Preservation of Trees by semidirect Products

Abstract

We show that the semidirect product of a group $C$ by $A{\ast }_{D}B$ is isomorphic to the free product of $A⋊C$ and $B⋊C$ amalgamated at $D⋊C$, where $A$, $B$ and $C$ are arbitrary groups. Moreover, we apply this theorem to prove that any group $G$ that acts without inversion on a tree $T$ that possesses a segment $\Gamma$ for its quotient graph, such that, if the stabilizers of the vertex set $\{P,Q\}$ and edge $y$ of a lift of $\Gamma$ in $T$ are of the form $G_P⋊H$, $G_Q⋊H$ and $G_y⋊H$, then $G$ is isomorphic to the semidirect product of $H$ by $(G_P{\ast }_{G_y}{G}_Q)$. Using our results we conclude with a non-standard verification of the isomorphism between $G{L}_2(\mathbb{Z})$ and the free product of the dihedral groups $D_4$ and $D_6$ amalgamated at their Klein-four group.