To any semigroup presentation
and base word
may be associated a nonpositively curved cube complex
,
called a
Squier complex, whose underlying graph consists of the words of
equal to
modulo
, where two
such words are linked by an edge when one can be transformed into the other by applying a
relation of
.
A group is a
diagram group if it is the fundamental group of a Squier complex. We
describe hyperplanes in these cube complexes. As a first application, we determine exactly
when
is a special cube complex, as defined by Haglund and Wise, so that the associated
diagram group embeds into a right-angled Artin group. A particular feature of Squier
complexes is that the intersections of hyperplanes are “ordered” by a relation
. As a strong consequence
on the geometry of
,
we deduce, in finite dimensions, that its universal cover isometrically embeds into a
product of finitely many trees with respect to the combinatorial metrics; in
particular, we notice that (often) this allows us to embed quasi-isometrically the
associated diagram group into a product of finitely many trees, giving information
on its asymptotic dimension and its uniform Hilbert space compression.
Finally, we exhibit a class of hyperplanes inducing a decomposition of
as a
graph of spaces, and a fortiori a decomposition of the associated diagram group
as a graph of groups, giving a new method to compute presentations of
diagram groups. As an application, we associate a semigroup presentation
to any finite
interval graph ,
and we prove that the diagram group associated
to
(for a given base word) is isomorphic to the right-angled Artin group
.
This result has many consequences on the study of subgroups
of diagram groups. In particular, we deduce that, for all
, the right-angled
Artin group
embeds into a diagram group, answering a question of Guba and Sapir.
Keywords
diagram groups, CAT(0) cube complexes, special groups,
Squier complexes, right-angled Artin groups
