Atiyah Conjecture Research Articles

On p‐adic limits of topological invariants

The purpose of this article is to define and study new invariants of topological spaces: the $p$-adic Betti numbers and the $p$-adic torsion. These invariants take values in the $p$-adic numbers and are constructed from a virtual pro-$p$ completion of the fundamental group. The key result of the article is an approximation theorem which shows that the $p$-adic invariants are limits of their classical analogues. This is reminiscent of L\"uck's approximation theorem for $L^2$-Betti numbers. After an investigation of basic properties and examples we discuss the $p$-adic analog of the Atiyah conjecture: When do the $p$-adic Betti numbers take integer values? We establish this property for a class of spaces and discuss applications to cohomology growth.

The strong Atiyah and Lück approximation conjectures for one-relator groups

It is shown that the strong Atiyah conjecture and the L\"uck approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.

The Bieri–Neumann–Strebel invariants via Newton polytopes

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav. We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a $3$-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincar\'e duality groups of type $\mathtt{F}$ in dimension $3$ and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl. We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the $L^2$-torsion polytope of Friedl-L\"uck is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-L\"uck-Tillmann.

The base change in the Atiyah and the Lück approximation conjectures

Let F be a free finitely generated group and $${A \in {\rm Mat}_{n \times m}(\mathbb{C}[F])}$$ . For each quotient G = F/N of F we can define a von Neumann rank function rkG(A) associated with the l2-operator l2(G)n → l2(G)m induced by right multiplication by A. For example, in the case where G is finite, $${{\rm rk}_G(A)=\frac{{\rm rk}_{\mathbb{C}}({{\bar{A}}})}{|G|}}$$ is the normalized rank of the matrix $${{{\bar{A}}} \in {\rm Mat}_{n \times m}(\mathbb{C}[G])}$$ obtained by reducing the coefficients of A modulo N. One of the variations of the Luck approximation conjecture claims that the function $${N\mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of marked groups. The strong Atiyah conjecture predicts that if the least common multiple lcm(G) of the orders of finite subgroups of G is finite, then $${{\rm rk}_G(A) \in \frac{1}{lcm (G)}\mathbb{Z}}$$ . In our first result we prove the sofic Luck approximation conjecture. In particular, we show that the function $${N \mapsto {\rm rk}_{F/N}(A)}$$ is continuous in the space of sofic marked groups. Among other consequences we obtain that a strong version of the algebraic eigenvalue conjecture, the center conjecture and the independence conjecture hold for sofic groups. In our second result we apply the sofic Luck approximation and we show that the strong Atiyah conjecture holds for groups from a class $${{\mathcal{D}}}$$ , virtually compact special groups, Artin’s braid groups and torsion-free p-adic analytic pro-p groups.

Localization, Whitehead groups and the Atiyah conjecture

Let Wh^w(G) be the K_1-group of square matrices over the integral group ring ZG which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let D(G) be the division closure of ZG in the algebra U(G) of operators affiliated to the group von Neumann algebra. Let C be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let G be a torsionfree group which belongs to C. Then we prove that Wh^w(G) is isomorphic to K_1(D(G)). Furthermore we show that D(G) is a skew field and henc K_1(\D(G)) is the abelianization of the multiplicative group of units in D(G).

The $L^2$-Torsion Polytope of Amenable Groups

We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the L^2 -torsion polytope among G -CW-complexes for these groups. As another application we prove that the L^2 -torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.

The Atiyah Conjecture and $$L^2$$ L 2 -cohomology computations

If G is a discrete group with bounded torsion, the Strong Atiyah Conjecture predicts that the $$L^2$$ -Betti numbers of any G-space are rational, with denominators determined by the orders of the torsion subgroups. We prove the conjecture for some new examples of groups, including the class of virtually sparse special groups. We then show that the conjecture can be used to compute the $$L^2$$ -cohomology of certain Coxeter groups that have highly symmetric nerves.

On the Center-Valued Atiyah Conjecture for $L^2$-Betti Numbers

The so-called Atiyah conjecture states that the \mathcal N(G) -dimensions of the L^2 -homology modules of finite free G -CW-complexes belong to a certain set of rational numbers, depending on the finite subgroups of G . In this article we extend this conjecture to a statement for the center-valued dimensions. We show that the conjecture is equivalent to a precise description of the structure as a semisimple Artinian ring of the division closure D(\Bbb{Q}[G]) of \Bbb{Q}[G] in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class \frak{C} , containing in particular free-by-elementary amenable groups. The center-valued Atiyah conjecture states that the center-valued L^2 -Betti numbers of finite free G -CW-complexes are contained in a certain discrete subset of the center of \Bbb{C}[G] , the one generated as an additive group by the center-valued traces of all projections in \Bbb{C}[H] , where H runs through the finite subgroups of G . Finally, we use the approximation theorem of Knebusch [15] for the center-valued L^2 -Betti numbers to extend the result to many groups which are residually in \frak{C} , in particular for finite extensions of products of free groups and of pure braid groups.

On computing homology gradients over finite fields

AbstractRecently the so-called Atiyah conjecture about l2-Betti numbers has been disproved. The counterexamples were found using a specific method of computing the spectral measure of a matrix over a complex group ring. We show that in many situations the same method allows to compute homology gradients, i.e. generalisations of l2-Betti numbers to fields of arbitrary characteristic. As an application we point out that (i) the homology gradient over any field of characteristic different than 2 can be an irrational number, and (ii) there exists a finite CW-complex with the property that the homology gradients of its universal cover taken over different fields have infinitely many different values.

Lamplighter groups and von Neumann‘s continuous regular ring

Let Γ \Gamma be a discrete group. Following Linnell and Schick one can define a continuous ring c ( Γ ) c(\Gamma ) associated with Γ \Gamma . They proved that if the Atiyah Conjecture holds for a torsion-free group Γ \Gamma , then c ( Γ ) c(\Gamma ) is a skew field. Also, if Γ \Gamma has torsion and the Strong Atiyah Conjecture holds for Γ \Gamma , then c ( Γ ) c(\Gamma ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ = Z 2 ≀ Z \Gamma =\mathbb {Z}_2\wr \mathbb {Z} . It is known that C ( Z 2 ≀ Z ) \mathbb {C}(\mathbb {Z}_2\wr \mathbb {Z}) does not even have a classical ring of quotients. Our main result is that if H H is amenable, then c ( Z 2 ≀ H ) c(\mathbb {Z}_2\wr H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930s.

Geometric Topology

Geometric topology has seen significant advances in the understanding and application of infinite symmetries and of the principles behind them. On the one hand, for advances in (geometric) group theory, tools from algebraic topology are applied and extended; on the other hand, spectacular results in topology (e.g., the proofs of new cases of the Novikov conjecture or the Atiyah conjecture) were only possible through a combination of methods of homotopy theory and new insights in the geometry of groups. This workshop focused on the rich interplay between algebraic topology and geometric group theory.

Freely independent random variables with non-atomic distributions

We examine the distributions of non-commutative polynomials of non-atomic, freely independent random variables. In particular, we obtain an analogue of the Strong Atiyah Conjecture for free groups thus proving that the measure of each atom of any $n \times n$ matricial polynomial of non-atomic, freely independent random variables is an integer multiple of $n^{-1}$. In addition, we show that the Cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real-analytic except at a finite number of points.

The Atiyah conjecture for the Hecke algebra of the infinite dihedral group

We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W , replacing the group von Neumann algebra \mathcal N W with the Hecke–von Neumann algebra \mathcal N_qW .

K–theory, LQEL manifolds and Severi varieties

We use topological K-theory to study non-singular varieties with quadratic entry locus. We thus obtain a new proof of Russo's Divisibility Property for locally quadratic entry locus manifolds. In particular we obtain a K-theoretic proof of Zak's theorem that the dimension of a Severi variety must be 2, 4, 8 or 16 and so resolve a conjecture of Atiyah and Berndt. We also show how the same methods applied to dual varieties recover the Landman parity theorem.

The strong Atiyah conjecture for virtually cocompact special groups

We provide new conditions for the strong Atiyah conjecture to lift to finite group extensions. In particular, we show that fundamental groups of compact special cube complexes satisfy these conditions, so the conjecture holds for finite extensions of these groups.

The Atiyah Conjecture and Artinian Rings

Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N(G), and let D(KG,U(G)) denote the division closure of KG in U(G); thus D(KG,U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, and \alpha \in \Mat_n(KG). Then \alpha induces a bounded linear map \alpha: l^2(G)^n \to \l^2(G)^n, and \ker\alpha has a well-defined von Neumann dimension \dim_{N(G)} (\ker\alpha). This is a nonnegative real number, and one version of the Atiyah conjecture states that d \dim_{N(G)}(\ker\alpha) \in Z. Assuming this conjecture, we shall prove that if G has no nontrivial finite normal subgroup, then D(KG,U(G)) is a d \times d matrix ring over a skew field. We shall also consider the case when G has a nontrivial finite normal subgroup, and other subrings of U(G) that contain KG.

The strong Atiyah conjecture for right-angled Artin and Coxeter groups

We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.

Von Neumann Algebras and Ergodic Theory of Group Actions

The workshop \emph{Von Neumann Algebras and Ergodic Theory of Group Actions} was organized by Dietmar Bisch (Vanderbilt University, Nashville), Damien Gaboriau (ENS Lyon), Vaughan Jones (UC Berkeley) and Sorin Popa (UC Los Angeles). It was held in Oberwolfach from October 26 to November 1, 2008. This workshop was the first Oberwolfach meeting on von Neumann algebras and orbit equivalence ergodic theory. The organizers took special care to invite many young mathematicians and more than half of the 28 talks were given by them. The meeting was very well attended by over 40 participants, leading senior researchers and junior mathematicians in the field alike. Participants came from about a dozen different countries including Belgium, Canada, Denmark, France, Germany, Great Britain, Japan, Poland, Switzerland and the USA. The first day of the workshop featured beautiful introductory talks to orbit equivalence and von Neumann algebras (Gaboriau), Popa's deformation/rigidity techniques and applications to rigidity in II _1 factors (Vaes), subfactors and planar algebras (Bisch), random matrices, free probability and subfactors (Shlyakhtenko), subfactor lattices and conformal field theory (Xu) and an open problem session (Popa). There were many excellent lectures during the subsequent days of the conference and many new results were presented, some for the first time during this meeting. A few of the highlights of the workshop were Vaes' report on a new cocycle superrigidity result for non-singular actions of lattices in SL(n, \mathbb R ) on {\mathbb R}^n and on other homogeneous spaces (joint with Popa), Ioana's result showing that every sub-equivalence relation of the equivalence relation arising from the standard SL(2, \mathbb Z )-action on the 2 -torus {\mathbb T}^2 is either hyperfinite, or has relative property (T), and Epstein's report on her result that every countable, non-amenable group admits continuum many non-orbit equivalent, free, measure preserving, ergodic actions on a standard probability space. Other talks discussed new results on fundamental groups of II _1 factors, L ^2 -rigidity in von Neumann algebras, II _1 factors with at most one Cartan subalgebra, subfactors from Hadamard matrices, a new construction of subfactors from a planar algebra and new results on topological rigidity and the Atiyah conjecture. Many interactions and stimulating discussions took place at this workshop, which is of course exactly what the organizers had intended. The organizers would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing the splendid environnment for holding this conference. Special thanks go to the very helpful and competent staff of the institute.

Galois cohomology of completed link groups

. In this paper we compute the Galois cohomology of the pro-p completion of primitive link groups. Here, a primitive link group is the fundamental group of a tame link in S 3 whose linking number diagram is irreducible modulo p (e.g. none of the linking numbers is divisible by p). The result is that (with Z/pZ-coefficients) the Galois cohomology is naturally isomorphic to the Z/pZ-cohomology of the discrete link group. The main application of this result is that for such groups the Baum-Connes conjecture or the Atiyah conjecture are true for every finite extension (or even every elementary amenable extension), if they are true for the group itself.

Finite group extensions and the Atiyah conjecture

The Atiyah conjecture for a discrete groupGGstates that theL2L^2-Betti numbers of a finite CW-complex with fundamental groupGGare integers ifGGis torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups ofGG. Here we establish conditions under which the Atiyah conjecture for a torsion-free groupGGimplies the Atiyah conjecture for every finite extension ofGG. The most important requirement is thatH∗(G,Z/p)H^*(G,\mathbb {Z}/p)is isomorphic to the cohomology of thepp-adic completion ofGGfor every prime numberpp. An additional assumption is necessary e.g. that the quotients of the lower central series or of the derived series are torsion-free. We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin’s pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does. As a consequence, if such an extensionHHis torsion-free, then the group ringCH\mathbb {C}Hcontains no non-trivial zero divisors, i.e.HHfulfills the zero-divisor conjecture. In the course of the proof we prove that if these extensions are torsion-free, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin’s full braid group, therefore answering question B6 on http://www.grouptheory.info. Our methods also apply to the Baum-Connes conjecture. This is discussed by Thomas Schick in his preprint “Finite group extensions and the Baum-Connes conjecture”, where for example the Baum-Connes conjecture is proved for the full braid groups.