You are currently browsing opsiser’s articles.  Our goal is to
understand the structure of subgroups of graphs of groups. This
will enable us to prove things like: F \ast_{z} F is LERF.
Exercise 22. A group is coherent if every fg subgroup is fp.
Prove that \pi_1 of a finite graph of groups with coherent
vertex groups and cyclic edge group is coherent.  Recall, that
if H \subseteq \pi_1 \mathcal{G}=G is fg then H has an induced
graph of groups structure. \mathcal{H} =H \backslash T_H
\subseteq H \backslash T where T is the Bass-Serre tree of
\mathcal{G}.  Topologically, H defines a covering space X^H
\rightarrow X_g, which inherits a graph of spaces structure with
underlying graph H\backslash T.  \Gamma' =H \backslash T_H
\subseteq H \backslash T is a finite connected subgraph and the
image of \Gamma' in X^H is a connected subcomplex X' such that
\pi_1 X' = \pi_1 X^H =H. X' has the structure of a graph of
space with underlying graph \Gamma'  There is an explicit
algebraic description of H\backslash T, which