Divergence and quasimorphisms of right-angled Artin groups

0.22 MB·English

by  Jason Behrstock

DIVERGENCE AND QUASIMORPHISMS OF RIGHT-ANGLED
ARTIN GROUPS
0
1 JASONBEHRSTOCKANDRUTHCHARNEY
0
2
Abstract. We give a group theoretic characterization of geodesics with su-

n perlinear divergence in the Cayley graph of a right-angled Artin group AΓ
a with connected defining graph. We usethis todetermine when two points in
J anasymptoticconeofAΓ areseparatedbyacut-point. Asanapplication,we

6 showthatifΓdoesnotdecomposeasthejoinoftwosubgraphs,thenAΓ has
2 an infinite-dimensional space of non-trivial quasimorphisms. By the work of
BurgerandMonod,thisleadstoasuperrigiditytheorem forhomomorphisms

] fromlattices intoright-angledArtingroups.
R
G
.
h
t 1. Introduction
a
m The divergence of a geodesic, γ: [−∞,∞] →X, in a metric space, can roughly
[ be thought of as the growth rate of a function from N to R, which for each N ∈N

gives the length of the shortest path in X\B (γ(0)) from γ(−N) to γ(N), where
2 N
B (γ(0)) is the open ball of radius N about γ(0). We refer to the divergence of
v N
7 a finitely generated group to mean the largest divergence over all geodesics in a
8 Cayley graph of G.
5 Thedivergencefunctionhasproventobeausefultoolinstudyingthelargescale
3
geometry of groups. Gersten classified geometric 3–manifolds by their divergence
.
1 [Ger] which allows one to distinguish quasi-isometry classes of 3–manifold groups
0
containing hyperbolic pieces from graph manifold groups [KL]. In addition, diver-
0
gencefunctionsarecloselyrelatedtocut-pointsintheasymptoticconesofagroup.
1
Interest in the existence of such cut-points arose from Dru¸tu–Osin–Sapir’s result
:
v
that a group is relatively hyperbolic with respect to a collection of subgroups H if
i
X and only if every asymptotic cone has a collection of cut-points with the property
r thatthe maximalsubsetsofthe asymptotic conenotseparatedbyanyone ofthese
a cut-points arise from asymptotic cones of the subgroups H [DS].
On the other hand, cut-points in asymptotic cones also arise in groups which
are not relatively hyperbolic. To prove that any point in an asymptotic cone of a
mapping class group is a cut-point, the first author showed that axes of pseudo-
Anosov elements in a mapping class group have super-linear divergence. This also
implies that these directions are quasi-geodesically stable, or equivalently, Morse
geodesics. [Beh]. (Alternate proofs have since been obtained by [DMS] and [DR]).
More recently, Dru¸tu–Mozes–Sapir showed in [DMS] that a group has superlinear
divergence if and only if its asymptotic cones contain cut-points, and that this
occurs if and only if the group contains Morse geodesics.
Date:January2010.
R.CharneywaspartiallysupportedbyNSFgrantDMS0705396.
J.BehrstockwaspartiallysupportedbyNSFgrantDMS0812513.
1
2 JASONBEHRSTOCKANDRUTHCHARNEY
In this paper we discuss divergence in right-angledArtin groups. Given a finite,
simplicial graph Γ, the right-angled Artin group A is the finitely presented group
Γ
with generators corresponding to vertices of Γ and relators of the form x−1y−1xy
whenever the vertices x and y of Γ are connected by an edge. Right-angled Artin
groupsformarichfamilyofgroupsinterpolatingbetweenZn,thegroupcorrespond-
ing to the complete graph on n vertices, and the free group F , corresponding to
n
the graph with n vertices and no edges.
IfΓ andΓ aretwographs,theirjoin isthegraphobtainedbyconnectingevery
1 2
vertex of Γ to every vertex of Γ by an edge. Subgraphs of Γ that decompose as
1 2
joins are central to understanding divergence of geodesics. We define a notion of
join length of a geodesic, which measures the number of cosets of join subgroups
the geodesic passes through(see Section 3 for the precise definition) and we prove,
Theorem 4.4 (Divergence and join length). Let Γ be a connected graph and
let α be a bi-infinite geodesic in A . Then α has linear divergence if and only if the
Γ
join length of α is finite.
The proof uses the action of A on a CAT(0) cube complex, X , the universal
Γ Γ
cover of the Salvetti complex of A . We show that the join length of a geodesic α
Γ
determines the behavior of the walls in X crossed by α.
Γ
Fromthe divergencetheorem,we obtainthefollowingcompletecharacterization
of when two points in an asymptotic cone of a right-angled Artin group can be
separated by a cut-point.
Theorem 4.6 (Classification of pieces). Let Γ be a connected graph. Fix a pair
of points a,b∈Aω. The following are equivalent.
Γ
(1) No point of Aω separates a from b.
Γ
(2) There exist points a′,b′ arbitrarily close to a,b, respectively for which the
join length between a′,b′ is finite.
In the terminology of [DS], cut-points in an asymptotic cone give rise to a tree-
grading whosepieces arethe maximalsubsets thatcannotbe separatedbyapoint.
The above result gives a complete description of the pieces in Aω. Since right-
Γ
angled Artin groups are not relatively hyperbolic [BDM], these pieces do not arise
by taking asymptotic cones of subgroups of A [DS].
Γ
In [BDM], Behrstock–Dru¸tu–Mosher introduce a notion of algebraic thickness
of a group. Theorem 4.6 shows that for a connected graph Γ, A is algebraically
Γ
thick of order zero if Γ is a join, and otherwise it is algebraically thick of order at
least one with respect to the set of maximal join subgroups. It was established in
[BDM,Corollary10.8],that,exceptforZ,right-angledArtingroupswithconnected
presentationgrapharethickoforderatmostone. Together,thesetworesultsshow
thatifA isajoin,thenitisalgebraicallythickoforderexactlyzero,andotherwise
Γ
it is algebraically thick of order exactly one.
Our main application of divergence is to show that subgroups of right-angled
Artin groups have many non-trivial quasi-morphisms. A function φ: G → R is
a homogeneous quasimorphism if φ(gn) = nφ(g) for all n > 0, and there exists a
constant D ≥0 such that
|φ(gh)−φ(g)−φ(h)|≤D
for every g,h∈G. The vector space of homogeneous quasimorphisms, modulo the
subspaceoftruehomomorphisms,isdenotedQH(G)andisrelatedtothebounded
g
DIVERGENCE AND QUASIMORPHISMS OF RIGHT-ANGLED ARTIN GROUPS 3
cohomology of G. Bestvina and Fujiwara [BF2] have shown that for group actions
ona CAT(0) space,satisfyinga weakdiscontinuity property,the existence ofrank-
one isometries (i.e., hyperbolic isometries with an axis not bounding a half-plane)
gives rise to non-trivial quasimorphisms. Using their results we prove,
Theorem 5.2 (Rank-one geodesics and Quasimorphisms). If G ⊆ A is
Γ
any non-cyclic, finitely generated subgroup which is not contained in a conjugate of
a join subgroup, then G contains an element which acts as a rank-one isometry of
X . In this case, QH(G) is infinite dimensional.
Γ g
Right-angled Artin groups have been shown to have an extremely rich family
of subgroups, cf. [BB], [HW], [CW]. In contrast, using Theorem 5.2 and Burger-
Monod’s result on nonexistence of quasimorphisms on higher rank lattices [BM1,
BM2, Mon1], we deduce:
Corollary 5.3 (Superrigidity with RAAG image). Let Λ be an irreducible
lattice in a connected semisimple Lie group with finite center, no compact factors,
and rank at least 2. Then for any right-angled Artin group A , every homomor-
Γ
phism ρ: Λ→A is trivial.
Γ
To the best of our knowledge this is the most general statement of superrigid-
ity for right-angled Artin groups, although many special cases follow from other
methods. For example, for lattices satisfying Kazhdan’s Property (T), Niblo–
Reeves[NR]showedthateveryactiononafinitedimensionalCAT(0)cubecomplex
has a global fixed point. Since A acts freely on the cube complex X , any ho-
Γ Γ
momorphism of such a lattice into A must be trivial. For non-uniform lattices,
Γ
superrigidity follows from the Margulis Normal Subgroup Theorem [Mar, Zim],
since every solvable subgroup of A is virtually abelian. Other special cases follow
Γ
from the work of Shalom [Sha], Monod [Mon2], and Gelander–Karlsson–Margulis
[GKM].
From Theorem 5.2 we also deduce the following, which P. Dani informed us she
hasindependently establishedinjointworkwithA.Abrams,N.Brady,M.Duchin,
A. Thomas and R. Young.
Corollary 5.4 (Quadratic divergence). Let Γ be a connected graph. A has
Γ
linear divergence if and only if Γ is a join; otherwise its divergence is quadratic.
The authors would like to thank Koji Fujiwara and Yehuda Shalom for helpful
conversations. Also, Behrstock would like to thank Brandeis University and Char-
ney wouldlike to thank the Forschungsinstitutfu¨r Mathematik atETH,Zurichfor
their hospitality during the development of this paper.
2. Walls
Let Γ be a finite, simplicial graph with vertex set V. The right-angled Artin
group associated to Γ is the group A with presentation
Γ
A =hV |vw =wv if v and w are connected by an edge in Γi.
Γ
Associated to any right-angled Artin group A is a CAT(0) cubical complex X
Γ Γ
withafreeactionofA . InthissectionwedescribeX andinvestigatethestructure
Γ Γ
of walls in this complex.
Let T be a torus ofdimension |V| with edges labelled by the elements of V. Let
X denote the subcomplex of T consisting of all faces whose edge labels span a
Γ
4 JASONBEHRSTOCKANDRUTHCHARNEY
complete subgraph in Γ (or equivalently, mutually commute in A ). X is called
Γ Γ
the Salvetti complex for A . It is easy to verify that the Salvetti complex has
Γ
fundamental group A and that the link of the unique vertex is a flag complex.
Γ
It follows that its universal cover, X , is a CAT(0) cube complex with a free,
Γ
cocompact action of A .
Γ
If Γ′ is a full subgraph of Γ, then the inclusion Γ′ → Γ induces an injective
homomorphism AΓ′ → AΓ and an embedding XΓ′ → XΓ. This embedding is
locallygeodesic,andhence(sinceX CAT(0))itisgloballygeodesic. Wemaythus
Γ
view XΓ′ as a convex subspace of XΓ.
We now recall some basic facts about walls in a CAT(0) cube complex. A wall
(orhyperplane) inaCAT(0) cube complex, X,is anequivalence classofmidplanes
ofcubeswheretheequivalencerelationisgeneratedbytherulethattwomidplanes
arerelatediftheyshareaface. EachwallisageodesicsubspaceofX andseparates
X into two components. Moreover, if a wall contains a (positive length) segment
of a geodesic γ, then it contains the entire geodesic γ.
In the discussion that follows, we are interested in the relation between non-
intersecting pairs of walls. The following terminology will be convenient.
Definition2.1. TwowallsH ,H inaCAT(0)cubecomplexarestronglyseparated
1 2
if H ∩H =∅ and no wall intersects both H and H .
1 2 1 2
Consider this definition in the context ofa right-angledArtin groupA and the
Γ
cube complex X . For example, suppose Γ consists of two disjoint edges, so A is
Γ Γ
thefreeproductZ2∗Z2. Inthiscase,the Salvetticomplexisthewedgeoftwotori,
and its universal cover X consists of flats which pairwise intersect in at most one
Γ
vertex. Since walls never contain vertices of X , they remain entirely in one flat.
Γ
It follows that two walls are strongly separated if and only if they lie in different
flats.
At the other extreme, suppose Γ is a square, in which case A = F ×F , the
Γ 2 2
product of two free groups of rank 2, and X is the product of two trees T ×T .
Γ 1 2
The walls consist of trees of the form T ×{y} and {x}×T where x and y are
1 2
midpointsofedgesinT andT respectively. Itisnoweasytoseethatnotwowalls
1 2
are strongly separated.
Now let A be an arbitrary right-angledArtin group and let H and H be two
Γ 1 2
walls in X . Consider the set of all minimal length geodesics from H to H . It
Γ 1 2
followsfrom[BH,PropositionII.2.2]thattheunionofallsuchpathsformsaconvex
subspace of X which we call the bridge between H and H .
Γ 1 2
Lemma 2.2. If H and H are strongly separated, then the bridge B between them
1 2
consists of a single geodesic from H to H .
1 2
Proof. It suffices to show that B∩H (and by symmetry B∩H ) is a single point.
1 2
We first show that B ∩ H does not intersect any other wall H. For suppose
1
x∈B∩H ∩H. Sincex∈B,itistheinitialpointofsomeminimallengthgeodesic
1
γ fromH toH . Theinitialsegmentofγ liesinsomecubeσ ofX whichcontains
1 2 Γ
midplanes in both H and H. These midplanes span σ, hence the initial segment
1
ofγ,whichis orthogonalto H , mustlie inH. Itfollowsthat allofγ lies inH and
1
hence H ∩H 6= ∅. This contradicts the assumption that H and H are strongly
2 1 2
separated.
DIVERGENCE AND QUASIMORPHISMS OF RIGHT-ANGLED ARTIN GROUPS 5
Now every wall H has an open neighborhood N(H) isometric to H × (0,1),
namely the neighborhood consisting of the interiors of all cubes containing a mid-
planeinH. Thenthesameargumentasabove(usingparallelcopiesofH inN(H))
shows that B∩H ∩N(H) must also be empty for all H 6= H . The only convex
1 1
subsetsofH disjointfromeveryN(H)aresinglevertices,soitfollowsthatB∩H
1 1
is a single point. (cid:3)
Lemma 2.3. There are universal constants C,D > 1 (depending only on the di-
mension of X ) such that if H and H are strongly separated and B is the bridge
Γ 1 2
between them, then
(1) for any x∈H and y ∈H ,
1 2
d(x,y)≥C−1(d(x,B)+d(y,B))−d(H ,H )−4
1 2
(2) for any geodesic α in X , and any point c on α, if H and H intersect α
Γ 1 2
inside the ball of radius r about c, then the bridge B is contained in the ball
of radius Dr about c.
Proof. (1)For any twopoints x,y inX , define d (x,y) to be the number ofwalls
Γ H
separatingx andy, orequivalently,the number ofwallscrossedby a geodesicfrom
xtoy. Thisdistancefunctionisquasi-isometrictothegeodesicmetricinX . More
Γ
precisely,
d(x,y)−C ≤d (x,y)≤Cd(x,y)+C
H
where C is the diameter of a maximal cube.
By Lemma 2.2, B consists of a single geodesic γ from H to H . Let h ,h be
1 2 1 2
the endpoints of γ. Let α be the geodesic from h to x, and β the geodesic from
1
y to h . Note that α lies in H and β lies in H . Since the strongly separated
2 1 2
hypothesisguaranteesthatno wallcrossesboth αandβ, the pathαγβ crossesany
givenwallatmosttwiceandd (x,y)isthenumberofwallsitcrossesexactlyonce.
H
It follows that
d (x,y)≥d (x,h )+d (y,h )−d (h ,h ).
H H 1 H 2 H 1 2
Applying the inequalities above, we obtain
d(x,y) ≥C−1d (x,y)−1
H
≥C−1(d (x,h )+d (y,h )−d (h ,h ))−1
H 1 H 2 H 1 2
≥C−1(d(x,h )+d(y,h )−2C)−d(h ,h )−2
1 2 1 2
≥C−1(d(x,B)+d(y,B))−d(H ,H )−4.
1 2
(2)Supposex=H ∩αandy =H ∩αareintheballofradiusr aboutc. Then
1 2
every point in B is within k = 1(d(x,B)+d(y,B))+d(H ,H ) of either x or y
2 1 2
and hence within k+r of c. By part (1), d(x,B)+d(y,B) is bounded by a linear
function of d(x,y), and by hypothesis, d(H ,H ) ≤ d(x,y) ≤ 2r so k is bounded
1 2
by a linear function of r. (cid:3)
We now introduce the notion of divergence for bi-infinite geodesics and discuss
how the existence of strongly separated walls affects the divergence.
Definition 2.4. Let X be a geodesic metric space. Let α: R→X be a bi-infinite
geodesic in X and let ρ be a linear function ρ(r) = δr −λ with 0 < δ < 1 and
λ≥0. Define div(α,ρ)(r) = length of the shortest path from α(−r) to α(r) which
stays outside the ball of radius ρ(r) about α(0) (or div(α,ρ)(r) = ∞ if no such
6 JASONBEHRSTOCKANDRUTHCHARNEY
path exists). We say α has linear divergence if for some choice of ρ, div(α,ρ)(r) is
boundedbyalinearfunctionofr,andthatαhassuper-linear divergence otherwise.
It is not difficult to verify that the definition of linear divergence is independent
of the choice of basepoint α(0). We leave this as an exercise for the reader.
Theorem 2.5. Let α be a bi-infinite geodesic in X and let H be the sequence of
Γ
walls crossed by α. Suppose H contains an infinite subsequence {H ,H ,...} such
0 1
that for all i, H is strongly separated from H . Then H is strongly separated
i i+1 i
from H for all i6=j and α has superlinear divergence.
j
Proof. Let α+ and α− denote the limit points of α in ∂X . Since H is disjoint
Γ i
fromH , the half spacesH+ containingα+ formadirectedsetH+ ⊂H+ ⊂....
i+1 i 0 1
Hence no two of these walls intersect and if some wall H intersects both H and
i
H , i<j, then it must cross H , contradicting the strong separation of H from
j i+1 i
H . It follows that H and H are strongly separated for any i<j.
i+1 i j
Let r′ =ρ(r) andconsiderthe ball Br′ ofradiusr′ aboutα(0). Letxi =Hi∩α.
By Lemma 2.3, for any n, we can choose r large enough so that Br′/2 contains xi
for all i≤ n, as well as the bridge between H and H . Let β be any path from
i−1 i
α(−r) to α(r) which stays outside Br′. Then β must cross H0,H1,...,Hn in a
sequenceofpoints y ,y ,...y . Note thateachy is distanceatleastr′/2fromthe
0 1 n i
bridges to the adjacentwalls, hence by Lemma 2.3, there is a universalconstantC
such that
d(y ,y ) ≥ r′ −(d(H ,H )−4
i−1 i C i−1 i
≥ r′ −d(x ,x )−4
C i−1 i
It follows that the length of β satisfies
|β| ≥Pd(y ,y )
i−1 i
≥ nr′ −4n−d(x ,x )
C 0 n
≥ nr′ −4n−r′
C
Since n→∞ as r →∞, this proves the superlinear divergence of α. (cid:3)
The following example shows that the converse of the above theorem does not
hold in complete generality. However, when Γ is a connected graph, we will give a
complete characterization of geodesics with superlinear divergence in Theorem 4.4
below.
Example 2.6. Suppose Γ is disconnected, then A splits as a free product and
Γ
X splits as a wedge of spaces. Take any point p∈X whose removal disconnects
Γ Γ
X , and any pair of geodesic rays γ and γ emanating from p for which γ \{p}
Γ 1 2 1
and γ \{p} are in distinct components of X \{p}. Then the union of γ and
2 Γ 1
γ is a bi-infinite geodesic with super-linear divergence (indeed infinite divergence,
2
since γ and γ can not be connected in the complement of any ball around p).
1 2
If we choose each of the γ to be contained in (the cube neighborhood of) a wall,
i
then α=γ ∪γ is a geodesic with superlinear divergence such that no three walls
1 2
crossed by α are pairwise strongly separated.
3. Joins
Inthissectionwegiveagroup-theoreticinterpretationofTheorem2.5. Choosing
a vertex x in X as a basepoint, we may identify the 1–skeleton of X with the
0 Γ Γ
Cayley graph of A so that vertices are labeled by elements of A and edges by
Γ Γ
DIVERGENCE AND QUASIMORPHISMS OF RIGHT-ANGLED ARTIN GROUPS 7
elementsofthestandardgeneratingset(namelythevertexsetofΓ). Foragenerator
v, let e denote the edge from the basepoint 1 to the vertex v. Any edge in X
v Γ
determines a unique wall, namely the wall containing the midpoint of that edge.
Denote by H the wall containing the midpoint of e .
v v
For a cube in X , all of the parallel edges are labelled by the same generator v.
Γ
It followsthat all ofthe edges crossinga wall H have the same label v, and we call
this a wallof type v. Since A acts transitivelyonedgeslabeled v, awallis oftype
Γ
v if and only if it is a translate of the standard wall H .
v
Let lk(v) denote the subgraph of Γ spanned by the vertices adjacent to v and
st(v) the subgraph spanned by v and lk(v). The stabilizer of the wall H is the
v
group generated by lk(v), which we denote by L . To see this, note that in any
v
cube containing the edge e , all other edges labeled v are of the form ge for some
v v
g ∈ L . An induction on the number of cubes between e and e now shows that
v v
the same holds for any edge e which crosses H .
v
In what follows,for two subgroupsK and L of A , we will use the notationKL
Γ
to mean the set of elements of A which can be written as a product kl for some
Γ
k ∈K,l∈L. In general, KL is not a subgroup.
Lemma 3.1. Let H =g H and H =g H . Then
1 1 v 2 2 w
(1) H intersects H ⇐⇒ v,w commute and g−1g ∈L L .
1 2 1 2 v w
(2) ∃ H intersecting both H and H ⇐⇒ ∃ u ∈ st(v)∩st(w) such that
3 1 2
g−1g ∈L L L .
1 2 v u w
Here (2) includes the case in which H is equal to H or H , hence H and H are
3 1 2 1 2
strongly separated if and only if the conditions in (2) are not satisfied.
Proof. Without loss of generality, we may assume that H =H and H =gH .
1 v 2 w
(1) If v,w commute, they span a cube in X , hence H and H intersect. Sup-
Γ v w
poseg =ab,witha∈L ,b∈L . ThenH =a−1H andH =bH ,sotranslating
v w v v w w
by a, we see that H intersects gH .
v w
Conversely, suppose H intersects gH in a cube C. Then C contains edges of
v w
type v and of type w hence v and w must commute. Moreover, C is a translate
C = hC′ of a cube C′ at the basepoint containing the edges e and e . Since e
v w v
and he both intersect H , h lies in L . Since ge and he both intersect gH ,
v v v w w w
h−1g lies in L . Thus, g ∈L L .
w v w
(2) If u∈st(v)∩st(w) and g =abc∈L L L , then H and bH =bcH both
v u w v w w
intersectH =bH . Translatingbya,weseethatH andgH bothintersectaH .
u u v w u
Conversely, suppose that H = hH intersects both H and H . By part (1), u
3 u 1 2
mustcommutewithbothv andw,sou∈st(v)∩st(w). Alsobypart(1),h∈L L
v u
and h−1g ∈L L , so g ∈L L L . (cid:3)
u w v u w
For two walls H and gH to satisfy the conditions of (2), both w and the
v w
letters in g must lie in a 2–neighborhood of v. The converse is not true. Consider
for example the case of the Artin group associated to a pentagon Γ with vertices
labeled (in cyclic order) a,b,c,d,e. Every vertex lies in a 2-neighborhoodof a, but
we claim that H and daH are strongly separated. This follow from the lemma
a c
since st(a)∩st(c)={b} and da does not lie in L L L =he,biha,cihb,di.
a b c
ToguaranteethatnotwowallsinX arestronglyseparated,weneedastronger
Γ
condition. Let Γ and Γ be (non-empty) graphs. The join of Γ and Γ is the
1 2 1 2
graph formed by joining every vertex of Γ to every vertex of Γ by an edge. The
1 2
Artin group associatedto such a graphsplits as a direct product, A =A ×A
Γ Γ1 Γ2
8 JASONBEHRSTOCKANDRUTHCHARNEY
and X splits as a metric product X = X ×X . The walls in X are thus of
Γ Γ Γ1 Γ2 Γ
the formH ×X or X ×H forsome wall H inX . Clearly,everywallofthe
1 Γ2 Γ1 2 i Γi
first type intersects every wall of the second type, and it follows that no two walls
are strongly separated.
Let g ∈ A and let v v ...v be a minimal length word representing g. For
Γ 1 2 k
i < k, set g = v v ...v . Then the set of walls crossed by the edge path in X
i 1 2 i Γ
from x to gx labelled v v ...v is given by
0 0 1 2 k
H={H ,g H ,g H ...g H }.
v1 1 v2 2 v3 k−1 vk
A different choice of minimal word gives the same set of walls, changing only the
order in which they are crossed. Define the separation length of g to be
ℓ (g)=max{k |H contains k walls which are pairwise strongly separated}.
S
IfJ isacompletesubgraphofΓwhichdecomposesasanon-trivialjoin,thenwe
call A a join subgroup of A . Define the join length of g to be
J Γ
ℓ (g)=min{k|g =a ...a where a lies in a join subgroup of A }.
J 1 k i Γ
If α is a (finite) geodesic in A , we can approximate α by an edge path which
Γ
crosses the same set of walls as α. The word labeling this edge path determines
an element g ∈ A . We define ℓ (α) = ℓ (g ) and ℓ (α) = ℓ (g ). If α is a
α Γ S S α J J α
bi-infinite geodesic, and α denotes the restriction of α to the interval [−n,n], we
n
define the separation and join lengths of α to be
ℓ (α)=lim ℓ (α ) ℓ (α)=lim ℓ (α ).
S n→∞ S n J n→∞ J n
Lemma 3.2. A bi-infinite geodesic α in A has finite join length if and only if
Γ
both the positive and negative rays of α eventually stay in a single join. If every
bi-infinite periodic geodesic in A has finite join length, then Γ is itself a join.
Γ
Proof. For any join J in Γ, X is a convex subspace of X so once α leaves X ,
J Γ J
it will never return, and similarly for translates of X . If α has finite join length
J
it lies entirely in some finite set of these join subspaces and hence each ray must
eventually remain in a single join. The reverse implication is obvious.
For the second statement, suppose Γ is not a join. Let J be a maximal join in
Γ and let v be a vertex not in J. Let g ∈ A be the product of all the vertices in
J
J and consider the bi-infinite geodesic α = ...gvgvgvgv.... Note that no vertex
w ∈ J ∪v commutes with both J and v since otherwise, we would have J ∪v
contained in the join st(w), contradicting the maximality of J. It follows that the
tails of α must involve every vertex of J ∪v, hence by the first statement of the
lemma, α has infinite join length. (cid:3)
In the proof of the previous lemma, we used the fact that for any vertex v of
Γ, st(v) is always a join, namely it is the join of {v} and lk(v). This fact plays a
crucial role in the next lemma as well as in the proof of Theorem 4.4 below.
Lemma 3.3. For any g ∈A ,
Γ
ℓ (g)≤ℓ (g)≤2ℓ (g)+1.
S J S
Thus a bi-infinite geodesic has infinite join length if and only if it has infinite
separation length.
DIVERGENCE AND QUASIMORPHISMS OF RIGHT-ANGLED ARTIN GROUPS 9
Proof. The first inequality follows from the observation above that no two walls
in a join are strongly separated. For the second inequality, fix a minimal word for
g and let H be the sequence of walls crossed by the corresponding edge path as
listed above. Set H = H and let H′ = g H be the first wall in the sequence
v1 i vi+1
strongly separated from H. Then by Lemma 3.1, g lies in the product of three
i
link subgroups, L L L , for some u , hence g =g v lies in a product of
v1 u1 vi+1 1 i+1 i i+1
the three join groups generated by st(v ), st(u ), and st(v ). Now repeat this
1 1 i+1
argument starting with H = g H and taking H′ = g H to be the next
i vi+1 j vj+1
strongly separated wall (or H′ = the last wall in H if no more strongly separated
walls exist), to conclude that g lies in the product of join subgroups
j+1
hst(v )ihst(u )ihst(v )ihst(u )ihst(v )i.
1 1 i+1 2 j+1
Continuingthisprocess,eachnewstronglyseparatedwalladdstwostarsubgroups.
Since we encounter at most ℓ (g) strongly separated walls, the inequality follows.
S
(cid:3)
4. The asymptotic cone
The goal of this section is to understand the structure of the asymptotic cones
of A . We begin by recalling some preliminaries on asymptotic cones, tree graded
Γ
spaces, and divergence; we refer the reader to [dDW], [DS], and [DMS] for more
details.
Let (X,d) be a geodesic metric space. Let ω be a non-principal ultrafilter,
(o ) a sequence of observation points in X, and (d ) a sequence of scaling con-
n n
stants such that lim d = ∞. Then the asymptotic cone, Cone (X,(o ),(d )),
ω n ω n n
is the metric space consisting of equivalence classes of sequences (a ) satisfy-
n
ing lim d(o ,a )/d < ∞, where two such sequences (a ),(a′ ) represent the
ω n n n n n
same point a if and only if lim d(a ,a′ )/d = 0, and the metric is given by
ω n n n
d (a,b)=lim d(a ,b )/d .
ω ω n n n
We will assume the observationpoints and scaling constants are fixed and write
Xω for Cone (X,(o ),(d )). In general, Xω is a complete geodesic metric space.
ω n n
In the case where X has a cocompactgroupaction, Xω is independent of choice of
observation points (but not, in general, of scaling constants) and is homogeneous.
Acompletegeodesicmetricspaceistreegraded ifitcontainsacollectionofclosed
subsets, P, called pieces such that the following three properties are satisfied: in
eachP ∈P,everypairofpointsisconnectedbyageodesicinP;anysimplegeodesic
triangle is contained in some P ∈ P; and each pair P,P′ ∈ P is either disjoint or
intersects in a single point. Dru¸tu–Osin–Sapir proved that a group is relatively
hyperbolic if and only if all of its asymptotic cones are tree-gradedwith respect to
pieces obtained by taking asymptotic cones of the peripheral subgroups. On the
other hand, Behrstock–Dru¸tu–Mosher proved that right-angled Artin groups are
relatively hyperbolic if and only if their defining graph is disconnected [BDM]. In
this section,we show that for connected defining graphs,althoughthe Artin group
A is not relatively hyperbolic, the asymptotic cones of A still have a non-trivial
Γ Γ
tree grading providing Γ is not a join. Moreover, although the pieces do not come
from asymptotic cones of subgroups,they can be characterizedgroup-theoretically
(see Theorem 4.6).
We beginby recallingthe workofDrutu–Mozes–Sapir[DMS] ondivergenceand
cut-points.
10 JASONBEHRSTOCKANDRUTHCHARNEY
Definition4.1. Letρ(k)=δk−λwith0<δ <1andλ≥0. Forpointsa,b,c∈X,
set k = d(c,{a,b})) and define div(a,b,c;ρ) to be the length of the shortest path
in X from a to b which lies outside the ball of radius ρ(k) about c. The divergence
of X with respect to ρ is the function
Div(X,ρ)(r)=sup{div(a,b,c;ρ)|a,b∈X,d(a,b)≤r}.
For a biinfinite geodesic α, the divergence function introduced in Section 2 can
be written as,
div(α,ρ)(r)=div(α(−r),α(r),α(0);ρ).
In particular, if X has linear divergence, then every bi-infinite geodesic in X has
linear divergence.
Drutu–Mozes–Sapir establish the following correspondence between cut-points
and divergence functions [DMS, Lemma 3.14].
Proposition 4.2 ([DMS]). Let a=(a ),b=(b ), c=(c ) be three points in Xω,
n n n
and let k = d (c,{a,b}). Then c is a cut-point separating a from b if and only if
ω
for some ρ,
lim div(a ,b ,c ; ρ)
ω n n n k =∞.
d
n
In particular, for a bi-infinite geodesic α in X, taking a =α(−d ),b =α(d ),
n n n n
and c =α(0), the proposition implies that c is a cut-point separating a from b if
n
and only if α has super-linear divergence.
WesaythatX iswide ifnoasymptoticconeofX hascut-points. Inthecasethat
X is the Cayley graph of a group G, the proposition above leads to the following
criterion for G to be wide (see [DMS, Proposition 1.1]).
Proposition 4.3 ([DMS]). A group G is wide if and only if Div(G,ρ)(r) is linear
for ρ(r)= 1r−2.
2
In the case of A , the divergence of a bi-infinite geodesic is determined by its
Γ
join length.
Theorem 4.4 (Divergence and join length). Let Γ be a connected graph and let α
be a bi-infinite geodesic in X . Then α has linear divergence if and only if the join
Γ
length of α is finite.
Proof. If the join length of α is infinite, then by Lemma 3.3, so is the separation
length. By Theorem 2.5, it follows that α has super-linear divergence.
Now suppose that the join length of α is finite. We will show that α lies in a
subspaceofX whoseimageinanyasymptotic coneXω hasnocut-points. Itthen
Γ Γ
follows from the remarks following Proposition 4.2 that α has linear divergence.
By Lemma 3.2, α lives entirely in a finite union of join subspaces, that is, sub-
spaceswhicharetranslatesofX forsomejoinJ. SinceA decomposesasadirect
J J
product of infinite groups, X is wide. Hence in any asymptotic cone Xω, the
J Γ
cone on gXJ gives rise to a subspace with no cut-points. If g′XJ′ is another join
subspacewhichsharesageodesiclinewithgX ,thentheunionoftheirasymptotic
J
cones contains a line in Xω hence also has no cut-points.
Γ
Thus, it suffices to show that any two join subspaces gXJ and g′XJ′ are con-
nectedbyasequenceofjoinsubspacessuchthatconsecutivesubspacessharealine.
We will call this a connecting sequence. By hypothesis, the graph Γ is connected,