COARSE OBSTRUCTIONS TO POSITIVE SCALAR CURVATURE IN ...

COARSE OBSTRUCTIONS TO POSITIVE SCALAR CURVATURE IN NONCOMPACT
ARITHMETIC MANIFOLDS
STANLEY S. CHANG
ABSTRACT. Block and Weinberger show that an arithmetic manifold can be endowed with a positive
scalar curvature metric if and only if its É-rank exceeds ¾. We show in this article that these
metrics are never in the same coarse class as the natural metric inherited from the base Lie group.
Furthering the coarse £ -algebraic methods of Roe, we find a nonzero Dirac obstruction in the Ã-
theory of a particular operator algebra which encodes information about the quasi-isometry type of
the manifold as well as its local geometry.
I. Introduction
In the course of showing that no manifold of non-positive sectional curvature can be endowed
with a metric of positive scalar curvature, Gromov and Lawson [9] were led to consider what we
would now call restrictions on the coarse equivalence type of complete noncompact manifolds
of such positively curved metrics. In particular, they showed that such metrics cannot exist in
manifolds for which there exists a degree one proper Lipschitz map from the universal cover to
Ê Ò , now understood to be essentially a coarse condition. Block and Weinberger [2] investigate
the situation in which no coarse conditions are imposed upon the complete metric, focusing on
quotients ÒÃ of symmetric spaces associated to a lattice in an irreducible semisimple Lie
group . They show that the space Å ÒÃ can be given a complete metric of uniformly
positive scalar curvature ¼ if and only if is an arithmetic group of É-rank exceeding ¾.
Note that the theorem of Gromov and Lawson [9] mentioned above establishes this theorem
in the case of rankÉ ¼. In the higher rank cases, for which the resulting quotient space is
noncompact, the metrics constructed by Block and Weinberger are however wildly different in the
large when compared to the natural one on Å inherited from the base Lie group . In fact, their
examples are all coarse quasi-isometric to rays. Their theory evokes a natural question: Can the
metric be chosen so that it is simultaneously uniformly positively curved and coarsely equivalent
to the natural metric induced by
One of the important developments in analyzing positive scalar curvature in the context of noncompact
manifolds, especially when restricted to the coarse quasi-isometry type, is introduced by
Roe [15], [14], who considers a higher index, analogous to the Novikov higher signature, that lives
naturally in the Ã-theory of the £ -algebra £´Å µ of operators on Å with finite propagation
speed. He describes a map from the Ã-theory group à £´ £´Å µµ to the Ã-homology à £´Åµ
of the Higson corona space which admits a dual transgression map À £´Åµ À £´Å µ. If
the Dirac operator on Å is invertible, then the image of its index in à £´Åµ vanishes, leading to
vanishing theorems for the index paired with coarse classes from the transgression of Å. Roe’s
1

2 STANLEY S. CHANG
construction is used to show that a metric on a noncompact manifold cannot be uniformly positively
curved if the Higson corona of the manifold contains an essential ´Ò ½µ-sphere. Such
spaces are called ultraspherical manifolds.
The usual Roe algebra, however, is unsuited to provide information about the existence of positive
scalar curvature metrics that exist on arithmetic manifolds, in particular because the corona is
too anemic. For example, the space at infinity of a product of punctured two-dimensional tori is a
simplex and therefore contractible. As a coarse object, the Ã-theory of the Roe algebra associated
to this multi-product space can be identified with à £´ £´Ê Ò ¼ µµ. Yet Higson, Roe and Yu [11] have
shown that the Euclidean cone È on a single simplex È must satisfy à £´ £´È µµ ¼. Since the
Euclidean hyperoctant Ê Ò ¼ is simply the cone on an ´Ò ½µ-simplex, we find that à £´ £´Ê Ò ¼ µµ
is the trivial group and hence no obstructions are detectable. Even by considering the fundamental
group of the manifold by tensoring the Roe algebra with £ ½´Å µ we find this detection process
unfruitful, since the Ã-theory group à £´ £´Å µ Å £ ½´Å µµ vanishes as well. What seems to be
critical is how different elements of the fundamental group at infinity can be localized to different
parts of the space at infinity.
In this article, we shall provide coarse indicial obstructions in the following noncompact manifolds:
a finite product of punctured two-dimensional tori, a finite product of hyperbolic manifolds,
the double quotient space ËÄ Ò´µÒËÄ Ò´ÊµËÇ Ò´Êµ of unit volume tori, and more generally the
double quotient space ÒÃ, where is an irreducible semisimple Lie group, Ã its maximal
compact subgroup and
an arithmetic subgroup of . Note that the first two do not correspond
to irreducible quotients, but an analysis of these spaces gives us the proper insight to attack the
more general cases. A further research project will analyze this problem without the irreducibility
assumption. The key feature in these particular manifolds Å is that they contain hypersurfaces Î
that are coarsely equivalent to a product ¢Í of Euclidean space with some iterated circle bundle
Í (i.e. a torus, Heisenberg group, or more generally a group of unipotent matrices). Moreover
such a hypersurface decomposes the manifold Å into a coarsely excisive pair ´ µ for which
Å and Î . A generalized form of the Mayer-Vietoris sequence constructed by
Higson, Roe and Yu [11] provides the following:
¡¡¡ Ã £´ ´µµ £ ¨ Ã £´ ´µµ £ Ã £´ ´Å £ µµ Ã £
£ ½´ ´Î µµ ¡¡¡
The boundary map à £´ £ ´Å µµ à £ ½´ £ ´ ¢ Í µµ sends Ind Å ´µ, the index of the
spinor Dirac bundle on the universal cover lifted from that on Å, to Ind ¢Í ´µ. To see that
these indices are indeed nonzero, we note that there is a boundary map Ã
£ £ ½´ ´ ¢ Í µµ
Ã
£ £ Ñ ´ ´Ê ¢ Í µµ, which sends index to index. We show that the index of the Dirac operator
in the latter group, however, is nonzero by noting that the Gromov-Lawson-Rosenberg conjecture
is true for nilpotent groups and hence provides an appropriate nonzero obstruction.
I would like to thank Alex Eskin, Benson Farb, Nigel Higson, Thomas Nevins, Mel Rothenberg,
John Roe, Stephan Stolz and Guoliang Yu for very useful conversations. In particular, I would like
to acknowlege the role of my advisor Shmuel Weinberger in pointing out the strength of certain
tools in the realization of these theorems.
II. The Generalized Roe Algebra
The coarse category is defined to contain metric spaces as its objects and maps ´ µ
´ µ between metric spaces as its morphisms satisfying the following expansion and properness

COARSE OBSTRUCTIONS TO POSITIVE SCALAR CURVATURE IN NONCOMPACT ARITHMETIC MANIFOLDS 3
conditions: (a) for each ʼ there is a corresponding ˼ such that, if ´Ü ½ Ü ¾ µ Ê in ,
then ´ ´Ü ½ µ´Ü ¾ µµ Ë, (b) the inverse image ½´µ under of each bounded set is
also bounded in . Such a function will be designated a coarse map, and two coarse maps
are said to be coarsely equivalent if their mutual distance of separation ´ ´Üµ´Üµµ is
uniformly bounded in Ü. Naturally two metric spaces are coarsely equivalent if there exist maps
from one to the other whose compositions are coarsely equivalent to the appropriate identity maps.
Two metrics ½ and ¾ on the same space Å are said to be coarsely equivalent if ´Å ½ µ and
´Å ¾ µ are coarsely equivalent metric spaces.
Following Roe [14], we recall a Hilbert space À is an Å-module for a manifold Å if there is a
representation of ¼´Å µ on À, thatis,a £ -homomorphism ¼´Å µ ´Àµ. We will say that
an operator Ì À À is locally compact if, for all ³ ¾ ¼´Å µ, the operators ̳ and ³Ì are
compact on À. Wedefinethesupport of ³ in an Å-module À to be the smallest closed set à Å
such that, if ¾ ¼´Å µ and ³ ¼,then à is not identically zero. Consider the Å-module
À Ä ¾´ Å µ, where Å is the universal cover of Å endowed with the appropriate metric lifted
from the base space. Let Å Å be the usual projection map and for any ³ ¾ ¼´ Å µ
consider the collection ´³ µ of paths ­ ¼ ½℄ Å in Å originating in Supp ´³µ and ending
in Supp ´µ. Denote by Ä ­ ℄, for­ ¾ ´³ µ, the maximum distance of any two points on the
projection of the curve ­ in Å by ,i.e.Ä ­ ℄ ×ÙÔ Üݾ¼½℄ ´ Æ ­´ÜµÆ ­´Ýµµ.
Definition: Let Å be a manifold with universal cover Å. We say that an operator Ì on ľ´ Å µ
has generalized finite propagation if there is a constant ʼ such that ³Ì is identically zero in
´Àµ whenever ³ ¾ ¼´ Å µ satisfies
Ò Ä ­ ℄ Ê
­¾ ´³µ
The infimum of all such Ê will be the generalized propagation speed of the operator Ì .If
½´Å µ is the fundamental group of Å, we denote by £ ´Å µ to be the norm closure of the £ -
algebra of all locally compact, -equivariant, generalized finite propagation operators on À.
Let Å be a manifold and Å its universal cover. Let Ì À À be an operator on À ľ´ Å µ.
Consider the subset É Å ¢ Å of pairs ´Ñ Ñ
¼ µ for which there exist functions ³ ¾ ¼´ Å µ
such that ³´Ñµ ¼, ´Ñ ¼ µ ¼and ³Ì does not identically vanish. We will say that the support
of Ì is the complement in Å ¢ Å of É. For such two points Ñ Ñ¼ ¾ Å,let­ÑÑ ¼ ¼ ½℄ Å
be the path of least length joining Ñ and Ñ ¼ in Å. We consider the projection of this path into Å
by and take the greatest distance between two points on this projected path. Then it is easy to
see that an operator Ì has generalized finite propagation, as previously defined, if
×ÙÔ
ÑÑ ¼
×ÙÔ ´ Æ ­ ÑÑ ¼´ÜµÆ ­ ÑÑ ¼ ´Ýµµ ½
Üݾ¼½℄
Definition: Consider the norm closure Á of the ideal in
£
´Å µ generated by operators Ì whose
matrix representation, parametrized by Å ¢ Å, satisfies the condition that ´ ¢ µ´Supp Ì µ is
bounded in Å ¢ Å. Then the generalized Roe algebra, denoted by
£
´Å µ, is obtained as the
quotient
£
´Å µÁ. Two operators in £ ´Å µ belong to the same class in £ ´Å µ if their nonzero
entries differ on at most a bounded set when viewed from the perspective of the base space.

4 STANLEY S. CHANG
Examples:
(1) Let Ì Ä ¾´Êµ Ä ¾´Êµ be operator on Ä ¾ -functions on the real line given by ´Ìµ´Üµ
´Ü ·½µfor all ¾ Ä ¾´Êµ and Ü ¾ Ê. Then for any ³ ¾ ¼´Êµ, ´³Ì µ´Üµ ³´Ü ·½µ´Ü ·
½µ´Üµ. If³ is supported at Ñ ½and is supported at Ñ ¼ ¼,then´³Ì µ is nonzero for any
supported at Ü ½. Hence ´¼ ½µ ¾ Supp Ì . It is easy to see that ´Ñ Ñ ¼ µ ¾ Supp Ì if and only
if Ñ ¼ Ñ ½. The propagation speed of Ì is ½. If we write Ì as a matrix parametrized by Ê ¢ Ê,
all the nonzero entries will lie at distance one from the diagonal.
(2) Let Å be the cylinder Ë ½ ¢ Ê with its universal cover Å Ê ¾ . An operator in the algebra
£ ´Å µ will be some Ì À À on ľ´Ê ¾ µ, which is of finite propagation speed (in the usual
sense) in the direction projecting down to the noncompact direction in Å, but has no such condition
in the orthogonal direction corresponding to the compact direction of Å. In this direction, however,
the operator is controlled by the condition that it be -equivariant. It is apparent that the operator,
when restricted to individual fibers, has finite propagation speed, although there is no requirement
that the speed to be uniformly bounded across all fibers.
(3) Let Å
Æ
ÊÈ Ò , Ò ¿, the once-punctured real projective space, expressible as the quotient
´Ë Ò ½ ¢ ʵ ¾ . Certainly Å is coarsely equivalent to the ray ¼ ½µ and is covered by the space
Å Ë Ò ½ ¢ Ê, where the points ´× Öµ and ´ × Öµ are identified by the projection map to
Å. Let Ì Ä ¾´ Å µ Ä ¾´ Å µ be given by the reflection ´Ìµ´× Öµ ´× Öµ. Consider
³ ¾ ¼´ Å µ compactly supported on Ë
Ò ½
¢ ½ ℄ and Ë Ò ½ ¢ ·½℄, respectively.
Notice that ³Ì will never be identically zero, and yet the length Ä ­ ℄ associated to ³ and
will always be at least . Hence the operator Ì is not of generalized finite propagation speed and
therefore not an element of the generalized Roe algebra
£
´Å µ.
The notion of a generalized elliptic operator is available in [14], [15] and [16], but we include
its definition here for completeness.
Definition: Let Å be a space and let À be an Å-module. If is an unbounded self-adjoint
operator on À, then we say that is a generalized elliptic operator on À if
(a) there is a constant ¼ such that, for all Ø ¾ Ê, the unitary operator Ø has bounded
propagation on À and and its propagation bound is less than Ø, and
(b) there is Ò¼ such that ´½ · ¾ µ Ò is locally traceable.
Lemma 1: Let a generalized elliptic operator in Ä ¾´Å˵. Suppose that is the lifted operator
on Å. If ¨ Ê Ê is compactly supported, then ¨´ µ lies in the generalized Roe algebra
£
´Å µ.
Proof: (cf. [14], [6]) Suppose that ¨ has compactly supported Fourier transform and denote by ¨
the Fourier transform of ¨. We may write
¨´ µ
½
¾
½
½
¨´Øµ Ø
Ø
It is known that Ø has finite propagation speed, and since ¨ is compactly supported, the integral
is defined and has a generalized propagation bound. Moreover, by construction is ½´Å µ-
equivariant. So ¨´ µ is ½´Å µ-equivariant as well. Therefore if ¨ is compactly supported, then
¨´ µ lies in
£
´Å µ and passes to an element of the quotient
£
´Å µ. However, functions with

COARSE OBSTRUCTIONS TO POSITIVE SCALAR CURVATURE IN NONCOMPACT ARITHMETIC MANIFOLDS 5
compactly supported Fourier transform form a dense set in ¼´Êµ and the functional calculus map
´ µ is continuous, so the result holds for all ¨ ¾ ¼´Êµ. £
Let Ê Ê be a chopping function on Ê, i.e. an odd continuous function with the property
that ´Üµ ¦½ as Ü ¦½. In addition, denote by
£
´Å µ the multiplier algebra of £ ´Å µ,
that is, the collection of all operators Ë such that ËÌ and ÌËbelong to
£
´Å µ for all Ì ¾ £ ´Å µ.
Then
£
´Å µ contains £ ´Å µ as an ideal. If is a generalized elliptic operator on Å and
its lift to Å,then´ µ belongs to
£
´Å µ. In addition, since ¾ ½ ¾ ¼´Êµ, wehave
´ µ
¾
½ ¾ £ ´Å µ. Moreover, since the ¾-grading renders the decompositions
´ µ
¼ ´ µ
´ µ· ¼
½
¼
¼ ½
it follows that ´ µ·´ µ ¼. By the discussion in [14], it follows that ´ µ is
a Fredholm operator and admits an index Ind ¾ Ã
£ ¼´ ´Å µµ. In addition, any two chopping
functions ½ and ¾ differ by an element of ¼´Êµ. By the lemma above, we have ½´ µ
¾´ µ ¾
£
´Å µ, so they define the same elements of Ã-theory. The common value for Ind
is denoted Ind ´µ and called the generalized coarse index of . We write
£
´Å µ and Ind ´µ
instead of £ ´ Å µ and Ind ´ µ to indicate that the construction is initiated by a generalized Dirac
operator on the base space. The following statements are standard results of index theory; one may
consult [14] and [15] for the essentially identical proof in the nonequivariant case.
Proposition 1: Let be a generalized elliptic operator in Ä ¾´Å˵. If ¼ does not belong to the
spectrum of , then the generalized coarse index Ind vanishes in ü´
£
´Å µµ.
Proposition 2: Let the lift of a generalized elliptic operator in ľ´ Å˵. In the ungraded case,
if there is a gap in the spectrum of , then the index Ind vanishes in ý´
£
´Å µµ.
Corollary: Let Å be a complete spin manifold. If Å has a metric of uniformly positive scalar
curvature in some coarse class, then the generalized coarse index of the spinor Dirac operator
vanishes.
We now embark on the task of computing the Ã-theory of this algebra and of coarse indices.
Let ´Åµ be a proper metric space. For any subset Í Å and ʼ, we denote by Pen´Í ʵ
the open neighborhood of Í consisting of points Ü ¾ Å for which ´Ü Í µ Ê.Let and be
closed subspaces of Å with Å . We then say that the decomposition ´ µ is a coarsely
excisive pair if for each ʼ there is an ˼ such that
Pen´ ʵ Pen´Êµ Pen´ ˵
We wish to analyze this decomposition in the following context.
Given general £ -algebras , and Å for which Å · , we have the Mayer-Vietoris
sequence
¡¡¡ Ã ·½´ Å µ Ã ´ µ Ã ´ µ ¨ Ã ´ µ Ã ´ Å µ ¡¡¡
The standard proof for the existence of such a sequence is developed from the isomorphism
à £´Ì µ ã ½´Åµ, whereÌ is the suspension of Å. A short discussion of this construction
is given in [11]. We are in particular interested in exploiting the boundary map à ´ Å µ

6 STANLEY S. CHANG
à ½´ µ to transfer information about the index of the Dirac operator on a complete noncompact
manifold Å to information about that on some hypersurface Î . For our purposes, we wish to
set Å to be the generalized Roe algebra
£
´Å µ on Å, while and represent analogous operator
algebras on closed subsets and ,where´ µ form a coarsely excisive decomposition of
Å. To construct the boundary map in question, we require a few technical lemmas and notion of
equivariant operators with generalized finite propagation on a subset of Å. The proof of the first
lemma follows the same argument as that in [11] and is stated without proof.
Definition: Let be a closed subspace of a proper metric space Å. Denote by
£
´ Å µ the
£ -algebra of all operators Ì in
£
´Å µ such that Supp Ì Pen´ ½´µÊµ¢Pen´ ½´µÊµ,
for some ʼ. Let
£
´ Å µ be the quotient £ ´ Å µÁ.
Lemma 2: Let ´ µ be a decomposition of Å. Then
(1)
£
´ Å µ·£ ´Åµ £ ´Å µ.
(2)
£
´ Å µ£ ´Åµ £ ´Åµ if in addition we assume that ´ µ is coarsely
excisive.
Lemma 3: Suppose that the inclusion Î Å induces an injection ½´Î µ ½´Å µ on the level
of fundamental groups. There is an isomorphism à £´ ´Î £ µµ ã´ ´Îŵµ.
£
Proof: Let Å Å be the projection map. Consider the £ -algebra £´Pen´ ½´Î µÒµ ½´Å µµ
given by the quotient by Á of the £ -algebra of locally compact, ½´Å µ-equivariant operators on
the Ò-neighborhood penumbra Pen´ ½´Î µÒµ. Then
£ ´Îŵ ÐÑ £´Pen´ ½´Î µÒµ ½´Å µµ
The inclusion map ½´Î µ ¸ Pen´ ½´Î µÒµ is a coarse equivalence. Since by the construction
the generalized Roe algebra its operators are defined up to their bounded parts, the map
induces a series of isomorphisms
à £´ £´ ½´Î µ ½´Å µµµ ã´ £´Pen´ ½´Î µÒµ ½´Å µµ
à £´ ÐÑ £´Pen´ ½´Î µÒµ ½´Å µµ
à £´ ´Îŵµ
£
Since ½´Î µ ¸ ½´Å µ is an injection, the inverse image ½´Î µ Å is a disjoint union
of isomorphic copies of Î , parametrized by the coset space ½´Å µ ½´Î µ. Therefore, there
is a one-to-one correspondence between ½´Å µ-equivariant operators on ½´Î µ and ½´Î µ-
equivariant operators on Î . Hence £´ ½´Î µ ½´Å µµ
£
´Î µ. Wethenhaveà £´ £ ´Î µµ
à £´
£
´Îŵµ, as desired. £
Let ´ µ be a coarsely excisive decomposition of Å such that Î satisfies ½´Î µ ¸
The boundary operator à ´ ´ £ Å µ · £ ´Åµµ à ½´ ´ £ Å µ
½´Å µ.
£
´Åµµ arising from the coarse Mayer-Vietoris sequence is by the previous lemmas truly a
map
à £´ ´Å £ µµ à £
£ ½´ ´Î µµ
Theorem: (Boundary of Dirac is Dirac) Consider a coarsely excisive decomposition ´ µ of
Å and let Î . If à £´ ´Å £ µµ à £
£ ½´ ´Î µµ is the boundary map from the
Mayer-Vietoris sequence derived above, then we have ´ Ind Å ´µµ Ind Î ´µ.

COARSE OBSTRUCTIONS TO POSITIVE SCALAR CURVATURE IN NONCOMPACT ARITHMETIC MANIFOLDS 7
Remark: Here Ind Å ´µ and Ind Î ´µ represent the generalized coarse indices of the spinor
Dirac operators on Å and Î , respectively. We will continue to use a subscript if the space to
which the index is related is ambiguous. The “boundary of Dirac is Dirac” principle is essentially
equivalent to Bott periodicity in topological Ã-theory. In all cases considered here, there are
commutative diagrams relating topological boundary to the boundary operator arising in the Ã-
theory of £ -algebras, and, on the topological side, a consideration of symbols suffices. See [16],
[10], [15] and [21].
Theorem 1: The Ò-fold product Å of punctured two-dimensional tori does not have a metric of
uniform positive scalar curvature in the same coarse equivalence class as the positive hyperoctant
with its standard Euclidean metric.
Proof: Consider the projection map Ô Å Ê Ò Æ Æ
¼
from the product Å Ì ¢¡¡¡¢ Ì to the
positive hyperoctant, where each component Ô is the quasi-isometric projection of the punctured
torus onto the positive reals numbers. Take a hypersurface Ë Ê Ò ¼
sufficiently far from the
origin so that the inverse image of every point on Ë is an Ò-torus, and so the space Î is coarsely
equivalent to the ´¾Ò ½µ-dimensional noncompact manifold Ê Ò ½ ¢ Ì Ò . The complement of
the hypersurface Î consists of two noncompact components. Define to be the closure of the
component containing the inverse image Ô ½´¼µ of the origin in Ê Ò ¼
. Take the closure of
ÅÒ Æ . Then the pair ´ µ forms a coarsely excisive decomposition of the space Å whose
intersection is Î .
Ë
¹
FIGURE 1. The hypersurface Ë in Ê Ò ¼ .
Consider the generalized coarse index Ind Å ´µ ¾ Ã
£ £´ ´Å µµ of the lifted classical Dirac
operator on the pullback spinor bundle of the universal cover Å. Note that ½´Å µ is the Ò-
fold product ¾ ¢¡¡¡ ¢ ¾ of free groups, and that ½´Î µ ½´Ê Ò ½ ¢ Ì Ò µ Ò . Hence
there is an injection ½´Î µ ¸ ½´Å µ and the Ã-theoretic Mayer-Vietoris sequence applies. The
boundary map of this sequence satisfies ´ Ind Å ´µµ Ind Î ´µ ¾ Ã
£ £´ ´Î µµ. However,Î
is coarsely equivalent to the hypersurface Ê Ò ½ ¢Ì Ò , so the index Ind Î ´µ can be taken to live in
à £ ½´ £ ´ÊÒ ½ ¢ Ì Ò µµ. Note that Ò will be taken to be at least ¾. There is yet another boundary
map à £ ½´ £ ´ÊÒ ½ ¢ Ì Ò µµ à £ Ò·½´ £ ´Ê ¢ Ì Ò µµ by peeling off Ò ¾ copies of the real
line. This boundary map (or composition of Ò ¾ boundary maps) preserves index.
Recall that £ ´Å µ is the norm closure of the £ -algebra of all locally compact, ½´Å µ-
equivariant, generalized finite propagation operators on Ä ¾´ Å µ, andÁ
£
´Å µ is the closure

8 STANLEY S. CHANG
of the ideal of such operators Ì that satisfies the condition that ´ ¢ µ´Supp Ì µ is bounded in
Å ¢ Å. The short exact sequence ¼ Á £ ´Ê ¢ Ì Ò µ £ ´Ê ¢ Ì Ò µ ¼ gives rise
to the six-term exact sequence in Ã-theory:
à ¼´Áµ à ¼´ £ ´Ê ¢ Ì Ò µµ à ¼´ £ ´Ê ¢ Ì Ò µµ
à ½´ ´Ê £ ¢ Ì Ò µµ
à ½´ ´Ê £ ¢ Ì Ò µµ
à ½´Áµ
Notice that the map à £´Áµ à £´ £ ´Ê ¢ Ì Ò µµ induced by the inclusion is the zero map by
an Eilenberg swindle argument. Hence both maps à £´ £ ´Ê ¢ Ì Ò µµ à £´ £ ´Ê ¢ Ì Ò µµ are
injections.
If Ò is even, the generalized coarse index IndÊ¢Ì Ò´µ of the Dirac operator resides in
à ½´ ´Ê¢Ì £ Ò µµ. Certainly the image of this index under the boundary map à ½´ ´Ê¢Ì £ Ò µµ
à ¼´ ´Ì £ Ò µµ is the index Ind Ì Ò´µ of on the Ò-torus. Since Ì Ò does not have a metric
of positive scalar curvature at all, the obstruction «´Ì Ò µ ¾ Ã £´ £´ Ò µµ, where Ì Ò
Ò is the classifying map, is nonvanishing. This “index” is constructed by Rosenberg in [17].
This special case of the Gromov-Lawson-Rosenberg conjecture holds for the group Ò [5]. This
index maps to our generalized coarse index Ind Ì Ò under the isomorphism à £´ £´ Ò µµ
à £´ ´Ì £ Ò µµ. Hence the index of in à ½´ ´Ê £ ¢ Ì Ò µµ is nonzero, and its projection onto
the group à ½´ ´Ê £ ¢ Ì Ò µµ is nonzero as well. This argument gives us the necessary index obstruction.
If Ò is odd, we apply the same argument as above with respect to the map à ¼´ ´Ê £ ¢ Ì Ò µµ
à ¼´ ´Ê £ ¢ Ì Ò µµ. £
The extension of this method to multifold products of hyperbolic manifolds involves the Margulis
lemma, which states that in such a space there exists a small positive constant Ò such
that the subgroup ´ÎÚµ ½´ÎÚµ generated by loops of length less than or equal to based
at Ú ¾ Î is almost nilpotent, i.e. it contains a nilpotent subgroup of finite index. It can be shown
that there exist such cusps, or submanifolds Î with compact convex boundary containing Ú,
such that is diffeomorphic to the product ¢ Ê·, where is diffeomorphic to an ´Ò ½µdimensional
nilmanifold with fundamental group containing ´ÎÚµ. Here a nilmanifold signifies
a quotient Æ of a nilpotent Lie group by a cocompact lattice . The nilmanifolds that arise in
this context as boundaries of pseudospheres will have a naturally flat structure.
Theorem 2: An Ò-fold product of hyperbolic manifolds has no uniform positive scalar curvature
metric coarsely equivalent to the usual Euclidean metric on the positive Euclidean hyperoctant.
Proof: Without loss of generality, it suffices to consider the case in which the noncompact hyperbolic
spaces have only one cusp. Let Ñ be the dimension of this product manifold. As in the
multifold product of tori, there is a positive ¾ Ê such that on each hyperbolic space À the inverse
image of each point Ü under the projection À Ê ¼ is by Margulis’ lemma a flat
compact connected Riemannian manifold of finite dimension. Consider the inverse image Î under
the induced product map Ô À ½ ¢¡¡¡¢À Ñ Ê Ñ ¼
of the same hypersurface as described in the
previous theorem. By Bieberbach’s theorem, every flat compact connected Riemannian manifold
admits a normal Riemannian covering by a flat torus of the same dimension. Hence Î is covered
by some product of Euclidean space and a higher-dimensional torus. Any metric of positive scalar

COARSE OBSTRUCTIONS TO POSITIVE SCALAR CURVATURE IN NONCOMPACT ARITHMETIC MANIFOLDS 9
curvature on Î would certainly lift to such a metric in this covering space. Using the same induction
argument as before, we show that such a metric is obstructed by the presence of a nonzero
Dirac class. £
III. Noncompact Quotients of Symmetric Spaces: A Special Case
We wish to apply the above techniques to the irreducible case ËÄ Ò´µÒËÄ Ò´ÊµËÇ Ò´Êµ. This
space is not locally symmetric because ËÄ Ò´µ does not act freely on ËÄ Ò´ÊµËÇ Ò´Êµ, sothe
quotient is not a Riemannian manifold. Let £ be any finite-sheeted branched cover of the double
quotient ËÄ Ò´µÒËÄ Ò´ÊµËÇ Ò´Êµ, i.e. a manifold corresponding to a subgroup of finite index in
ËÄ Ò´µ.
The Iwasawa decomposition gives a unique way of expressing the group ËÄ Ò´Êµ asaproduct
ËÄ Ò´Êµ ÆÃ,whereÆ is the subgroup of standard unipotent matrices (upper triangular
matrices with all diagonal entries equal to ½), the subgroup of ËÄ Ò´Êµ consisting of
diagonal matrices with positive entries, and à the orthogonal subgroup ËÇ Ò´Êµ. The quotient
Ò´Ò ½µ
ËÄ Ò´µÒËÄ Ò´ÊµËÇ Ò´Êµ can then be trivially seen to have compact directions arising
from Æ, an additional Ò ½ noncompact directions from ,andan´Ò ¾µ-dimensional simplex
¾
as its boundary. In short, the bordified space is coarsely an ´Ò ½µ-simplex.
Theorem 3: Let Ò ¿ and let ËÄ Ò´µ £ be a torsion-free subgroup of ËÄ Ò´µ of finite index. Then
the manifold £ ËÄ Ò´µ £ ÒËÄ Ò´ÊµËÇ Ò´Êµ lacks a uniform positive scalar curvature metric
that is coarsely equivalent to the natural one inherited from ËÄ Ò´Êµ.
Proof: To build the appropriate hypersurface in £ , consider first the orbifold , which by the
above discussion can be expressed as ÒÆÃÃ,where ËÄ Ò´µ and à ËÇ Ò´Êµ. Consider
the Weyl chamber corresponding to the subset · of positive diagonal matrices with
decreasing entries, i.e.
¼
½
½
¼ ¼
¼
·
¾
¼
.
. . ..
. ½ ¾ Ò ½ Ò
¼ ¼
Ò
Here Ò ´ ½ · ¾ · ¡¡¡ · Ò ½ µ. The coordinates ½ ¾ Ò ½ parametrize the Ò ½
noncompact directions of .
The closure of this Weyl chamber corresponding to ½ ¾ ¡¡¡ Ò is a simplex with one
boundary face at infinity. One can construct a closed, convex subset ·
· À with the following
properties (see Figure 1): the set ·
À is bounded away from the boundary faces ·½ and the
quotient ÒÆ·
ÀÃÃ of the resulting Siegel set Æ·
ÀÃ has a boundary Ï which
is coarsely equivalent to an iterated circle bundle over ´Ò ¾µ-dimensional Euclidean space. More
precisely, there is a coarse equivalence Ï Ê Ò ¾ such that the fiber over each point is a compact
arithmetic quotient of the group of unipotent matrices. (Notice that in general arithmetic subgroups
are not unipotent, but we create this hypersurface Ï for the explicit purpose of exploiting the
unipotent parts of .) The reader may wish to consider the Ò ¿ case and construct the subset

10 STANLEY S. CHANG
·
À · given by
·
À
´½ ½ µ ¼ ½ ¾ Ä
where Ä is sufficiently large. Refer to the following section for a discussion about the action of
on Ï .
Since £ is a finite-sheeted branch cover of , there is a natural projection map Ô £ .
The set Ô ½´Ï µ £ will be a disjoint union of copies of Ï . Let Ï £ be just one connected
component. This noncompact space Ï £ partitions the space £ into a coarsely excisive pair whose
closures ´µ satisfy the equalities Å £ and Ï £ .IfInd £´µ denotes the
generalized coarse index of the classical spinor Dirac operator on £ , then the Mayer-Vietoris map
à £´ £ ´£ µµ à £ ½´ £ ´Ï £ µµ defined in the previous chapter satisfies ´ Ind £´µµ
Ind Ï £´µ IndÊ Ò ¾ ¢Í Ñ´µ, whereÍ Ñ is the compact fiber of the iterated circle bundle of
Ò´Ò ½µ
dimension Ñ . Applying the same argument as before, we need only to show that the
¾
index of the Dirac operator in à £´ ´Í £ Ñ µµ is nonzero. However, the compact space Í Ñ is
a quotient of a nilpotent group by a cocompact lattice, and hence by Gromov and Lawson [9]
has no metric of positive scalar curvature at all. As with the theorem for punctured tori, there
is a nonvanishing Rosenberg index «´Í Ñ µ ¾ Ã £´ £ Ö ´µµ, where ½´Í Ñ µ, which maps to
the generalized coarse index in à £´ £ ´Í Ñ µµ, as desired. Here the Gromov-Lawson-Rosenberg
conjecture is true since Í Ñ is a nilmanifold. £
Remark: One can avoid the detailed construction of Ï by accepting the coarse simplicial picture
of ËÄ Ò´µÒËÄ Ò´ÊµËÇ Ò´Êµ. To define an appropriate hypersurface that stays a bounded distance
from the simplicial faces, except for the face at infinity, one can merely take the open cone on
an ´Ò ¿µ-dimensional sphere in the interior of the boundary face, and close this cone under the
action of Æ´Æ µ.
IV. The General Noncompact Arithmetic Case
Let be an affine algebraic group defined over É. We say that is semisimple if its radical
(i.e. its greatest connected normal solvable subgroup) is trivial. For such a , we denote its real
locus ´Êµ by , which is a semisimple Lie group with finitely many connected components. It
is well known that the spherical Tits building ¡É´µ associated with over É is a connected
infinite simplicial complex if rank É´µ ½. The simplices of ¡É´µ correspond bijectively
to the proper rational parabolic subgroups of . If ´Éµ is an arithmetic subgroup of
´Éµ, then there are only finitely many -conjugacy classes of rational parabolic subgroups, so the
quotient Ò¡É´µ is a finite simplicial complex, called the Tits complex of ÒÃ and denoted
¡´ Òõ. Hereà is a maximal compact subgroup of . See [13] for details.
Such a real semisimple Lie group has a decomposition ÈÃ where È is a parabolic subgroup
of and à is maximal compact. This parabolic È satisfies the relation È ´µÆ, where
È is a connected maximal split torus with centralizer ´µ and Æ is the unipotent radical
of È . The Langlands decomposition of a parabolic È gives È ÆÅ,whereÅ ´µ,the
quotient Å´Å µ is semisimple and ´Å µ is compact. Of course this decomposition depends on
È and the point Ü ¼ ¾ fixed by Ã.
Recall that a subgroup of is an arithmetic lattice if there exist

COARSE OBSTRUCTIONS TO POSITIVE SCALAR CURVATURE IN NONCOMPACT ARITHMETIC MANIFOLDS 11
(1) a closed subgroup ¼ of some SL ´Êµ such that ¼ is defined over É,
(2) compact normal subgroups à and à ¼ ¼ ,and
(3) an isomorphism à ¼ à ¼
such that ´ µ is commensurable with ¼ ,where and ¼ are the images of and ¼ in Ã
and ¼ Ã ¼ , respectively.
The effect of an arithmetic lattice on the components of the Langlands decomposition is as
follows. Let È ÆÅ be a minimal parabolic É-subgroup, and let Ì be a maximal É-split torus
of . ThenÅ satisfies the equality ´Ì µÌÅ.SinceÌ is maximal, the subgroup Å contains
no É-split tori. By definition, we have rank ÉÅ ¼. Hence, the arithmetic subgroups of Å
are cocompact in Å. Since the intersection of with Å is an arithmetic subgroup of Å, every
quotient of Å by an arithmetic subgroup of yields a compact quotient. A similar argument holds
for the subgroup Æ of .
To understand the coarse type of ÒÃ, we appeal to Ji and MacPherson [13] in their proof
of a conjecture of Siegel. In particular, let È ¼ È ½ È Ò be representatives of the
conjugacy classes of rational parabolic subgroups of . For each , letÈ
ÆÈ
ÅÈ
È
be
the Langlands decomposition of È . Then there exists bounded ÆÈ
ÅÈ
and Siegel sets
¢ È Ø ÆÈ
ÅÈ
¢ È
such that
(1) each Siegel set ¢ È Ø is mapped injectively into ÒÃ;
(2) the image of in ´ È µÒÆÈ
ÅÈ
is compact;
(3) if we identify ¢ È Ø with its image in ÒÃ, then ÒÃ can be decomposed in
the following disjoint union
ÒÃ
Ò
¼
¢ È Ø
Here the subset È Ø ¾ È
« ´ÐÓ µ Ø ½Ö is a shift of the positive chamber
·
¾ È
« ´ÐÓ µ ¼ ½Ö, where the « are the associated set of simple
È
roots and Ø is sufficiently large. Using this so-called precise reduction theory and identifying È Ø
with a cone in the Lie algebra , one can endow it with the simplicial metric Ë defined by
the Killing form through the exponential map. Then ´È Ø Ë µ is a metric cone over the open
simplex ·
È ´½µ in the Tits building ¡É´µ associated with È ,when·
È ´½µ is endowed with
a suitable simplicial metric. We can glue these metric cones ´È Ø Ë µ to form a local distance
Ò
function Ð Ë on
¼ Ò
È Ø. If ind is the distance function on the subspace
¼ È Ø induced by
ÒÃ, then it is not hard to show that the tangent cone Ì ½´Ò
¼ È Ø ind µ at infinity exists
and is equal to ´Ò
¼ È ØÐ Ë µ. The tangent cone Ì ½´ Òõ therefore exists and is equal to a
metric cone over the Tits complex ¡´ Òõ [13]. The resulting fact that the Gromov-Hausdorff
distance between ÒÃ and ´Ò
¼ È ØÐ Ë µ is finite allows us to build a map ÒÃ
´Ò
¼ È ØÐ Ë µ whose properties are captured in the following description.
Picture from Reduction Theory: Let Å ÒÃ. There is a compact polyhedron É and a
Lipschitz map Å É,whereÉ is the open cone on É so that (1) every point inverse deform
retracts to an arithmetic manifold, (2) respects the radial direction, and (3) all point inverses have
uniformly bounded size.

12 STANLEY S. CHANG
Again, the polyhedron É is the geometric realization of the category of proper É-parabolic
subgroups of , modulo the action of . The inverse image ½´£µ of the barycenter of a simplex
is the arithmetic symmetric space associated to that parabolic. Concretely, for SL Ò´µ SL Ҵʵ,
the space É is an Ò ¾ simplex, the parabolics correspond to flags, and the associated arithmetic
groups have a unipotent normal subgroup with quotient equal to a product of SL Ñ´µ, where the
Ñ are sizes of the blocks occurring in the flag. As one goes to infinity, the unipotent directions
shrink in diameter and are responsible for the finite volume property of the lattice quotient, while
the other parabolic directions remain of bounded size. Alternatively, for any choice of basepoint
in the homogeneous space, there are constants and that satisfy the following condition: if Ü
is a given point and É Ü is the largest parabolic subgroup associated with a simplex whose cone
contains Ü within its -neighborhood, then the orbit of Ü under É Ü has diameter less than . Note
the empty simplex means that there is a compact core which is stabilized by the whole group. In
addition to the proof in [13], this picture can be ascertained from [4], [18]; the fact that ÒÃ
has finite Gromov-Hausdorff distance from É is first asserted in [8].
As a guide the reader should consider the picture suggested by a product of hyperbolic manifolds.
In the compact case, each hyperbolic manifold contributes to É a point. In the case of
cusps, it contributes the open cone on a finite set of points. Thus É is a join of some number of
finite sets. Using this model, we find that the inverse image of any point in the interior of any
simplex is exactly a product of closed hyperbolic manifolds, cores of hyperbolic manifolds, and
flat manifolds.
To build an appropriate hypersurface in ÒÃ, we require a key estimate of Eskin [7] about
the “coarse isotropy” of our space (see also [3] and [13]). Some details are provided as follows.
Let È ÆÅ be a minimal parabolic É-subgroup of and write ÆÅÃ. Consider
the chamber decomposition of , the Lie algebra of . The corresponding Weyl group Ï acts on
these chambers via the hyperplanes. If Û¾Ï
Û is the Bruhat decomposition of , let
­ ¾ Û for some È Û ¾ Ï .If Ò, we write ­ Ò ¼ ¼ ¼ . Denote by È· the set of positive
roots of £ and the set of negative roots. Let Ê È
Û È· be the set of roots that are
positive but are negated under the action of Û. For some positive reals constants « ,Abelsand
Margulis [1] provide the following equation:
(£) ¼ Û «¾Ê
« «´µ · Ó´½µ
where and ¼ are viewed as elements of the Lie algebra . The implications of this equation
are as follows. Consider an element in the positive Weyl chamber ´µ. Then the intersection
´µ ´µ of the orbit ´µ of under and the Weyl chamber ´µ containing has a bounded
diameter, uniformly in . In other words, if ­´µ stays in the same positive Weyl chamber, then Û
is the identity and Ê is empty. Hence ¼ · Ó´½µ, implying that ¼ can be found at a uniformly
bounded distance from itself. In this event, the action of ­ corresponds to a translation of to
(possibly) the compact fiber direction of ÒÃ. It is also a general fact that the image of a vertex
of any subsector under the action of ­ ¾ is the vertex of an analogous subsector. With this
machinery, we are able to prove the following.

COARSE OBSTRUCTIONS TO POSITIVE SCALAR CURVATURE IN NONCOMPACT ARITHMETIC MANIFOLDS 13
Theorem 4: Let be a semisimple irreducible Lie group, Ã its maximal compact subgroup and
an arithmetic lattice with rankÉ ¾. If £ is any torsion-free subgroup of of finite index,
then the manifold £ £
ÒÃ lacks a uniform positive scalar curvature metric that is coarsely
equivalent to the natural one inherited from .
Proof: Let ÈÃ,whereÈ ÆÅ is a minimal parabolic É-subgroup of . The precise
reduction theory provides a compact polyhedron É and a Lipschitz map £ É from
£ to an open cone É on É. Consider one maximal simplex in É corresponding to some Weyl
chamber ·, and construct a hypersurface in · as in Figure 1 (this hypersurface is coarsely a
cone on a sphere Ë, whereË lies in the interior of the simplicial face at infinity). This subset À
can be oriented so that the distance from À to any hyperplane « ¼ will exceed the quantity
×ÙÔ ¾· diam ´ £´µ ·µ, which is finite by (£). Let Ï ½´Àµ be the corresponding hypersurface
in £ ;thisÏ induces a coarsely excisive decomposition ´µ of £ . The fundamental
group ½´Ï µÆÅ £ injects into ½´ £ µ £ , so the hypothesis of Lemma 3 is satisfied. In
the most general case, the space Ï is coarsely equivalent to a bundle over Euclidean space whose
fiber consists of two components: a nilmanifold Æ £ and (possibly) a compact homogeneous manifold
Å £ . If Å £ is trivial, the argument follows exactly as in Theorem 3. In the presence of a
compact homogeneous manifold, we may pass to the coarse index of the Dirac operator to Ê ¢ Å £
and use the usual Rosenberg obstruction on Å £ as in Theorem 1 to obtain our desired result. The
proof that the Gromov-Lawson conjecture holds for compact, locally symmetric manifolds is found
in []. £
REFERENCES
[1] H. Abels and G. Margulis, Reductive groups as metric spaces, preprint.
[2] J. Block and S. Weinberger, Arithmetic manifolds of positive scalar curvature, J. Differential Geom. 52 (1999) no.
2, 375–406.
[3] A. Borel and L. Ji, Compactifications of locally symmetric space, preprint, 2000.
[4] A. Borel and J.P. Serre, Corners and arithmetic groups, Comm. Math. Helv. 48 (1973) 436–491.
[5] B. Botvinnik and P. Gilkey, The Gromov-Lawson-Rosenberg Conjecture: the Twisted Case, Houston J. of Math.
(1997) 23, no. 1, 143–160.
[6] J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace
operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), no. 1, 15–53.
[7] A. Eskin, private communication.
[8] M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, vol. 2 (Sussex, 1991), 1–295.
[9] M. Gromov and H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds,
Inst. Hautes Études Sci. Publ. Math. no. 58 (1983), 83–196 (1984).
[10] N. Higson, £ -algebra extension theory and duality, J. Funct. Anal. 129 (1995), no. 2, 349–363.
[11] N. Higson, J. Roe and G. Yu, A coarse Mayer-Vietoris principle, Math. Proc. Cambridge Philos. Soc. 114 (1993),
no. 1, 85–97.
[12] L. Ji, Metric compactifications of locally symmetric spaces, International Journal of Mathematics 9 (1998), no. 4,
465–491.
[13] L. Ji and R. MacPherson, Geometry of compactifications of locally symmetric spaces, preprint, 2000.
[14] J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104
(1993), no. 497.
[15] J. Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics,
90. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American
Mathematical Society, Providence, RI, 1996.
[16] J. Roe, Partitioning noncompact manifolds and the dual Toeplitz problem, Operator algebras and applications, vol.
1, 187–228, London Math. Soc. Lecture Note Ser., 135, Cambridge Univ. Press, Cambridge-New York, 1988.

14 STANLEY S. CHANG
[17] J. Rosenberg, £ -algebras, positive scalar curvature and the Novikov conjecture II, Geometric methods in operator
algebras (Kyoto, 1983), 341–374, Pitman Res. Notes Math. Ser., 123, Longman Sci. Tech., Harlow, 1986.
[18] L. Saper, Tilings and finite energy retractions of locally symmetric spaces, Comment. Math. Helv. 72 (1997), no.
2, 167–202.
[19] S. Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 136 (1992), no. 3, 511–540.
[20] D. Witte, Arithmetic Groups and Locally Symmetric Spaces, preprint.
[21] F. Wu, The higher index theorem for manifolds with boundary, J. Funct. Anal. 103 (1992), no. 1, 160–189.
RICE UNIVERSITY,HOUSTON, TX 77005
E-mail address: sschang@math.rice.edu