Preservation of Trees by semidirect Products

Abstract: We show that the semidirect product of a group $C$ by $A*<em>D B$ is isomorphic to the free product of $A\rtimes C$ and $B\rtimes C$ amalgamated at $D\rtimes C$, 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 $G</em>{P}!\rtimes H$, $G_{Q}!\rtimes H$ and $G_{y}! \rtimes H$, then $G$ is isomorphic to the semidirect product of $H$ by $(\,G_P \,*_{G_y} \,G_Q \,)$. Using our results we conclude with a non-standard verification of the isomorphism between $GL_2(\mathbb{Z})$ and the free product of the dihedral groups $D_4$ and $D_6$ amalgamated at their Klein-four group.