. Also, we give here some illustrative trees of this characterization.