Relative Rigidity, Quasiconvexity and C-Complexes
arXiv:0704.1922v4  [math.GT]  4 Aug 2008
RELATIVE RIGIDITY, QUASICONVEXITY AND
C-COMPLEXES
MAHAN MJ
Abstract. We introduce and study the notion of relative rigidity for
pairs (X,J ) where
1) X is a hyperbolic metric space and J a collection of quasiconvex sets
2) X is a relatively hyperbolic group and J the collection of parabolics
3) X is a higher rank symmetric space and J an equivariant collection
of maximal flats
Relative rigidity can roughly be described as upgrading a uniformly
proper map between two such J ’s to a quasi-isometry between the cor-
responding X’s. A related notion is that of a C-complex which is the
adaptation of a Tits complex to this context. We prove the relative
rigidity of the collection of pairs (X,J ) as above. This generalises a re-
sult of Schwarz for symmetric patterns of geodesics in hyperbolic space.
We show that a uniformly proper map induces an isomorphism of the
corresponding C-complexes. We also give a couple of characterizations
of quasiconvexity of subgroups of hyperbolic groups on the way.
AMS subject classification = 20F67(Primary), 22E40,
57M50(Secondary)
Contents
1. Introduction 2
1.1. Relative Rigidity and Statement of Results 2
1.2. Relative Hyperbolicity and Electric Geometry 6
1.3. Height of Subgroups and C-Complexes 8
2. Characterizations of Quasiconvexity 9
2.1. Limit Sets and Quasiconvexity 10
2.2. Quasiconvexity and Relative Hyperbolicity 11
3. Relative Rigidity 13
3.1. Pairing of Limit Sets by Quasi-isometries 13
3.2. C-Complexes 16
3.3. Cross Ratios, Annular Systems and a Dynamical Formulation 17
3.4. Axiomatisation, Relative Hyperbolicity 18
3.5. Symmetric Spaces of Higher Rank 20
References 21
http://arxiv.org/abs/0704.1922v4
2 MAHAN MJ
1. Introduction
1.1. Relative Rigidity and Statement of Results. In this paper, we
study a rigidity phenomenon within the framework of coarse geometry. We
call it relative rigidity. Much of the work on quasi-isometric rigidity (e.g.
Farb-Schwartz [FS96] Kleiner-Leeb [KL97b] Eskin-Farb [EF 3] and Mosher-
Sageev-Whyte [MSW03] [MSW04] ) contains a crucial step showing that a
self quasi-isometry of a space X coarsely preserves a family J of distin-
guished subsets of X. The family J again has a coarse intersection pattern
that may be combinatorially coded and these proofs of quasi-isometric rigid-
ity often show that the intersection pattern is preserved by a quasi-isometry.
In this note, we investigate a sort of a converse to this:
When does a uniformly proper map between two families J1 and J2 come
from a quasi-isometry φ between X1 and X2? Does such a map preserve
intersection patterns?
We show that the answer is affirmative when
(1) Xi’s are (Cayley graphs of) hyperbolic groups and Ji’s correspond
to cosets of a quasiconvex subgroup
(2) Xi’s are (Cayley graphs of) relatively hyperbolic groups and Ji’s
correspond to parabolic subgroups
(3) Xi’s are symmetric spaces of non-positive curvature and Ji’s corre-
spond to lifts of a maximal torus in a compact locally symmetric
space modeled on Xi.
If in addition one can show that a quasi-isometry preserving intersection
patterns is close to an isometry, we would be able to conclude that a uni-
formly proper map between the Ji’s is induced by an isometry. This latter
phenomenon has been investigated by Mosher, Sageev and Whyte [MSW04]
and has been termed pattern rigidity. Thus, in a sense, the notion of relative
rigidity complements that of pattern rigidity.
Some further examples where a family of distinguished subsets of a space
and the resulting (combinatorial) configuration yields information about the
ambient space are:
1) Collection of flats in a symmetric space of higher rank (Mostow [Mos73])
2) Collection of maximal abelian subgroups of the mapping class group
(Behrstock-Drutu-Mosher [BDM05] )
3) Collection of hyperbolic spaces in the Cayley complex of the Baumslag-
Solitar groups (Farb-Mosher [FM98] , [FM 3] ; see also [FM00] )
4) Quasi-isometric rigidity of sufficiently complicated patterns of flats in the
universal cover of a Haken 3 manifold (Kapovich-Leeb [KL97a] )
5) We were most influenced by a beautiful result of Schwarz [Sch97] which
shows that a uniformly proper map from a symmetric pattern of geodesics in
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 3
Hn to another symmetric pattern of geodesics in Hn (for n > 2) is induced
by an isometry. Again as in Mostow, there are two parts to this. A first step
is to construct a quasi-isometry of Hn inducing the given pairing. Schwarz
terms this ambient extension. The second is to construct an isometry.
Let us look at a general form of the situation that Schwarz considers.
(X1, d1), (X2, d2) are metric spaces. Let J1,J2 be collections of closed sub-
sets of X1,X2 respectively. Then di induces a pseudo-metric (which, by
abuse of notation, we continue to refer to as di) on Ji for i = 1, 2. This
is just the ordinary (not Hausdorff) distance between closed subsets of a
metric space. In [Sch97], X1 = X2 = H
n, and Ji are lifts (to the universal
cover) of finite collections of closed geodesics in two hyperbolic manifolds.
Also, the hypothesis in Schwarz’s paper [Sch97] is the existence of a uni-
formly proper map φ between symmetric patterns of geodesics J1 and J2. A
uniformly proper map may be thought of as an isomorphism in the so-called
coarse category in the sense of John Roe [Roe95]. Thus, we can re-interpret
the first step of Schwarz’s result as saying that an isomorphism φ in the
coarse category between Ji implies the existence of a quasi-isometry from
Hn to itself inducing φ. In the language of [Sch97], uniformly proper pairings
come from ambient extensions.
In Mostow’s proof of rigidity for higher rank symmetric spaces, he obtains
in a crucial step, an isomorphism of Tits complexes [Mos73]. We would like
to associate to a pair (X,J ) some such complex just as a Tits complex
is associated to a higher rank locally symmetric space and its collection of
maximal parablic subgroups. We propose the notion of a C-complex in this
paper as the appropriate generalization of a Tits complex to coarse geometry.
Then what we would hope for (as a conclusion) is an isomorphism of these
C-complexes. This transition from the existence of a uniformly proper map
between Ji’s to the existence of a a quasi-isometry between Xi’s inducing an
isomorphism of C-complexes is what we term relative rigidity. Schwarz
proves the relative rigidity of pairs (X,J ) where X is hyperbolic space and
J a symmetric collection of geodesics. Much of what he does extends to the
case whereX is a higher rank symmetric space and J a symmetric collection
of maximal periodic flats or a symmmetric collection of maximal parabolic
subgroups in a non-uniform lattice.
The main point of this paper is illustrated first in the context of rela-
tive rigidity of the category of pairs (Γ,J ), where Γ is (the Cayley graph
of) a hyperbolic group, and J the set of cosets of a quasiconvex subgroup.
Throughout this paper we shall assume that the quasiconvex sub-
groups are of infinite index in the big groups.
Note that the upgrading of a uniformly proper map between J ’s to a
quasi-isometry between the Γ’s is the most we can hope for in light of the
fact that the Cayley graph of a finitely generated group is only determined
up to quasi-isometry. (See Paulin [Pau96] for a proof of this fact.)
4 MAHAN MJ
We start with a pair of hyperbolic groups G1, G2 with Cayley graphs
Γ1,Γ2, and quasiconvex subgroups H1,H2. Let Λ1, Λ2 be the limit sets of
H1,H2 in ∂G1, ∂G2 respectively. For convenience we consider the collection
Ji of translates of Ji the join of Λi in Γi rather than cosets of Hi. Recall
that the join of Λi is the union of bi-infinite geodesics in Γi with end-points
in Λi. This is a uniformly quasiconvex set and lies at a bounded Hausdorff
distance from the Cayley graph of the subgroup Hi (Since H has finite index
in its commensurator, only finitely many cosets of H are at a finite Hausdorff
distance from it. Since Ji is at a bounded Hausdorff distance from Hi the
same is true for elements of Ji.) The main theorems of this paper are as
follows.
Theorem 3.5:Let φ be a uniformly proper (bijective, by definition) map
from J1 → J2. There exists a quasi-isometry q from Γ1 to Γ2 which pairs
the sets J1 and J2 as φ does.
The construction of the quasi-isometry q proceeds by constructing a ”coarse
barycenter” of some infinite diameter sets (reminiscent of the celebrated
measure-theoretic barycenter method discovered by Douady and Earle, and
extended greatly by Besson, Courtois, Gallot [BCG98] ).
We prove an analogous theorem for pairs (X,J ) when X is (strongly)
hyperbolic relative to the collection J .
Theorem 3.11:Let Xi be (strongly) hyperbolic relative to collections Ji
(i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from
J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the
sets J1 and J2 as φ does.
As a Corollary of Theorem 3.11 and work of Hruska and Kleiner [HK04],
we deduce relative rigidity for pairs (X,J ) where X is a CAT(0) space with
isolated flats and J is the collection of maximal flats.
The third main theorem of this paper is an analog for higher rank sym-
metric spaces.
Theorem 3.13:Let Xi be symmetric spaces of non-positive curvature, and
Ji be equivariant collections of lifts of a maximal torus in a compact locally
symmetric space modeled on Xi (i = 1, 2). Let φ be a uniformly proper
(bijective, by definition) map from J1 → J2. There exists a quasi-isometry
q from X1 to X2 which pairs the sets J1 and J2 as φ does.
In fact, combining Theorem 3.13 with the quasi-isometric rigidity theorem
of Kleiner-Leeb [KL97b] and Eskin-Farb [EF 3], we may upgrade the quasi-
isometry of Theorem 3.13 to an isometry.
Let Gi,Hi (i = 1, 2) be hyperbolic groups and quasiconvex subgroups re-
spectively. In Section 1.3, we shall construct simplicial complexes (termed C-
complexes) from the incidence relations determined by the cosets of Hi. Let
C(Gi,Hi) be the C-complexes associated with the pairs (Gi,Hi). Roughly
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 5
speaking, the vertices of C(Gi,Hi) are the translates g
iΛi of Λi by distinct
coset representatives g
i and the (n− 1)-cells are n-tuples {g
1Λ, · · · , g
1Λ} of
distinct translates such that ∩n1g
1Λ 6= ∅.
Theorem 3.7: Let φ : J1 → J2 be a uniformly proper map. Then φ
induces an isomorphism of C(G1,H1) with C(G2,H2).
On the way towards proving Theorems 3.5 and 3.7, we prove two Proposi-
tions characterizing quasiconvexity. These might be of independent interest.
The first is in terms of the Hausdorff topology on the collection C0c (∂G),
which is the collection of closed subsets of ∂G having more than one point.
Proposition 2.3: Let H be a subgroup of a hyperbolic group G with limit
set Λ. Let L be the collection of translates of Λ by elements of distinct
cosets of H (one for each coset). Then H is quasiconvex if and only if L is
a discrete subset of C0c (∂G).
The second characterization is in terms of strong relative hyperbolicity.
Definition 1.1. A subgroup H of a group G is said to be malnormal if for
all g ∈ G \H, gHg−1 ∩H is trivial. A subgroup H of a group G is said to
be almost malnormal if for all g ∈ G \H, gHg−1 ∩H is finite.
It was pointed out to us by the referee that the following result follows
from work of Farb [Far98], Bowditch ([Bow97] Theorem 7.11) and Drutu-
Sapir ([DS05] Lemma 4.15). We shall include a proof for completeness.
Proposition 2.9:[Far98] [Bow97] [DS05] Let G be a hyperbolic group and
H a subgroup. Then G is strongly relatively hyperbolic with respect to H if
and only if H is a malnormal quasiconvex subgroup.
The prototypical example is that of (fundamental groups of) a closed
hyperbolic manifold with a totally geodesic embedded submanifold.
Finally, we give an intrinsic or dynamic reformulation of Theorems 3.5
and 3.7 following Bowditch [Bow98], which makes use of the existence of a
cross-ratio on the boundary of a hyperbolic group. The cross-ratio in turn
induces a pseudometric on the collection L of translates of Λ.
Theorem 3.10: Let G1, G2 be uniform convergence (hence hyperbolic)
groups acting on compacta M1,M2 respectively. Also, let Åi (for i = 1, 2)
be Gi-invariant annulus systems and let (..|..)i denote the corresponding an-
nular cross-ratios.
Let H1,H2 be subgroups of G1, G2 with limit sets Λ1,Λ2. Suppose that the
set Li of translates of Λi (for i = 1, 2) by essentially distinct elements of Hi
in Gi forms a discrete subset of C
c (Mi).
Also assume that there exists a bijective function φ : L1 → L2 and that this
pairing is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2.
1) Hi is quasiconvex in Gi
2) There is a homeomorphism q : M1 → M2 which pairs L1 with L2 as φ
6 MAHAN MJ
does. Further, q is uniformly proper with respect to the cross-ratios (..|..)1
and (..|..)2 on M1, M2 respectively.
3) q (and hence also φ) induces an isomorphism of C-complexes C(G1,H1)
with C(G2,H2).
Acknowledgements: My interest in relative hyperbolicity and quasi-isometric
rigidity is largely due to Benson Farb. It is a pleasure to acknowledge his
help, support and camaraderie, both mathematical and personal. I would
also like to thank the referee for suggesting several corrections and for pro-
viding additional references.
1.2. Relative Hyperbolicity and Electric Geometry. We start off by
fixing notions and notation. Let G (resp. X) be a hyperbolic group (resp.
a hyperbolic metric space) with Cayley graph (resp. a net) Γ equipped with
a word-metric (resp. a simplicial metric) d.
Here a net N is a collection of distinct points xi ∈ X such that there exist
0 < C1 < C2 such that
1) d(xi, xj) ≥ C1 for all i 6= j
2) For all x ∈ X, there exists xi ∈ N such that d(xi, x) ≤ C2
For the net N we construct a graph GN with edges corresponding to pairs
xi 6= xj such that d(xi, xj) ≤ 4C2. The simplicial metric on N is obtained
by declaring that each edge of GN has length one.
Let the Gromov boundary of Γ be denoted by ∂G. (cf.[GdlH90]).
We shall have need for the fact that for hyperbolic metric spaces (in the
sense of Gromov [Gro85]) the notions of quasiconvexity and qi embeddings
coincide [Gro85].
We shall now recall certain notions of relative hyperbolicity due to Gro-
mov [Gro85] and Farb [Far98].
Let X be a path metric space. A collection of closed subsets H = {Hα}
of X will be said to be uniformly separated if there exists ǫ > 0 such that
d(H1,H2) ≥ ǫ for all distinct H1,H2 ∈ H.
The electric space (or coned-off space) X̂ corresponding to the pair
(X,H) is a metric space which consists of X and a collection of vertices vα
(one for each Hα ∈ H) such that each point of Hα is joined to (coned off at)
vα by an edge of length
Definition 1.2. [Far98] [Bow97] Let X be a geodesic metric space and H
be a collection of uniformly separated subsets. Then X is said to be weakly
hyperbolic relative to the collection H, if the electric space X̂ is hyperbolic.
Lemma 1.3. (See Bowditch [Bow97], generalizing Lemma 4.5 and Propo-
sition 4.6 of Farb [Far98]) Given δ, C,D there exists ∆ such that if X is a
δ-hyperbolic metric space with a collection H of C-quasiconvex D-separated
sets. then, the electric space X̂ is ∆-hyperbolic, i.e. X is weakly hyperbolic
relative to the collection H.
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 7
Definitions: Given a collection H of C-quasiconvex, D-separated sets
and a number ǫ we shall say that a geodesic (resp. quasigeodesic) γ is a
geodesic (resp. quasigeodesic) without backtracking with respect to ǫ
neighborhoods if γ does not return to Nǫ(H) after leaving it, for any H ∈
H. A geodesic (resp. quasigeodesic) γ is a geodesic (resp. quasigeodesic)
without backtracking if it is a geodesic (resp. quasigeodesic) without
backtracking with respect to ǫ neighborhoods for some ǫ ≥ 0.
Electric P -quasigeodesics without backtracking are said to have similar
intersection patterns if for β, γ electric P -quasigeodesics without backtrack-
ing both joining x, y, the following are satisfied.
(1) Similar Intersection Patterns 1: if precisely one of {β, γ} meets an
ǫ-neighborhood Nǫ(H1) of an electrocuted quasiconvex set H1 ∈ H,
then the length (measured in the intrinsic path-metric on Nǫ(H1) )
from the entry point to the exit point is at most D.
(2) Similar Intersection Patterns 2: if both {β, γ} meet some Nǫ(H1)
then the length (measured in the intrinsic path-metric on Nǫ(H1) )
from the entry point of β to that of γ is at most D; similarly for exit
points.
Definition 1.4. [Far98] [Bow97] Let X be a geodesic metric space and H
be a collection of mutually disjoint uniformly separated subsets such that X
is weakly hyperbolic relative to the collection H. If any pair of P - electric
quasigeodesics without backtracking starting and ending at the same point
have similar intersection patterns with horosphere-like sets (elements of H)
then quasigeodesics are said to satisfy Bounded Penetration and X is
said to be strongly hyperbolic relative to the collection H.
Definition 1.5. [Mj05] A collection H of uniformly C-quasiconvex sets in
a δ-hyperbolic metric space X is said to be mutually D-cobounded if for
all Hi,Hj ∈ H, πi(Hj) has diameter less than D, where πi denotes a nearest
point projection of X onto Hi. A collection is mutually cobounded if it
is mutually D-cobounded for some D.
Mutual coboundedness was proven by Farb for horoballs in finite volume
Hadamard manifolds of pinched negative curvature in Lemma 4.7 of [Far98].
The following generalization is due to Bowditch [Bow97].
Lemma 1.6. (See Bowditch [Bow97] Lemma 7.13 for a proof) Suppose X is
a δ-hyperbolic metric space with a collection H of C-quasiconvex K-separated
D-mutually cobounded subsets. Then X is strongly hyperbolic relative to the
collection H.
Gromov gave a different definition of strong relative hyperbolicity.
We give a condition below that is equivalent to a special case of Gromov’s
definition. Let X be a geodesic metric space with a collection H of uniformly
8 MAHAN MJ
separated subsets {Hi}. The hyperbolic cone cHi is the product of Hi
and the non-negative reals Hi ×R+, equipped with the metric of the type
2−tds2 + dt2. More precisely, Hi ×{n} is given the path metric of Hi scaled
by 2−n. The R+ direction is given the standard Euclidean metric. Let X
denote X with hyperbolic cones cHi glued to it along Hi’s. X
h will be
referred to as the hyperbolically coned off X. This is to be contrasted with
the coned off space X̂ in Farb’s definition.
Definition 1.7. X is said to be strongly hyperbolic relative to the collec-
tion H in the sense of Gromov if the hyperbolically coned off space Xh is a
hyperbolic metric space.
The equivalence of the two notions of strong relative hyperbolicity was
proven by Bowditch in [Bow97].
Theorem 1.8. ( Bowditch [Bow97] ) X is strongly hyperbolic relative to a
collection H of uniformly separated subsets {Hi} in the sense of Gromov if
and only if X is strongly hyperbolic relative to the collection H in the sense
of Farb.
1.3. Height of Subgroups and C-Complexes. The notion of height of
a subgroup was introduced by Gitik, Mitra, Rips and Sageev in [GMRS97]
and further developed by the author in [Mit04].
Definition 1.9. Let H be a subgroup of a group G. We say that the elements
{gi|1 ≤ i ≤ n} of G are essentially distinct if Hgi 6= Hgj for i 6= j. Con-
jugates of H by essentially distinct elements are called essentially distinct
conjugates.
Note that we are abusing notation slightly here, as a conjugate of H by
an element belonging to the normalizer of H but not belonging to H is still
essentially distinct from H. Thus in this context a conjugate of H records
(implicitly) the conjugating element.
Definition 1.10. We say that the height of an infinite subgroup H in G is
n if there exists a collection of n essentially distinct conjugates of H such
that the intersection of all the elements of the collection is infinite and n is
maximal possible. We define the height of a finite subgroup to be 0. We say
that the width of an infinite subgroup H in G is n if there exists a collection
of n essentially distinct conjugates of H such that the pairwise intersection
of the elements of the collection is infinite and n is maximal possible.
The main theorem of [GMRS97] states:
Theorem 1.11. If H is a quasiconvex subgroup of a hyperbolic group G,then
H has finite height and finite width.
In this context, a theorem we shall be needing several times is the following
result from [GMRS97] that is proved using a result of Short [Sho91].
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 9
Theorem 1.12. (Lemma 2.6 of [GMRS97]) Let G be a hyperbolic group and
Hi (for i = 1 · · · k ) be quasiconvex subgroups with limit sets Λi, i = 1 · · · k.
Then ∩Hi is a quasiconvex subgroup with limit set ∩Λi.
We now proceed to define a simplicial complex C(G,H) for a group G
and H a subgroup. For G hyperbolic and H quasiconvex, we give below
three equivalent descriptions of a complex C(G,H). In this case, let ∂G
denote the boundary of G, Λ the limit set of H, and J the join of Λ.
1) Vertices ( 0-cells ) are conjugates of H by essentially distinct elements,
and (n − 1)-cells are n-tuples {g1H, · · · , gnH} of distinct cosets such that
∩n1giHg
i is infinite (in fact by Theorem 1.12 an infinite quasiconvex sub-
group of G).
2) Vertices ( 0-cells ) are translates of Λ by essentially distinct elements,
and (n−1)-cells are n-tuples {g1Λ, · · · , gnΛ} of distinct translates such that
∩n1giΛ 6= ∅.
3) Vertices ( 0-cells ) are translates of J by essentially distinct elements,
and (n− 1)-cells are n-tuples {g1J, · · · , gnJ} of distinct translates such that
∩n1giJ is infinite.
We shall refer to the complex C(G,H) as the C-complex for the pair
G,H. (C stands for “coarse” or “Čech” or “cover”, since C(G,H) is like a
coarse nerve of a cover, reminiscent of constructions in Cech cochains.) Note
that if h(H) denote the height of H, then (h(H)+1) is the dimension of the
C-complex C(G,H). Also, if w(H) denote the width of H, then w(H) = w
is equal to the size of the largest complete graph Kw that is embeddable in
C(G,H). If C(G,H) is connected then its one-skeleton is closely related to
the coned off space Γ̂ with an appropriately chosen set of generators.
This definition is inspired by that of the Tits complex for a non-uniform
lattice in a higher rank symmetric space. Related constructs in the context
of codimension 1 subgroups also occur in work of Sageev [Sag95] where he
constructs cubings.
2. Characterizations of Quasiconvexity
Let G be a hyperbolic group. Let Cc(∂G) denote the collection of closed
subsets of the boundary ∂G equipped with the Hausdorff topology. Let
C0c (∂G) ⊂ Cc(∂G) denote the subset obtained from Cc(∂G) by removing
the singleton sets {{x} : x ∈ ∂G}. Next fix a subgroup H ⊂ G with limit
set Λ ⊂ ∂G. Consider the G-invariant collection L ={ gΛ } ⊂ C0c (∂G) with
g ranging over distinct cosets (one for each coset) of H in G. Note that L
is (strictly speaking) a multi-set as distinct elements of L may denote the
same element of C0c (∂G) in case two distinct translates of Λ coincide. One
extreme case is when Λ = ∂G, though H is of infinite index in G (e.g. if
H is normal of infinite index in in G.) Then L consists of infinitely many
copies of Λ.
10 MAHAN MJ
Definition 2.1. The join J(Λ) of Λ is defined as the union of all bi-infinite
geodesics whose end-points lie in Λ
It is easy to see that J(Λ) is 2δ-quasiconvex if G is δ-hyperbolic. In fact
this is true for any subset Λ of the boundary of a δ-hyperbolic metric space
X (no equivariance is necessary). For Λ the limit set of H, J(Λ) is H-
invariant. The visual diameter dia∂G(Λ) of a subset Λ of ∂G is the same as
the diameter in the metric on ∂G obtained from the Gromov inner product.
(See [GdlH90] Chapter 7 for details about the visual metric on ∂G.)
2.1. Limit Sets and Quasiconvexity. The next Lemma follows directly
from the fact that sets with visual diameter bounded below contain points
with Gromov inner product bounded above and conversely[GdlH90].
Lemma 2.2. For all ǫ > 0 there exists N such that if the diameter dia∂G(Λ) ≥
ǫ for a closed subset Λ of ∂G, then there exists p ∈ J(Λ) such that d(p, 1) ≤
N . Conversely, for all N > 0 there exists ǫ > 0 such that if there exists
p ∈ J(Λ) with d(p, 1) ≤ N , then dia∂G(Λ) ≥ ǫ.
The next Proposition gives our first characterisation of quasiconvex sub-
groups of a hyperbolic group.
Proposition 2.3. (Characterization of Quasiconvexity I) Let H be a
subgroup of a hyperbolic group G with limit set Λ. Let L be the collection of
translates of Λ (counted with multiplicity) by elements of distinct cosets of
H (one for each coset). Then H is quasiconvex if and only if L is a discrete
subset of C0c (∂G).
Proof: Suppose H is quasiconvex. We want to show that L is a discrete
subset of C0c (∂G). Thus it suffices to show that any limit of elements of L
is a singleton set. This in turn follows from the following.
Claim: For all ǫ > 0, Lǫ = {Li ∈ L : dia∂G(Li) ≥ ǫ} is finite.
Proof of Claim: Let N = N(ǫ) be as in Lemma 2.2. Since dia∂G(Li) ≥ ǫ,
therefore by Lemma 2.2, there exists pi ∈ J(Li) such that dG(pi, 1) ≤ N .
Also, there exists K > 0 depending on δ (recall that J(Li) is 2δ-qc) and
the quasiconvexity constant of H such that if Li = giΛ, then there exists
hi ∈ H with dG(pi, gihi) ≤ K. Hence, dG(1, gihi) ≤ K + N . Since G is
finitely generated, the number of such elements gihi is finite. Since gi are
picked from distinct cosets of H, we conclude that the set Lǫ is finite. ✷
Conversely, suppose that H is not quasiconvex. Assume, without loss of
generality, that a finite generating set of H is contained in a finite generating
set of G and that ΓH ,ΓG are Cayley graphs with respect to these generating
sets. Then there exist pi ∈ J(Λ) such that dG(pi,ΓH) ≥ i. Translating by an
appropriate element of H, we may assume that dG(pi,ΓH) = dG(pi, 1) ≥ i.
Further, we may assume (by passing to a subsequence if necessary) that
the sequence dG(pi, 1) is monotonically increasing. Then p
i J(Λ) has limit
set p−1i Λ. Further, as pi ∈ J(Λ), therefore, 1 ∈ p
i J(Λ). Since J(Λ)
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 11
is 2δ-qc, so is p−1i J(Λ) for all i. Hence, there exists ǫ > 0 by Lemma
2.2 such that dia∂Gp
i J(Λ) ≥ ǫ. Since dG(pi, 1) is monotonically strictly
increasing, we conclude that pi’s lie in distinbct cosets of H. Further, since
Cc(∂G) is compact, we conclude that the collection p
i J(Λ) has a convergent
subsequence, converging to a subset of diameter greater than or equal to
ǫ. Therefore, the collection L is not a discrete subset (strictly speaking a
multiset) of C0c (∂G). ✷
We next prove a result about projections of J(Li) on J(Lj). We start off
with an elementary fact about hyperbolic metric spaces. See [Mit98] for a
proof.
Lemma 2.4. [Mit98] Given δ > 0, there exist D,C1, k, ǫ such that if a, b, c, d
are points of a δ-hyperbolic metric space (Z, d), with d(a, [b, c]) = d(a, b),
d(d, [b, c]) = d(c, d) and d(b, c) ≥ D then [a, b] ∪ [b, c] ∪ [c, d] lies in a C1-
neighborhood of any geodesic joining a, d and is a (k, ǫ)-quasigeodesic.
Assume that H is quasiconvex and that Lk is the limit set gkΛ of gkH.
Let Pj denote the nearest point projection of ΓG onto J(Lj). Also, let
Hk = gkΓH be the left translate of ΓH by gk.
Proposition 2.5. There exists K > 0 such that Pj(ΓHi) lies in a K-
neighborhood of J(Li ∩ Lj) if (Li ∩ Lj) 6= ∅. Else, Pj(ΓHi) has diameter
less than K.
Proof: Since J(Li) is 2δ-qc and H is quasiconvex, it suffices to show that
Pj(J(Li)) lies in a K-neighborhood of J(Li ∩Lj) if the latter is non-empty.
By G-equivariance, we may assume that Lj = Λ and gi = 1. We represent
Pj by P in this case.
First note that by Theorem 1.12, Hi∩Hj is quasiconvex and the limit set
of Hi ∩Hj is Li ∩ Lj. Also, J(Li ∩ Lj) ⊂ J(Li).
Let a, b ∈ J(Li). Let P (a) = c, P (b) = d. Let D,C1, k, ǫ be as in Lemma
2.4. If dG(c, d) ≥ D, then [a, c]∪ [c, d]∪ [d, b] is a (k, ǫ)-quasigeodesic lying in
a C1 neighborhood of [a, b]. Since J(Li), J(Lj) are both 2δ-qc, [a, b] lies in a
2δ -neighborhood of J(Li), and [c, d] lies in a 2δ -neighborhood of J(Lj). In
particular c, d lie in a (C1 + 2δ)-neighborhood of J(Lj). Translating by an
element of H, we may assume that c = 1. (Note that the argument in this
paragraph works independent of whether J(Li) ∩ J(Lj) is empty or not.
We proceed now by contradiction. Suppose there exists a sequence of Li’s
and bi ∈ J(Li) such that P (bi) = di lies at a distance greater than i from
J(Li ∩Lj) (resp. c = 1) according as J(Li)∩ J(Lj) is non-empty or empty.
This shows that the sequence Li has a limit point on Λ disjoint from Li ∩Λ
for all i and further that J(Li) passes through a bounded neighborhood
of 1. Hence the sequence Li is not discrete in C
c (∂G). This contradicts
Proposition 2.3 and proves our claim. ✷
2.2. Quasiconvexity and Relative Hyperbolicity. As an immediate
corollary of Proposition 2.5 in conjunction with Theorem 1.12 of Short
[Sho91], we immediately conclude
12 MAHAN MJ
Corollary 2.6. Let H be a malnormal quasiconvex subgroup of a hyperbolic
group G with Cayley graph Γ and limit set L. Then the set of joins J
of distinct translates of L is a uniformly cobounded collection of uniformly
quasiconvex sets in Γ.
Combining Lemma 1.6 with Corollary 2.6 above, we have the following
Proposition due to Bowditch [Bow97].
Proposition 2.7. (Characterization of Quasiconvexity II) [Bow97]
Let H be a malnormal quasiconvex subgroup of a hyperbolic group G. Then
G is strongly relatively hyperbolic with respect to H.
In fact the converse to Proposition 2.7 is also true. We came to learn from
the referee that this follows by combining work of Farb [Far98], Bowditch
[Bow97] and Drutu-Sapir [DS05]. We provide a proof below for completeness
(and because it is easily done).
Malnormality of strongly relatively hyperbolic subgroups is due to Farb
[Far98]. In fact this does not require G to be hyperbolic.
Lemma 2.8. (Farb [Far98]) Let G be strongly relatively hyperbolic with
respect to H. Then H is malnormal in G.
It remains to show that H is quasiconvex if a hyperbolic group G be
strongly relatively hyperbolic with respect toH. We use Gromov’s definition
of strong relative hyperbolicity. Attach hyperbolic cones cH to distinct
translates of ΓH in ΓG to obtain the hyperbolically coned off Cayley graph
ΓhG. Then Γ
G is hyperbolic by Gromov’s definition.
If H is not quasi-isometrically embedded in G then for all i ∈ N, there
exist pi1, pi2 ∈ ΓH such that
dH(pi1, pi2) ≥ idG(pi1, pi2)
. Also from the metric dcH on cH, we find that dcH(pi1, pi2) is of the order
of log2dH(pi1, pi2). Hence, we can further assume that
dH(pi1, pi2) ≥ idcH(pi1, pi2)
. Join pi1, pi2 by shortest paths αi, βi in cH, ΓG respectively. Then αi∪βi =
σi is a closed loop in Γ
G with total length l(σi) = (dcH(pi1, pi2)+dG(pi1, pi2)).
Therefore il(σi) ≤ 2dH(pi1, pi2).
Since any (combinatorial) disk Di in Γ
G spanning σi must contain a path
γi in ΓH joining p1i, p2i, therefore the area A(Di) of Di must be at least that
of N1(γi), the 1-neighborhood of γi in Di.
Therefore there exists C > 0 such that for all i,
A(Di) ≥ A(N1(γi)) ≥
dH(pi1, pi2)
il(σi)
Since i is arbitrary, this shows that ΓhG cannot satisfy a linear isoperi-
metric inequality. Hence ΓhG cannot be a hyperbolic metric space. This is a
contradiction. Hence H must be quasi-isometrically embedded in G. Hence
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 13
(see for instance [Gro85] ), H is quasiconvex in G. This completes our proof
of the following characterisation of strongly relatively hyperbolic subgroups
of hyperbolic groups.
Proposition 2.9. Let G be a hyperbolic group and H a subgroup. Then
G is strongly relatively hyperbolic with respect to H if and only if H is a
malnormal quasiconvex subgroup.
3. Relative Rigidity
3.1. Pairing of Limit Sets by Quasi-isometries. We now consider two
hyperbolic groupsG1, G2 with quasiconvex subgroupsH1,H2, Cayley graphs
Γ1,Γ2. Let Lj for j = 1, 2 denote the collection of translates of limit sets
(counted with multiplicity as before) of H1,H2 in ∂G1, ∂G2 respectively.
Individual members of the collection Lj will be denoted as L
i . Let Jj
denote the collection {J
i = J(L
i ) : L
i ∈ Lj}. Following Schwarz [Sch97],
we define:
Definition 3.1. A bijective map φ from J1 → J2 is said to be uniformly
proper if there exists a function f : N → N such that
1) dG1(J(L
i ), J(L
j )) ≤ n ⇒ dG2(φ(J(L
i )), φ(J(L
j ))) ≤ f(n)
2) dG2(φ(J(L
i )), φ(J(L
j ))) ≤ n ⇒ dG1(J(L
i ), J(L
j )) ≤ f(n).
When Ji consists of all singleton subsets of Γ1,Γ2, we shall refer to φ as
a uniformly proper map from Γ1 to Γ2.
Note: We observe that if Ji is just the collection of singleton sets in Γi,
then a uniformly proper map between J ’s is the same as a quasi-isometry
between Γi’s. This can be seen by putting n = 1 in conditions 1 and 2 above
and then using the fact that graphs have edge length one. Hence what is
important here is that J ’s are infinite diameter sets.
Definition 3.2. A map q from Γ1 to Γ2 is said to pair the sets J1 and J2
as φ does if there exists a function h : N → N such that
dG(p, J
j )) ≤ n ⇒ dG(q(p), φ(J(L
j ))) ≤ h(n).
The following Lemma generalises Lemma 3.1 of Schwarz [Sch97], where
the result is proven in the special case of a symmetric pattern of geodesics
in Hn. The referee pointed out to us that the Lemma follows from Lemma
7 of [NR03] by Niblo and Reeves.
Lemma 3.3. For M,m > 0, there exists R > 0, such that the following
holds.
Let L1, · · · , LM be distinct translates of the limit set of a quasiconvex sub-
group H of a hyperbolic group G, such that dG(Ji, Jj) ≤ m for all i, j =
1 · · · ,M and Ji = J(Li). Then there exists a ball of radius R meeting Ji
for all i = 1 · · · ,M .
Proof: If ∩M1 Li 6= ∅, choose any point p ∈ J(∩
1 Li). Then B1(p) intersects
all Ji and we are through.
14 MAHAN MJ
Suppose therefore that ∩m1 Li = ∅. We proceed by induction on M . There
exists RM−1 such that a ball of radius RM−1 meets Ji for i = 1 · · ·M − 1.
We now proceed by contradiction. If no such R exists for M , we have
collections {Lk1 , · · · , L
M}, k ∈ N such that a ball of radius RM−1 meets
Jki , i = 1 · · ·M−1 but no ball of radius k meets J
i , i = 1 · · ·M . In particular,
(since J(∩M−11 L
i ) ⊂ ∩
i )), if ∩
i 6= ∅, then Nk(J(∩
i )) ∩
JMi = ∅.
For all i, j, k, choose points pkij ∈ J
i such that dG(p
ij, p
ji) ≤ m.
Assume by G-invariance of J that the ball of radius RM−1 centered at
1 ∈ ΓG meets J
i , i = 1 · · ·M − 1. Therefore J
M lies outside a k-ball about
Since the collection of Ji’s through 1 is finite, therefore assume after pass-
ing to a subsequence if necessary, that
1) {Jki }k is a constant sequence for i = 1 · · ·M − 1. Hence, {L
i }k is a con-
stant sequence Li (say) for i = 1 · · ·M − 1.
2) pkiM → piM ∈ ∂G for i = 1 · · ·M − 1. Hence p
Mi → piM ∈ ∂G. Further,
by (1) above, piM ∈ Li.
3) LkM converges to a closed set Z ⊂ ∂G. By Proposition 2.3, Z must be a
singleton set {z}.
4) JkM lies outside Bk(1) ∪ Nk(J(∩
i )). If ∩
1 Li 6= ∅, then assume
further by G-invariance, that 1 ∈ J(∩M−11 L
i ). Also, using Theorem 1.12
due to Short [Sho91], and translating by an appropriate element of ∩M−11 H
we may assume that 1 ∈ J(∩M−11 L
i ) is closest to J
Now, pkMi ∈ J
M and hence by (3) above, p
Mi → z ∈ ∂G. Combining
this with (2) above, we get z = piM for all i = 1 · · ·M − 1. Therefore,
z ∈ ∩M−11 Li 6= ∅.
But dG(1, J
M ) = dG(J(∩
i ), J
M ) ≥ k. Let zk ∈ J
M such that
dG(1, J
M ) = dG(1, zk) = dG(J(∩
i ), J
M ) ≥ k.
Then the Gromov inner product (zk, p
iM )1 is uniformly bounded above.
Therefore (zk, piM )1 is uniformly bounded above. Hence finally (z, piM )1 is
bounded above. In particular z 6= piM . This is the contradiction that proves
the Lemma. ✷
Definition of q
Let φ be a uniformly proper (bijective, by definition) map from J1 → J2.
We shall now show tha there exists a quasi-isometry q from Γ1 to Γ2 which
pairs the sets J1 and J2 as φ does.
We will define a map q : Γ1 → Γ2 which pairs J1 with J2 as φ does and
prove that q is a quasi-isometry as promised.
Choose K > 0 such that the K neighborhood BK(g) of g ∈ Γ1 has greater
than w2 ( the width of H2 in G2 ) J
i ’s passing through it.
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 15
Let J
(for j = 1, 2 ) denote the collection of J
i ’s passing through
NK(g) for g ∈ Γj, j = 1, 2.
By the proof of Proposition 2.3, there exists M = M(K) (independent of
g ∈ Γ1) such that at most M J
i ’s in J
K,g pass through NK(g). Since φ is a
bijective pairing, φ(J
K,g) has at least (w2 +1) and at most M(K) elements
in it. By definition of w2, and by Theorem 1.12 at least two of the limit sets
of the φ(J1i )’s are disjoint. Let L
1 and L
2 denote these limit sets. Hence, by
Corollary 2.6, for any K1 ≥ f(K), there exists D such that the collection of
points
{p ∈ Γ2 : d2(p, J
2 ) ≤ K1, d2(p, J
2 ) ≤ K1}
has diameter less than D.
Also, by uniform properness of φ,
d2(φ(J
m), φ(J
n)) ≤ f(2K)
for J1m, J
n passing through NK(g) (independent of g).
Summarising,
1) L21 and L
2 are disjoint.
2) But, by Lemma 3.3, using m = f(2K) and M = M(K), there exists
R = R(K) and a ball of radius R meeting each φ(J1i ).
3) For anyK1, there existsD, such that {p ∈ Γ2 : d2(p, J
2 ) ≤ K1, d2(p, J
2 ) ≤
K1} has diameter less than D.In particular, we may choose K1 = R.
Define q(g) to be the center of the ball of radius R obtained in (2) above.
By (3), q(g) is thus defined upto a uniformly bounded amount of discrepancy
for all g ∈ Γ1.
Lemma 3.4. q is uniformly proper with respect to the metrics d1, d2.
Proof: The proof is an almost exact replica of Lemma 3.2 of Schwarz [Sch97]
and we content ourselves with reproducing the heuristics of his argument
here.
If x, y are close in Γ1, then the pairwise minimal distances between ele-
ments of JxK1 and J
is uniformly bounded above. Hence, by Lemma 3.3,
there exists a uniform upper bound to the radius of a minimal radius ball
intersecting all elements of φ(JxK1) as well as φ(J
). Also, since the center
w of such a ball is defined upto a bounded amount of discrepancy, it must be
at a bounded distance from both q(x) as well as q(y). Hence d2(q(x), q(y))
is uniformly bounded, i.e. close.
Conversely, suppose that q(x), q(y) are close. First, by Lemma 3.3, there
exists a uniform upper bound R on radius of minimal radius balls B1, B2
centered at q(x), q(y), intersecting all elements of φ(J xK1), φ(J
K1) respec-
tively. Then the (R + d2(q(x), q(y))) ball about q(x) (or q(y)) meets every
element of φ(J xK1) as well as φ(J
K1). Since φ is uniformly proper, this means
that there is a uniform upper bound on the minimal radius of a ball meeting
16 MAHAN MJ
every element of (J xK1) as well as (J
K1). As before, d1(x, y) is uniformly
bounded, i.e. x, y are close. ✷
Similarly, we can construct q−1 using the bijective pairing φ−1 such that
q−1 is uniformly proper. Also, from Lemma 3.3 q, q−1 composed with each
other in either direction is close to the identity.
Since φ pairs L1, L2 bijectively and is uniformly proper from J1 to J2,
therefore by invariance of J2 under G2, every point of Γ2 lies close to the
image of q. Therefore q is uniformly proper, by Lemma 3.4 above, from Γ1
onto a net in Γ2. Hence q is a quasi-isometry. This concludes the proof of
the main theorem of this subsection.
Theorem 3.5. Let φ be a uniformly proper (bijective, by definition) map
from J1 → J2. There exists a quasi-isometry q from Γ1 to Γ2 which pairs
the sets J1 and J2 as φ does.
We have thus shown one aspect of relative rigidity, viz. upgrading a
uniformly proper map between Ji’s to a quasi-isometry between Γi’s. In
the next subsection, we shall deduce the second aspect, viz. isomorphism of
C-complexes.
3.2. C-Complexes. By Theorem 3.5 we obtain a quasi-isometry q from Γ1
to Γ2 which pairs J1 and J2 as φ does. Since q is a quasi-isometry, it extends
to a quasiconformal homeomorphism from ∂G1 to ∂G2. Also, for all α > 0,
there exists β > 0 such that
d1(x, J
i ) ≤ α ⇒ d2(q(x), φ(J
i )) ≤ β
and conversely,
d2(y, J
i ) ≤ α ⇒ d1(x, φ
−1(J2i )) ≤ β
In particular, ∂q maps the limit set L1i homeomorphically to the limit set
of φ(J1i )). Hence, ∂q preserves intersection patterns of limit sets. Since φ
pairs J1 with J2 as q does, summarising we get:
Lemma 3.6. The following are equivalent.
1) ∩ki=1L
i = ∅
2) ∩ki=1∂q(L
i ) = ∅
3) ∩ki=1φ(L
i ) = ∅
Hence by the definition of the C-complexes C(G1,H1) and C(G2,H2),
we find that ∂q induces an isomorphism of C(G1,H1) with C(G2,H2). We
conclude:
Theorem 3.7. Let φ : J1 → J2 be a uniformly proper map. Then φ induces
an isomorphism of C(G1,H1) with C(G2,H2).
Note: In Theorem 3.5 and Theorem 3.7 we start with the assumption that
there exists a uniformly proper pairing of the collections J1 and J2. This can
be translated to a pairing of collections of limit sets L1 and L2. Theorem 3.5
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 17
then says that the pairing of the Ji’s (or Li’s) is induced by a quasi-isometry
from Γ1 to Γ2. Thus, the existence of a uniformly proper pairing implies
the existence of a quasi-isometry between the Γi’s, i.e. an ambient extension
(or, equivalently, a quasiconformal homeomorphism between ∂Gi’s).
Also Theorem 3.7 shows that a uniformly proper pairing induces an iso-
morphism of the C-complexes C(Gi,Hi). This is reminiscent of the initial
step in the proof of rigidity theorems for higher rank symmetric spaces,
where Tits complexes replace C-complexes.
3.3. Cross Ratios, Annular Systems and a Dynamical Formulation.
In this subsection, we give a more intrinsic formulation of Theorems 3.5 and
3.7. The hypothesis of these theorems is given in terms of distances between
elements of Ji. A more intrinsic way of formulating this hypothesis would
be in terms of the action of Gi on ∂Gi, i = 1, 2. In this case, the distance
between J il , J
m can be approximated by the hyperbolic cross-ratio of their
limit sets. This was described in detail by Bowditch [Bow98]. We give the
relevant definitions and Theorems below and then dynamically reformulate
Theorems 3.5 and 3.7.
Let M be a compactum.
Definition 3.8. An annulus A is an ordered pair (A−, A+) of disjoint
closed subsets of M such that M \ (A− ∪ A+) 6= ∅. An annulus system
is a collection of such annulii. If A = (A−, A+), then −A = (A+, A−). An
annulus system is symmetric if A ∈ A ⇒ −A ∈ A.
Given a closed set K ⊂ M and an annulus A, we say that K < A if
K ⊂ intA−. Also, A < K if K < −A.
If A,B are annulii, we say that A < B if M = intA− ∪ intB+.
Fix an annulus system A. Given closed sets K,L ⊂ M , we say that the
annular cross-ratio (K|L)A ∈ N∪∞ for the maximal number n ∈ N such
that we can find annulii A1, · · ·An ∈ A such that
K < A1 < · · · < An < L
. We set (K|L)A = ∞ if there is no such bound.
Thus (K|L)A is the length of the maximal chain of nested annulii sep-
atrating K,L. For two point sets {x, y} = K and {z, w} = L, we write
(K|L)A as (xy|zw)A.
One of the crucial results of [Bow98] is:
Theorem 3.9. (Bowditch [Bow98]) Suppose a group G acts as a uniform
convergence group on a perfect metrizable compactum M . Then there exists
a symmetric G-invariant annulus system A such that if x, y, z, w are distinct
elements in M , then the theree quantities (xy|zw)A, (xz|yw)A, (xw|zy)A are
all finite and at least two of them are zero. Also, if x 6= y, then (x|y)A >
0. Further, G is hyperbolic, and dG(J(K), J(L)) differs from (K,L)A upto
bounded additive and multiplicative factors.
18 MAHAN MJ
Combining Theorems 3.5 , 3.7 with Proposition 2.3 and Theorem 3.9,
we get the dynamical formulation we promised. Let C0c (M) denote the
collection of closed subsets of M containing more than one point. (Replacing
dGi by cross-ratios (..|..)i in Definition 3.1 we get the corresponding notion
of a map being uniformly proper with respect to the cross-ratios (..|..)1 and
(..|..)2 in the theorem below. Similarly for the homeomorphism q.)
Theorem 3.10. Let G1, G2 be uniform convergence (hence hyperbolic) groups
acting on compacta M1,M2 respectively. Also, let Ai (for i = 1, 2 ) be Gi-
invariant annulus systems and let (..|..)i denote the corresponding annular
cross-ratios.
Let H1,H2 be subgroups of G1, G2 with limit sets Λ1,Λ2. Suppose that the
set Li of translates of Λi (for i = 1, 2) by essentially distinct elements of Hi
in Gi forms a discrete subset of C
c (Mi).
Also assume that there exists a bijective function φ : L1 → L2 and that this
pairing is uniformly proper with respect to the cross-ratios (..|..)1 and (..|..)2.
(1) Hi is quasiconvex in Gi
(2) There is a homeomorphism q : M1 → M2 which pairs L1 with L2 as
φ does. Further, q is uniformly proper with respect to the cross-ratios
(..|..)1 and (..|..)2 on M1, M2 respectively.
(3) q (and hence also φ) induces an isomorphism of C-complexes C(G1,H1)
with C(G2,H2).
Thus from a uniformly proper map with respect to the pseudometrics on
Li’s induced by cross-ratios we infer a quasi-isometry that is an ambient
extension as also a (simplicial) isomorphism of C-complexes.
3.4. Axiomatisation, Relative Hyperbolicity. For classes of pairs (X,J ),
what did we really require to ensure relative rigidity? Assume (X, d) is a
metric space and let the induced pseudometric on J be also denoted by d.
1) For all k > 0 there exists M ∈ N such that for all x ∈ X, Nk(x) meets at
most M of the J ’s in J . (This is a coarsening of the notion of height.)
2) For all K ∈ N, there exists k = k(K) > 0 such that for all x ∈ X, Nk(x)
meets at least K of the J ’s in J . (This is the converse condition to (1).)
3) For all k > 0, n ∈ N there exists K > 0 such that for any collection
J1, · · · , Jn ∈ J with d(Ji, Jj) ≤ k, there exists a ball of radius at most K
meeting all the Ji’s.
4) There exists N ∈ N such that for all k > 0 there exists K = K(k) > 0
such that the following holds.
For all n ≥ N and J1, · · · , Jn ∈ J , the set of points {x ∈ X : Nk(x) ∩ Ji 6=
∅, i = 1 · · · n} is either empty or has diameter bounded by K.
Given (1)-(4), the construction of q : X1 → X2 from a uniformly proper
pairing φ : J1 → J2 goes through as in Theorem 3.5. In short, pick N from
(4). From (2), pick k = k(N). Now for all x ∈ X1, consider the collection
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 19
of J ’s in J1 that meet Nk(x). By (1) there is an upper bound M = M(k)
on the number of such J ’s. Map these over to J2. Any two of these are at a
distance of at most m apart where m depends on φ and k. From (3) choose
K = K(M,m) such that a ball of radius K meets all these. Set q(x) to be
the center of such a ball. By (4), q(x) is defined upto a uniformly bounded
degree of discrepancy. The rest of the proof goes through as before. Hence
(1)-(4) define sufficient conditions for relative rigidity for a class of pairs
(X,J ).
With these conditions, it is easy to extend Theorem 3.5 to pairs (X,J )
where X is (strongly) hyperbolic relative to the collection J . Conditions
(1) and (2) are trivial.
Condition (3) follows from “bounded penetration” (see Farb [Far98]). For
any subcollection J1 of J with d(Ji, Jj) ≤ C0 (for all Ji, Jj ∈ J1), fix any
two J1, J2 ∈ J and a geodesic γ12 of length ≤ C0 joining them. Construct
an electric triangle for triples J1, J2, J3 ∈ J1 of horosphere-like sets for arbi-
trary J3 ∈ J1, such that the hyperbolic geodesics γ13, γ32 joining J1, J3 and
J3, J2 respectively have lengths bounded by C0. Then γ12 and γ13 meet J1
at a uniformly bounded distance from each other by bounded penetration.
To see this, first note that J1, J2 can be joined by two paths, one consisting
of one side of the triangle and the other the union of the two remaining sides
of the triangle and both paths have electric length bounded by 2C0; in par-
ticular both paths are uniform quasigeodesics (with quasigeodesic constant
depending only on C0). They may be converted to quasigeodesics without
backtracking by not increasing lengths. Thus γ13 ∪ γ32 decomposes as the
union of a quasigeodesic without backtracking γ′12 joining J1, J2 and (possi-
bly) a uniformly bounded (≤ C0) number of loops of length not longer than
2C0. The entire quasigeodesic without backtracking γ
12 lies near γ12 for all
J3 ∈ J1. The same holds for the loops of bounded length (since they in turn
may be regarded as uniform quasigeodesics without backtracking starting
and ending at the same point.) In particular J3 lies at a uniformly bounded
distance D0 from γ12. Since γ12 has length bounded by C0, and J3 may be
chosen arbitrarily satisfying the hypothesis of (3) above, it follows that for
any x ∈ γ12, d(x, J3) ≤ (C0 + D0) for all J3 ∈ J1. Condition (3) follows.
(Results closely related to the proof of Condition (3) here occur as Lemma
3.11 of [Mj05] and Prop. 8.6 of [HW06].)
Condition (4) follows from the fact that for a pair of distinct Ji, Jj ,
Nk(Ji) ∩Nk(Jj) is either empty or has diameter bounded by some C(k).
We have thus shown:
Theorem 3.11. Let Xi be (strongly) hyperbolic relative to collections Ji
(i = 1, 2). Let φ be a uniformly proper (bijective, by definition) map from
J1 → J2. There exists a quasi-isometry q from X1 to X2 which pairs the
sets J1 and J2 as φ does.
20 MAHAN MJ
By work of Hruska and Kleiner [HK04], CAT(0) spaces with isolated flats
are (strongly) hyperbolic relative to maximal flats. Hence we have from
Theorem 3.11 above:
Corollary 3.12. Let Xi be CAT(0) spaces with isolated flats and let Ji
denote the collections of maximal flats (i = 1, 2). Let φ be a uniformly
proper (bijective, by definition) map from J1 → J2. There exists a quasi-
isometry q from X1 to X2 which pairs the sets J1 and J2 as φ does.
3.5. Symmetric Spaces of Higher Rank. We now consider CAT(0)
spaces which are at the other end of the spectrum. Let M be a compact lo-
cally symmetric space and T a totally geodesic torus with rank = rank(M).
Take X = M̃ and J to be the lifts of T to M̃ . As these are all equivariant
examples (i.e. J is invariant under a cocompact group action), it is enough
to check (1)-(4) at a point.
(1) and (2) are clear. To prove condition (4), we consider ∩iNk(Fi) and
it is easy to bound from below the N appearing in Condition (4) (Section
3.4) in terms of the size of the Weyl group and rank. In that case, ∩iNk(Fi)
has bounded diameter or is empty.
Finally, to prove (3), we proceed as in Lemma 3.3. As in Lemma 3.3
we assume by induction that any k flats {F1, · · · , Fk} that ”coarsely pair-
wise intersect at scale D” (i.e. ND(Fi) ∩ ND(Fj) 6= ∅ ) intersect coarsely
(i.e. ∩i=1···kND′(Fi) 6= ∅ for some D
′ = D′(D, k)). To get to the induc-
tive step, we suppose that for i = k + 1, we have collections of worse and
worse counterexamples. Consider a maximal collection F = {F1, · · · , Fk}
of maximal flats whose ”coarse intersection at scale D” ∩iND(Fi) = F is
non-null. Translate the collection by a group element so that a fixed point
0 (thought of as the origin) lies on the intersection F . Now take a sequence
of maximal flats F j whose D-neighborhoods ND(F
j) intersect each ND(Fi),
but dj = d(F
j , F ) = d(0, F ) ≥ j. We scale the metric on (X, d) by a factor
of dj to obtain a sequence of metric spaces (X,
) converging (via a non-
principal ultrafilter) to a Euclidean building X∞ (this fact is due to Kleiner
and Leeb [KL97b], but we shall only mildly need the exact nature of X∞).
Fi’s converge to flats F
i ⊂ X
∞ and F j ’s converge to a flat G∞ ⊂ X∞.
Then the collection G = F∞i , G
∞ satisfy the following:
(P1) Each element of G is a flat in X∞
(P2) By induction, the intersection of any i elements of G is non-empty and
convex for i ≤ k
(P3) The intersection of all the (k + 1) elements of G is empty.
Consider the subcomplex K = G∞
i of X
∞. Then K is a union of
r-flats, where r = rank(X). In particular, the homology groups Hn(K) = 0
for n > r. On the other hand, if we consider the nerve of the covering of K
by the sets G∞, F∞i , then using the properties (P1), (P2), (P3) to compute
Cech homology groups, we conclude that K has the same homology groups
as the boundary of a k-simplex. In particular, Hk(K) = Z. For k > r this
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 21
is a contradiction, finally proving Condition (3). Thus we conclude:
Theorem 3.13. Let Xi be symmetric spaces of non-positive curvature, and
Ji be equivariant collections of lifts of a maximal torus in a compact locally
symmetric space modeled on Xi (i = 1, 2). Let φ be a uniformly proper
(bijective, by definition) map from J1 → J2. There exists a quasi-isometry
q from X1 to X2 which pairs the sets J1 and J2 as φ does.
Combining Theorem 3.13 with the quasi-isometric rigidity theorem of
Kleiner-Leeb [KL97b] and Eskin-Farb [EF 3] we can upgrade the quasi-
isometry q to an isometry i.
Corollary 3.14. Let Xi be symmetric spaces of non-positive curvature, and
Ji be equivariant collections of lifts of a maximal torus in a compact locally
symmetric space modeled on Xi (i = 1, 2). Let φ be a uniformly proper
(bijective, by definition) map from J1 → J2. There exists an isometry i
from X1 to X2 which pairs the sets J1 and J2 as φ does.
Remark 3.15. The technique of using asymptotic cones and the nerve of
the covering by flats can be generalised easily to equivariant flats of arbitrary
(not necessarily maximal) rank.
We conclude this paper with two (related) questions:
Question 1: In analogy with a Theorem of Ivanov, Korkmaz, Luo (see for
instance [Luo00] ), regarding the automorphism group of the curve complex,
we ask:
If the C-Complex C(G,H) of a pair (G,H) (for G a hyperbolic group and
H a quasiconvex subgroup) is connected, is the automorphism group of
C(G,H) commensurable with G?
Question 2: Consider the pair (G,H), with G a hyperbolic group and H
a quasiconvex subgroup. Let (J , d) be the collection of joins as in Lemma
3.3 with the induced pseudometric. For a uniformly proper map φ from
(J , d) to itself, is there an isometry pairing the elements of J as φ? We
have proved in Theorem 3.5 that a quasi-isometry q exists pairing the J
as φ does. The question is whether q may be upgraded to an isometry, or
better, to an element of G? This question is related to the notion of pattern
rigidity introduced by Mosher, Sageev and Whyte in [MSW04].
References
[BCG98] G. Besson, G. Courtois, and S. Gallot. A real Schwarz lemma and some appli-
cations. Rend. Mat. Appl., VII. Ser. 18, No.2, pages 381–410, 1998.
[BDM05] J. Behrstock, C. Drutu, and L. Mosher. Thick metric spaces, relative hyper-
bolicity, and quasi-isometric rigidity. preprint, arXiv:math.GT/0512592, 2005.
[Bow97] B. H. Bowditch. Relatively hyperbolic groups. preprint, Southampton, 1997.
[Bow98] B. H. Bowditch. A topological characterization of hyperbolic groups. J. A. M.
S. 11, pages 643–667, 1998.
[DS05] C. Drutu and M. Sapir. Tree-graded spaces and asymptotic cones of groups.
Topology 44, pages 959–1058, 2005.
22 MAHAN MJ
[EF 3] A. Eskin and B. Farb. Quasi-flats and rigidity in higher rank symmetric spaces.
. J. Amer. Math. Soc., 10, pages pp. 653–692., (1997),no. 3.
[Far98] B. Farb. Relatively hyperbolic groups. Geom. Funct. Anal. 8, pages 810–840,
1998.
[FM98] B. Farb and L. Mosher. A rigidity theorem for the solvable Baumslag-Solitar
groups. Inventiones Math. , Vol. 131, No.2, pages 419–451, 1998.
[FM00] B. Farb and L. Mosher. On the asymptotic geometry of abelian-by-cyclic
groups. Acta Math. , Vol. 184, No.2, pages 145–202, (2000).
[FM 3] B. Farb and L. Mosher. Quasi-isometric rigidity for the solvable Baumslag-
Solitar groups, II. Inventiones Math. , Vol. 137, pages 273–296, (1999), No.
[FS96] B. Farb and R.E. Schwartz. The large-scale geometry of Hilbert modular
groups. J. Differential Geom. 44(3), page 435478, 1996.
[GdlH90] E. Ghys and P. de la Harpe(eds.). Sur les groupes hyperboliques d’apres
Mikhael Gromov. Progress in Math. vol 83, Birkhauser, Boston Ma., 1990.
[GMRS97] R. Gitik, M. Mitra, E. Rips, and M. Sageev. Widths of Subgroups. Trans.
AMS, pages 321–329, Jan. ’97.
[Gro85] M. Gromov. Hyperbolic Groups. in Essays in Group Theory, ed. Gersten,
MSRI Publ.,vol.8, Springer Verlag, pages 75–263, 1985.
[HK04] G. Christopher Hruska and Bruce Kleiner. Hadamard spaces with isolated
flats. arxiv:math.GR/0411232, preprint 2004.
[HW06] G. Christopher Hruska and D. T. Wise. Packing Subgroups in Relatively Hy-
perbolic Groups. arxiv:math.GR/0609369, preprint 2006.
[KL97a] M. Kapovich and B. Leeb. Quasi-isometries preserve the geometric decompo-
sition of Haken manifolds. Inventiones Math., Vol. 128, F. 2, pages 393–416,
(1997).
[KL97b] B. Kleiner and B. Leeb. Rigidity of quasi-isometries for symmetric spaces and
euclidean buildings. IHES Publ. Math. 86, page 115197, (1997).
[Luo00] F. Luo. Automorphisms of the complex of curves. Topology 39, pages 283–298.,
(2000).
[Mit98] Mahan Mitra. Cannon-Thurston Maps for Trees of Hyperbolic Metric Spaces.
Jour. Diff. Geom.48, pages 135–164, 1998.
[Mit04] Mahan Mitra. Height in Splittings of Hyperbolic Groups. Proc. Indian Acad.
of Sciences, v. 114, no.1, pages 39–54, Feb. 2004.
[Mj05] Mahan Mj. Cannon-Thurston Maps, i-bounded Geometry and a Theorem of
McMullen. preprint, arXiv:math.GT/0511041, 2005.
[Mos73] G. D. Mostow. Strong Rigidity of Locally Symmetric Spaces. Princeton Uni-
versity Press, 1973.
[MSW03] L. Mosher, M. Sageev, and K. Whyte. Quasi-actions on trees I.Bounded va-
lence. Annals of Mathematics, 158, arXiv:math.GR/0010136, page 115164,
(2003).
[MSW04] L. Mosher, M. Sageev, and K. Whyte. Quasi-actions on trees II.Finite depth
Bass-Serre trees. arXiv:math.GR/0405237, (2004).
[NR03] G. Niblo and L. Reeves. Coxeter groups act on cat(0) cube complexes. J. Group
Theory 6, pages 399–413, (2003).
[Pau96] F. Paulin. Un groupe hyperbolique est dtermine par son bord. J. London Math.
Soc. 54, pages 50–74, 1996.
[Roe95] J. Roe. Index Theory, Coarse Geometry, and Topology of Manifolds. CBMS
series, AMS, 1995.
[Sag95] M. Sageev. Ends of group pairs and non-positively curved cube complexes.
Proc. London Math. Soc. 71, pages 585–617, 1995.
[Sch97] R. E. Schwarz. Symmetric patterns of geodesics and automorphisms of surface
groups. Invent. math. Vol. 128, No. 1, page 177199, 1997.
RELATIVE RIGIDITY, QUASICONVEXITY AND C-COMPLEXES 23
[Sho91] H. Short. Quasiconvexity and a theorem of Howson’s. Group Theory from a
Geometrical Viewpoint (E. Ghys, A. Haefliger, A. Verjovsky eds.), 1991.
RKM Vivekananda University, Belur Math, WB-711 202, India
