The subgroups of a tree product of groups

Abstract:

Let $G = {\Pi ^ \ast }({A_i};{U_{jk}} = {U_{kj}})$ be a tree product with $H$ a subgroup of $G$. By extending the technique of using a rewriting process we show that $H$ is an HNN group whose base is a tree product with vertices of the form $x{A_i}{x^{ - 1}} \cap H$. The associated subgroups are contained in vertices of the base, and both the associated subgroups of $H$ and the edges of its base are of the form $y{U_{jk}}{y^{ - 1}} \cap H$. The $x$ and $y$ are certain double coset representatives for $G\bmod (H,{A_i})$ and $G\bmod (H,{U_{jk}})$, respectively, and the elements defined by the free part of $H$ are specified. More precise information about $H$ is given when $H$ is either indecomposable or $H$ satisfies a nontrivial law. Introducing direct tree products, we use our subgroup theorem to prove that if each edge of $G$ is contained in the center of its two vertices then the cartesian subgoup of $G$ is a free group. We also use our subgroup theorem in proving that if each edge of $G$ is a finitely generated subgroup of finite index in both of its vertices and some edge is a proper subgroup of both its vertices then $G$ is a finite extension of a free group iff the orders of the ${A_i}$ are uniformly bounded.