Preservation of Trees by semidirect Products

TL;DR

This paper establishes a new structural isomorphism for semidirect products involving free products and applies it to groups acting on trees, including a novel proof of the isomorphism between $GL_2(Z)$ and an amalgamated free product of dihedral groups.

Contribution

It introduces a theorem relating semidirect products and free products with amalgamation, and applies it to analyze groups acting on trees, including a new proof of a classical isomorphism.

Findings

Semidirect product of a group with a free product is isomorphic to an amalgamated free product of semidirect products.

Groups acting on trees with certain stabilizer conditions are isomorphic to specific semidirect products.

Confirmed the isomorphism between $GL_2(Z)$ and the free product of dihedral groups $D_4$ and $D_6$ amalgamated at Klein-four group.

Abstract

We show that the semidirect product of a group $C$ by $A*_D B$ 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 $G_{P}\!\rtimes H$, $G_{Q}\!\rtimes H$ 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.