Let
be a connected, triangle-free, planar graph with at least five
vertices that has no separating vertices or edges. If the graph
is
, we prove that the
right-angled Coxeter group
is virtually a Seifert manifold group or virtually a graph manifold group and we give
a complete quasi-isometry classification of these groups. Furthermore, we prove that
is hyperbolic relative to a
collection of
right-angled
Coxeter subgroups of
.
Consequently, the divergence of
is linear, quadratic or exponential. We also generalize right-angled Coxeter groups
which are virtually graph manifold groups to certain high-dimensional right-angled
Coxeter groups (our families exist in every dimension) and study the coarse geometry
of this collection. We prove that strongly quasiconvex, torsion-free, infinite-index
subgroups in certain graph of groups are free and we apply this result to our
right-angled Coxeter groups.
