Colouring hyperplanes in cube complexes
Victor Chepoi and Mark Hagen*
CAT(0) Spaces and Groups Kervaire Seminar, March 2012
Question
Which uniformly locally finite CAT(0) cube complexes embed in the
product of finitely many trees?
Druţu-Sageev: such embeddings exist for hyperbolic and relatively
hyperbolic cube complexes.
Cube complexes
Recall that a CAT(0) cube complex X is a simply-connected CW
complex such that
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Cells are Euclidean cubes of various dimensions, attached along
faces by combinatorial isometries.
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X satisfies Gromov’s nonpositive curvature condition on links.
Remark
The 1-skeleton X1 is a median graph. The class of median graphs
corresponds exactly to the class of 1-skeleta of CAT(0) cube complexes
(Chepoi, 2000). We always use the usual path-metric on X1 when
discussing metric properties of (subcomplexes of) X.
Hyperplanes I
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1-cubes c , c ′ are opposite if they are opposite 1-cubes in a
2-cube of X.
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The transitive closure of the relation ‘‘opposite’’ is called
‘‘parallelism’’ and is an equivalence relation on the 1-cubes.
Figure: Left: the red 1-cubes are opposite. Right: 1-cubes of the same colour
are parallel.
Hyperplanes II
▸
A hyperplane H is a subspace such that, for each cube
c ≅ [−1, 1]d , either H ∩ c = ∅ or H ∩ c is a midcube of c, i.e. a
subspace obtained by restricting exactly one coordinate to 0.
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Two 1-cubes are parallel if and only if they intersect the same
hyperplane.
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The carrier N (H ) of H is the smallest subcomplex containing H.
Figure: Hyperplanes and carriers.
Hyperplanes III
Theorem (Sageev, 1995)
Let H be a hyperplane of X. Then:
1. H is a CAT(0) cube complex of dimension at most dim(X) − 1.
2. H is convex.
3. H is two-sided, i.e. N (H ) ≅ H × [−1, 1].
4. H separates X into exactly two halfspaces, H + and H − .
Crossing hyperplanes
Definition
The hyperplanes H , H ′ cross if each of the ‘‘quarter-spaces’’ H ± ∩ (H ′ )±
is non-empty. Equivalently, there exists a 2-cube s ⊂ N (H ) ∩ N (H ′ ), two
of whose 1-cubes are dual to H and two of whose 1-cubes are dual to
H′.
Figure: Carriers of crossing hyperplanes.
The crossing graph
The crossing graph Γ# X of X has a vertex for each hyperplane.
Hyperplanes H , H ′ are adjacent if H and H ′ cross, i.e. H and H ′ are
adjacent in Γ# X if H ∩ H ′ ≠ ∅.
An example of a crossing graph
Figure: A CAT(0) cube complex and its crossing graph.
Crossing graphs of trees and products
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X is a tree if and only if Γ# X does not have edges.
Let X = X1 × X2 be a product of CAT(0) cube complexes. Then
Γ# X ≅ Γ# X1 ⋆ Γ# X2 is the join of the crossing graphs of the
factors. The converse also holds.
Embeddings in products of trees
Proposition
χ(Γ# X) denotes the chromatic number of Γ# X and τ (X) the smallest
t such that X embeds in the product of t trees. Then χ(Γ# X) = τ (X).
Embedding from colouring
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Partition the set H of hyperplanes into t disjoint families H1 , . . . Ht
such that each Hi is a pairwise disjoint (non-crossing) family.
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The wall structure on X induced by each Hi is dual to a tree Ti .
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The identity H → ⊔i Hi induces an isometric embedding X → ∏i Ti .
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Informally: colour hyperplanes so that crossing hyperplanes have
different colours, then force hyperplanes of different colours to
cross.
Example
Figure: Colouring hyperplanes to obtain an embedding in the product of trees.
Generalizing crossing
Definition (Contacting hyperplanes)
The distinct hyperplanes H , H ′ contact if no third hyperplane separates
them. Equivalently, N (H ) ∩ N (H ′ ) ≠ ∅.
Can happen in two ways:
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If H and H ′ cross, then H ∩ H ′ ≠ ∅, so H and H ′ contact.
If H and H ′ contact but do not cross, then they osculate: there is a
1-cube c dual to H and a 1-cube c ′ dual to H ′ , such that H , H ′
have a common 0-cube, and cc ′ does not form the corner of a
2-cube.
Osculating hyperplanes
Figure: A heuristic picture of the carriers of an osculating pair.
The contact graph
Definition (Contact graph)
The contact graph ΓX of X has a vertex for each hyperplane, and the
hyperplanes H , H ′ correspond to adjacent vertices if and only if they
contact.
Note:
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ΓX is a simplicial graph and contains Γ# X.
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The degree of X is equal to the cardinality of a largest clique in ΓX.
Colouring ΓX turns out to be more approachable than colouring Γ# X.
An example of a contact graph
Figure: A cube complex and its contact graph. Crossing-edges are black and
osculation-edges are purple.
Recubulating
Osculation and crossing are closely related:
Proposition (H., 2010; Chepoi-H., 2011)
Let X be a CAT(0) cube complex of dimension D and degree ∆. Then
there exists a CAT(0) cube complex X′ such that:
1. dim X′ = ∆.
2. X′ has degree at most ∆2 + ∆.
3. X isometrically embeds in X′ .
4. ΓX = Γ# X′ .
Recubulating
Figure: Top: the basic recubulation move. Bottom: recubulating a simple cube
complex. Black lines are 1-cubes; coloured lines are hyperplanes.
Colouring and embeddings into products of trees
Proposition
Let X be a CAT(0) cube complex, and let T = χ(ΓX). Then X
isometrically embeds in the product of at most T trees.
Proof.
ΓX contains Γ# X. Apply Proposition 0.5.
Proposition
Let X be a CAT(0) cube complex of finite degree, and let T = χ(ΓX).
Then there exists a CAT(0) cube complex Y, of finite degree and
dimension dim(X), that does not embed in the product of t trees for
t ≤ T.
Proof.
Recubulate and apply Proposition 0.5.
A negative result
Theorem (Chepoi-H.,2011)
There exists uniformly locally finite, 5-dimensional CAT(0) cube complex
X that does not embed in the product of finitely many trees.
1. Uses example due to Burling of a family of set systems in R3 whose
intersection graphs have bounded clique number but arbitrarily
large chromatic number.
2. Yields a CAT(0) cube complex whose ‘‘pointed contact graph’’
has infinite chromatic number (Chepoi, 2011).
3. Now recubulate to obtain X.
A positive result
Theorem (Chepoi-H.,2011)
Let X be a 2-dimensional CAT(0) cube complex of degree ∆. There
exists M such that X isometrically embeds in the product of at most
M ∆15 trees.
Proof: Colour ΓX with at most M ∆15 colours...
Strategy
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Fix a base hyperplane H0 (a vertex of ΓX). For each n ≥ 0, will
colour the n-sphere Sn ⊂ ΓX about H0 with χ colours. This gives a
2χ-colouring of ΓX.
Step 1: show by induction that the n-ball Bn is χn -colourable for
some χn < ∞.
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Step 2: show that χ(Sn ) ≤ χ4 for each n ≥ 0.
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Conclude that χ(ΓX) ≤ 2χ4 .
Step 2: Clusters I
Let H , H ′ ∈ Sn . Then H ∼ H ′ if there exists a path ρ in ΓX joining H to H ′
such that no vertex of ρ is in Sk for k ≤ n − 1.
Figure: Equivalent grade-n hyperplanes.
A grade-n cluster is a full subgraph of Sn generated by a ∼-class of
vertices.
Step 2: Clusters II
Theorem (H.,2010)
For any CAT(0) cube complex X, each cluster in ΓX has diameter at
most 4.
To colour Sn , it suffices to colour each cluster, by definition, and this
theorem tells us that for each cluster, we need at most χ4 colours.
Step 1: Grandfathers and weak combing
Let n ≥ 2. For each H ∈ Sn , choose a ‘‘minimal’’ geodesic γ(H ) in ΓX
joining H0 to H. (Minimality here is defined using ‘‘realizations’’ of such
paths in X, and, among minimal paths, one chooses γ(H ) arbitrarily.)
The hyperplane γ(H )(n − 2) is the grandfather of H.
Lemma (Weak combing)
If H , H ′ ∈ Sn contact, then their grandfathers either contact or coincide.
Step 1: Footprints and imprints
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If H contacts Hn−1 , then the footprint of H on Hn−1 is
N (H ) ∩ N (Hn−1 ).
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If H ′ also contacts Hn−1 , and H contacts H ′ , then the footprints of
H and H ′ on Hn−1 intersect (Helly property + convexity of
hyperplane carriers).
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The family of footprints on Hn−1 of the different H can be coloured
with 2∆ colours, since it corresponds to a collection of subtrees of
Hn . Let c1 (H ) be the colour of the footprint of H on Hn−1 .
Step 1: Fathers
For each H ∈ Sn , choose a hyperplane f (H ) ∈ Sn−1 in the following way:
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Let U be the grandfather of H, and fix a root vertex in the tree U.
Let f (H ) be a hyperplane that contacts H and U and has footprint
on U as close as possible to the root.
Choose f (H ) arbitrarily subject to these constraints. f (H ) is the
father of H.
Consequences for colouring
By induction, Sn−1 is coloured. If H , H ′ ∈ Sn , then:
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c1 (H ) is the colour of the grandfather of H.
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c2 (H ) is the colour of the father of H.
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c3 (H ) is the colour of the footprint of H on its father.
If H contacts H ′ , then one of the following happens:
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H and H ′ have different grandfathers. These contact, whence
c1 (H ) ≠ c1 (H ′ ).
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H and H ′ have the same father. Their footprints thus intersect,
whence c3 (H ) ≠ c3 (H ′ ).
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H and H ′ have different fathers, and the fathers contact, so
c2 (H ) ≠ c2 (H ′ ).
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H and H ′ have the same grandfather and distinct, non-contacting
fathers.
Step 1: the graph Υ(U )
If U ∈ Sn−2 , let Υ(U ) be the graph with:
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Vertices: hyperplanes H ∈ Sn with grandfather U.
Edges: H and H ′ are adjacent if they have different,
non-contacting fathers.
Remaining step is to colour Υ(U ): one distributes its edges over three
spanning subgraphs and colours each of these.
The final colouring
The grade-n hyperplane H has
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c1 (H ) -- the colour of its father, by induction on n.
c2 (H ) -- the colour of the footprint of H on its father, using facts
about colouring families of subtrees of a tree.
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c3 (H ) -- the colour of the grandfather, by induction.
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c4 (H ) -- the colour of H in Υ(U ), where U is the grandfather of H.
The colour of H is (c1 (H ), c2 (H ), c3 (H ), c4 (H )). This yields the desired
colouring of S4 .
Further questions
Let X be a uniformly locally finite CAT(0) cube complex. Can the
hyperplanes be finitely coloured when X 3- or 4-dimensional?
For which families F of convex subcomplexes of a cube complex X is
the intersection graph of F finitely colourable?