- A differential complex for CAT(0) cubical spaces
- with Jacek Brodzki and Nigel Higson
- preprint
- In the 1980'sPierre Julg and Alain Valette ...In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to C*-algebra K-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of p-adic groups. The main aim of this paper is to extend the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces. A secondary aim is to illustrate the utility of the extended construction by developing an application to operator K-theory and giving a new proof of K-amenability for groups that act properly on bounded-geometry CAT(0)-cubical spaces.
- Dynamic asymptotic dimension and controlled operator K-theory
- with Rufus Willett and Guoliang Yu
- preprint
- In earlier workwe introduced ...In earlier work the authors introduced dynamic asymptotic dimension, a notion of dimension for topological dynamical systems that applies to many interesting examples. In this paper, we use finiteness of dynamic asymptotic dimension to get information on the $K$-theory of the associated crossed product $C^*$-algebras: specifically, we give a new proof of the Baum-Connes conjecture for such actions. The key tool is controlled K-theory, as developed by Oyono-Oyono and the third author. Our main result is not new: it follows from work of Tu on amenable groupoids. Nonetheless, the proof is very different: it amounts to a computation of the $K$-theory of a crossed product which is quite independent of the topological formula posited by the Baum-Connes machinery.
We have tried to keep the paper as self-contained as possible: we hope the main part of the paper will be accessible to someone with the equivalent of a first course in operator $K$-theory. In particular, we do not assume prior knowledge of controlled $K$-theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant $K$-theory to set up.

- A Milnor sequence in operator K-theory
- with Guoliang Yu
- reference
- Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry and C*-algebras
- with Rufus Willett and Guoliang Yu
- to appear
- We introducedynamic asymptotic dimension ...We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact ́etale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lueck, and Reich in the context of controlled topology, and its connections with Gromov's theory of asymptotic dimension. We also show that dynamic asymptotic dimension gives bounds on the nuclear dimension of Winter and Zacharias for C*-algebras associated to dynamical systems. Dynamic asymptotic dimension also has implications for K-theory and manifold topology: these will be drawn out in subsequent work.
- Exactness and the Kadison-Kaplansky conjecture
- with Paul Baum and Rufus Willett
- in
*Operator algebras and their applications: a tribute to Richard V. Kadison*, in the AMS series Contemporary Mathematics. - We surveyresults connecting exactness ...We survey results connecting exactness in the sense of C*-algebra theory, coarse geometry, geometric group theory, and expander graphs. We summarize the construction of the (in)famous non-exact monster groups whose Cayley graphs contain expanders, following Gromov, Arzhantseva, Delzant, Sapir, and Osajda. We explain how failures of exactness for expanders and these monsters lead to counterexamples to Baum-Connes type conjectures: the recent work of Osajda allows us to give a more streamlined approach than currently exists elsewhere in the literature. We then summarize our work on reformulating the Baum-Connes conjecture using exotic crossed products, and show that many counterexamples to the old conjecture give confirming examples to the reformulated one; our results in this direction are a little stronger than those in our earlier work. Finally, we give an application of the reformulated Baum-Connes conjecture to a version of the Kadison-Kaplansky conjecture on idempotents in group algebras.
- Expanders, exact crossed products and the Baum-Connes conjecture
- with Paul Baum and Rufus Willett
- Annals of K-theory
**1**(2016) 155-208. - We reformulatethe Baum-Connes conjecture ...We reformulate the Baum-Connes conjecture with coefficients by introducing a new crossed product functor for C*-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to the original Baum-Connes conjecture become confirming examples for our reformulated conjecture.
- New C*-completions of discrete groups and related spaces
- with Nate Brown
- Bulletin of the London Math. Soc.
**45**(2013), 1181-1193. - Suppose G isa discrete group ...Suppose G is a discrete group and D is an ideal in the algebra of bounded functions on G. We introduce a C*-completion C*
_{D}(G) that encapsulates the unitary representations with matrix coefficients in D. This general framework unifies some classical results and leads to new insights. For example, we give the first C*-algebraic characterization of a-T-menability; provide new examples of 'exotic' quantum groups; and, after extending our construction to transformation groupoids, we improve and simplify a recent result of Nowak and Douglas. - Coarse non-amenability and coarse embeddings
- with Goulnara Arzhantseva and Jan Spakula
- Geometric and Functional Analysis (GAFA),
**22**(2012), 22-36; doi 10.1007/s00039-012-0145-z - slides from a related talk given at the conference Dubrovnik VII - Geometric Topology, June 2011 (pdf)
- slides from a related talk given at the conference surrounding the Clifford Lectures at Tulane, March 2012 (pdf)
- Coarse amenability(Property A) was introduced ...Coarse amenability (Property A) was introduced as a verifiable property implying coarse embeddability in Hilbert space. In the context of discrete groups coarse amenability is equivalent to a wide variety of other properties including exactness of the reduced group C*-algebra and amenability of the action on the Stone-Cech compactification. We construct the first example of a bounded geometry metric space that is coarsely embeddable in Hilbert space, and coarsely NON-amenable. Previously, locally finite spaces with these properties were constructed by Nowak; these examples, however, do not have bounded geometry.
- Coarse non-amenability and covers with small eigenvalues
- with Goulnara Arzhantseva
- Mathematische Annalen
**354**(2012), 863-870; online first with doi 10.1007/s00208-011-0759-8. - the final publication is available at springerlink.com
- Given aclosed Riemannian manifold ...Given a closed Riemannian manifold M and a (virtual) epimorphism of the fundamental group of M onto a free group of rank 2, we construct a tower of finite sheeted regular covers Mn of M such that λ1(Mn) tends to zero. This is the first example of such a tower which is not obtainable up to uniform quasi-isometry (or even up to uniform coarse equivalence) by the previously known methods where the fundamental group of M is supposed to surject onto an amenable group.
- Complexes and exactness of certain Artin groups
- with Graham Niblo
- Algebraic and Geometric Topology (AGT)
**11**(2011), 1471-1495. - In his workon the Novikov conjecture ...In his work on the Novikov conjecture, Yu introduced Property A as a readily verified criterion implying coarse embeddability. Studied subsequently as a property in its own right, Property A for a discrete group is known to be equivalent to exactness of the reduced group C*-algebra and to the amenability of the action of the group on its Stone-Cech compactification. In this paper we study exactness for groups acting on a finite dimensional CAT(0) cube complex. We apply our methods to show that Artin groups of type FC are exact. While many discrete groups are known to be exact the question of whether every Artin group is exact remains open.
- Operator norm localization for linear groups and its application to K-theory
- with Romain Tessera and Guoliang Yu
- Advances in Mathematics
**226**(2011), no. 4, 3495-3510. - We provethe operator norm localization ...We prove the operator norm localization property for linear groups. As an application we prove the coarse Novikov conjecture for box spaces of a linear group.
- A notion of geometric complexity and its application to topological rigidity
- with Romain Tessera and Guoliang Yu
- Inventiones Mathematicae,
**189**(2012), 315-357. doi 10.1007/s00222-011-0366-z - this paper and the next were originally one paper, which was cut in two at request of the referee
- the original (longer) paper is still available available on the arXiv.
- We introducea geometric invariant ...We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M x Rn is homeomorphic to N x Rn, for n large enough. This statement is known as the stable Borel conjecture. On the other hand, we show that the class of FDC groups includes all countable subgroups of GL(n,K), for any field K.
- Discrete groups with finite decomposition complexity
- with Romain Tessera and Guoliang Yu
- Groups, Geometry and Dynamics
**7**(2013), 377-402. - In a companionpaper we introduced ...In a companion paper we introduced a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. In that article we proved the stable Borel conjecture for a closed aspherical manifold whose universal cover, or equivalently whose fundamental group, has FDC. In this note we continue our study of FDC, focusing on permanence and the relation to other coarse geometric properties. In particular, we prove that the class of FDC groups is closed under taking subgroups, extensions, free amalgamated products, HNN extensions, and direct unions. As consequences we obtain further examples of FDC groups -- all elementary amenable groups and all countable subgroups of almost connected Lie groups have FDC.
- Permanence in coarse geometry
- in
*Recent Progress in General Topology III*, edited by K.P. Hart, J. van Mill and P. Simon, 2014, p 507-533. - The large scaleor coarse perspective ...The large scale, or coarse perspective on the geometry of metric spaces plays an important role in approaches to conjectures in operator algebras and the topology of manifolds. Coarse geometric properties having implications for these conjectures include, among others, finite asymptotic dimension, its weaker variant finite decomposition complexity, and coarse embeddability. In this paper, we survey the permanence characteristics of these and other properties. Rather than focus on the individual properties, however, we examine the general structure of permanence results in coarse geometry.
- Property A and CAT(0) cube complexes,
- with Jacek Brodzki, Sarah Campbell, Graham Niblo and Nick Wright
- Jour. Functional Anal.,
**256**(2009) 1408--1431. - The main resultsof the paper are ...The main results of the paper are: first, a finite dimensional CAT(0) cube complex has Property A; second, a discrete group acting properly on a CAT(0) cube complex fixing a vertex at infinity is amenable. Property A is a non-equivariant analog of amenability defined for metric spaces by Yu in his work on the Novikov conjecture. The first result above is proven by directly constructing the 'Folner sets' called for in Yu's original definition. The result applies to non-locally finite complexes. The same construction, suitably adapted to allow a basepoint at infinity, is the essential ingredient in the proof of the second result above.
- Weak amenability of CAT(0) cubical groups
- with Nigel Higson
- Geometriae Dedicata
**148**(2010), 137-156 (online first). - We provethat if G is a discrete ...We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations pi_z for which the quantities z^{l(g)} are matrix coefficients. Here l is a length function on G obtained from the combinatorial distance function on the complex X.
- Uniform embeddability of relatively hyperbolic groups
- with Marius Dadarlat
- Jour. fur die Reine und Angew. Math. (Crelle)
**612**(2007) 1-15. - Let G bea finitely generated group ...Let G be a finitely generated group which is hyperbolic relative to a finite family A, B, ..., C of subgroups. We prove that G is uniformly embeddable in a Hilbert space if and only if each subgroup is uniformly embeddable in a Hilbert space. A 'gluing' technique for proving uniform embeddability of metric spaces is introduced, and plays a fundamental role in the proof of the result. Further applications of the gluing technique will be considered elsewhere.
- Uniform embeddings of bounded geometry spaces into reflexive Banach space
- with Nate Brown
- Proceedings of the AMS,
**133**(2005), 2045-2050. - We showthat every metric space with bounded geometry ...We show that every metric space with bounded geometry uniformly embeds into a direct sum of l-p spaces (p's going off to infinity). In particular, every sequence of expanding graphs uniformly embeds into such a reflexive Banach space even though no such sequence uniformly embeds into a fixed l-p space. In the case of discrete groups we prove the analogue of a-T-menability -- the existence of a metrically proper affine isometric action on a direct sum of l-p spaces.
- Exactness and Uniform Embeddability of Discrete Groups
- with Jerry Kaminker
- Journal of the London Math. Society.,
**70**(2004), 703-718. - We definea numerical quasi-isometry invariant ...We define a numerical quasi-isometry invariant R(G), of a finitely generated group G, whose values parametrize the difference between G being uniformly embeddable in a Hilbert space and its reduced C*-algebra being exact. As an application we show that if a finitely generated group G admits a uniform embedding into Hilbert space H for which the image of G, in the metric induced from H, is a quasi-geodesic space then the reduced C*-algebra of G is exact.
- The Novikov Conjecture for Linear Groups
- with Nigel Higson and Shmuel Weinberger
- Publications Mathematiques de l'IHES,
**101**(2005), 243-268. - slides from a related talk given at the AMS Sectional Meeting in Bloomington, April 2003 (ps)
- slides from a talk given at the Clay Mathematics Institute - Vanderbilt University Summer School on Noncommutative Geometry, May 2003 (ps)
- Let K be afield. We show ...Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space; indeed, we show that such groups are exact, in the sense of C*-algebra theory. This implies that Novikov's higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. We conclude with an application to the problem of homotopy invariance of relative eta invariants.
- Constructions Preserving Hilbert Space Uniform Embeddability of Discrete Groups
- with Marius Dadarlat
- Transactions of the AMS,
**355**(2003), 3253-3275. - Uniform embeddability(in a Hilbert space) ...Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences to the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg-Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that as properties of countable discrete groups uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.
- Uniform Embeddability and Exactness of Free Products
- with X. Chen, M. Dadarlat and G. Yu
- Jour. Functional Anal.,
**205**(2003), 168-179. - Let A and Bbe countable discrete groups ...Let A and B be countable discrete groups and let G = A * B be their free product. We show that if both A and B are uniformly embeddable in a Hilbert space then so is G. We give two different proofs: the first directly constructs a uniform embedding of G from uniform embeddings of A and B; the second works without change to show that if both A and B are exact then so is G.
- Group C*-algebras and K-Theory
- with Nigel Higson
- In volume 1831 of the Springer LNM series. For the contents of the volume see Springer's web site.
- Geometric and Analytic Properties of Groups
- with Jerry Kaminker
- In volume 1831 of the Springer LNM series. For the contents of the volume see Springer's web site.
- Exactness of One Relator Groups
- Proceedings of the AMS
**130**(2002), 1087-1093. - slides from a related talk given at the AMS Meeting in San Francisco, Oct 2000 (ps)
- A discretegroup G is C*-exact ...A discrete group G is C*-exact if the reduced crossed product with G converts a short exact sequence of G-C*-algebras into a short exact sequence of C*-algebras. A one relator group is a discrete group G admitting a presentation G = < X | R > where X is a countable set and R is a single word over X. In this short note we prove that all one relator discrete groups are C*-exact. Using the Bass-Serre theory we also prove that a countable discrete group G acting without inversion on a tree is C*-exact if the vertex stabilizers of the action are C*-exact.
- Boundary Calculations in E-Theory for Operators of Dirac Type
- unpublished manuscript
- Exactness and the Novikov Conjecture
- with Jerry Kaminker
- Topology
**41**(2002), 411-418. - slides from a related talk given at the Groupoid Fest '99 in Iowa City (ps)
- slides from a related talk given at the Joint Meetings 2000 in Washington (ps)
- In this notewe will study a connection ...In this note we will study a connection between the conjecture that the reduced C*-agebra of G is exact and the Novikov conjecture for G. The main result states that if the inclusion of the reduced C*-algebra of a discrete group G into the uniform Roe algebra of G is a nuclear map then G is uniformly embeddable in a Hilbert space. By a result of G. Yu, this implies that G satisfies the Novikov conjecture. Note that the hypothesis is a slight strengthening of the usual notion of exactness since a group G is exact if and only if the inclusion of its reduced C*-algebra into the bounded operators on l2(G) is nuclear.
- Addendum to "Exactness and the Novikov Conjecture"
- with Jerry Kaminker
- Topology
**41**(2002), 419-420. - Berezin Quantization and K-Homology
- Communications in Math. Physics,
**240**(2003), 423-446. - The E-theorydefined by Connes and Higson ...The E-theory defined by Connes and Higson provides a realization of K-homology, the generalized homology theory dual to K-theory, based on the notion of asymptotic homomorphisms. With this realization it becomes possible to associate a K-homology element to a quantization scheme. In this article we associate an asymptotic homomorphism and K-homology element to the Berezin quantization of a bounded symmetric domain. Further, we identify this element with the element of K-homology defined by the Dolbeault operator of the domain.
- Equivariant E-Theory for C*-Algebras
- with Nigel Higson and Jody Trout
- Memoirs of the AMS
**702**(2000). - Wick Quantization and Asymptotic Morphisms
- Houston J. of Math.
**26**(2000), 361-375. - The E-theoryof Connes and Higson provides a new ...The E-theory of A. Connes and N. Higson provides a new realization of K-homology based on the notion of asymptotic morphisms. In this note we begin to develop the idea that through this realization K-homology becomes a receptacle for topological invariants of quantization schemes. We study the example of the Wick quantization, showing that it determines an element of the relative E-theory group E(C, S1), where C denotes the compactification of the complex plane by the 'circle at infinity.' By constructing an explicit homotopy we show that this element is equal to the one determined by the dbar-operator on the complex plane.
- Boundary Calculations in Relative E-Theory
- Michigan Math. Journal
**45**(1998), 159-188. - The E-theorydefined by Connes and Higson provides ...The E-theory defined by Connes and Higson provides a realization of K-homology based on the notion of asymptotic morphisms. A first order, elliptic differential operator D on a manifold M determines an element of the E-homology group [D] in E(C0(M)). If W is a compactification of M so that (W,bdy(W)) is a compact metric pair there is a boundary map in E-homology E0(C0(M)) to E1(C(bdy(W))). In this paper we use the relative E-homology groups E0(C(W),C0(M)) to determine explicitly the image of [D] under this boundary map in a number of cases of interest. We consider both manifolds with boundary, in which case we recover earlier results of P. Baum, R. Douglas and M. Taylor, and metric compactifications (in the sense of coarse geometry) of complete Riemannian manifolds.
- Relative E-Theory
- K-theory
**17**(1999), 55-93. - The E-theoryof Connes and Higson provides a new ...The E-theory of A. Connes and N. Higson provides a new realization of K-homology based on the notion of asymptotic morphisms. In this paper we define relative E-theory, associating to a C*-algebra A and an ideal I the abelian groups Enrel(A;I). These groups are related to the E-theory groups of A and I in the familiar way; by a long exact sequence and excision isomorphisms. The definition of relative E-theory is motivated by the properties of first order, elliptic differential operators on complete Riemannian manifolds. Applications will be considered in a future publication.
- A Note on Toeplitz Operators
- with Nigel Higson
- Intl. Journal of Math.
**7**(1996), 501-513. - We studyToeplitz operators on Bergman spaces ...We study Toeplitz operators on Bergman spaces using techniques from the analysis of Dirac-type operators on complete Riemannian manifolds, and prove an index theorem of Boutet de Monvel from this point of view.
- K-Homology and the Index Theorem
- Contemporary Mathematics
**148**(1993), 47-66. - Let M be acompact, even-dimensional, oriented Riemannian manifold ...Let M be a compact, even-dimensional, oriented Riemannian manifold. Let D be a Dirac-type operator associated to a Clifford bundle E on M. Following an argument of P. Baum we prove the Atiyah-Singer Index Theorem for such operators.