In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting
with a graph
,
we produce a new graph through a purely combinatorial procedure, and call it the extension
graph
of
. We produce a
second graph
, the
clique graph of
,
by adding an extra vertex for each complete subgraph of
. We prove that each
finite induced subgraph
of
gives rise to
an inclusion
.
Conversely, we show that if there is an inclusion
then
is an induced
subgraph of
.
These results have a number of corollaries. Let
denote the path on four
vertices and let
denote
the cycle of length
.
We prove that
embeds in
if and
only if
is an induced
subgraph of
. We
prove that if
is any
finite forest then
embeds in
.
We recover the first author’s result on co-contraction of graphs, and prove that if
has no triangles
and
contains
a copy of
for
some
, then
contains
a copy of
for some
.
We also recover Kambites’ Theorem, which asserts that if
embeds
in
then
contains an induced square. We show that whenever
is triangle-free and
then there is an
undistorted copy of
in
.
Finally, we determine precisely when there is an inclusion
and show that there is no “universal” two–dimensional right-angled Artin
group.
Keywords
right-angled Artin group, mapping class group, surface
group, co-contraction
