Novikov-Shubin Invariants Research Articles

Stability of L2-Invariants on Stratified Spaces

Abstract Let $\overline{M}$ be a compact smoothly stratified pseudo-manifold endowed with a wedge metric $g$. Let $\overline{M}_{\Gamma }$ be a Galois $\Gamma $-covering. Under additional assumptions on $\overline{M}$, satisfied for example by Witt pseudo-manifolds, we show that the $L^{2}$-Betti numbers and the Novikov–Shubin invariants are well defined. We then establish their invariance under a smoothly stratified codimension-preserving homotopy equivalence, thus extending results of Dodziuk, Gromov, and Shubin to these pseudo-manifolds.

Families torsion and Morse functions for covering spaces

In this paper we prove the Cheeger–Müller theorem for \(L^2\)-analytic torsion form under the assumption that there exists a fiberwise Morse function and the Novikov–Shubin invariants are positive.

Profinite invariants of arithmetic groups

AbstractWe prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of typeF. Our methods imply similar results for$\ell^2$-torsion as well as a strong profiniteness statement for Novikov–Shubin invariants.

Dyson’s Spike for Random Schroedinger Operators and Novikov–Shubin Invariants of Groups

We study Schroedinger operators with random edge weights and their expected spectral measures $\mu_H$ near zero. We prove that the measure exhibits a spike of the form $\mu_H(-\varepsilon,\varepsilon) \sim \frac{C}{| \log\varepsilon|^2}$ (first observed by Dyson), without assuming independence or any regularity of edge weights. We also identify the limiting local eigenvalue distribution, which is different from Poisson and the usual random matrix statistics. We then use the result to compute Novikov-Shubin invariants for various groups, including lamplighter groups and lattices in the Lie group Sol.

An interesting example for spectral invariants

AbstractIn [HL99], the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then shown that the associated heat operator converges to the Chern character of the index bundle of the operator. In [BH08], this result was improved by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.

L2-RHO FORM FOR NORMAL COVERINGS OF FIBER BUNDLES

We define the secondary invariants L2-eta and -rho forms for families of generalized Dirac operators on normal coverings of fiber bundles. On the covering family we assume transversally smooth spectral projections and Novikov–Shubin invariants bigger than 3( dim B + 1) to treat the large time asymptotic for the heat operator. In the case of a bundle of spin manifolds, we study the L2-rho class in relation to the space [Formula: see text] of positive scalar curvature vertical metrics.

L 2-Invariants of finite aspherical CW-complexes

Let X be a finite aspherical CW-complex whose fundamental group π 1(X) possesses a subnormal series $${\pi_1(X) \vartriangleright G_m \vartriangleright \cdots \vartriangleright G_0}$$ with a non-trivial elementary amenable group G 0. We investigate the L 2-invariants of the universal covering of such a CW-complex X. The main result is the proof of the vanishing of the L 2-torsion $${\rho^{(2)}({\tilde X})}$$ under the condition that π 1(X) has semi-integral determinant. We further show that the Novikov–Shubin invariants $${\alpha_n({\tilde X})}$$ are positive.

Index Theory and Non-Commutative Geometry II. Dirac Operators and Index Bundles

AbstractWhen the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K—theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the Novikov-Shubin invariants of the operator is improved.

L 2-invariants of Riemannian foliations

For a Riemannian foliation \(\mathcal F\) on a closed manifold M, we define L 2-spectral sequence Betti numbers and spectral sequence Novikov–Shubin invariants. The spectral sequence of the lift of \(\mathcal F\) to the universal covering of M is used in the definitions. These invariants are natural extensions of the L 2-Betti numbers and the Novikov–Shubin invariants of differentiable manifolds. It is shown that these numbers are invariant by foliated homotopy equivalences, and they are computed for several examples.

Secondary Novikov-Shubin invariants of groups and quasi-isometry

We define new L 2 -invariants which we call secondary Novikov-Shubin invariants.We calculate the first secondary Novikov-Shubin invariants of finitely generated groups by using random walk on Cayley graphs and see in particular that these are invariant under quasi-isometry.

A Cheeger–Müller theorem for delocalized analytic torsion

We study the delocalized L 2-analytic torsion introduced by John Lott, and define the delocalized L 2-combinatorial torsion. By using the method of Bismut–Zhang, under the conditions of positive Novikov–Shubin invariants and finite conjugacy class, we get the Cheeger–Muller type relation between the delocalized L 2-analytic torsion and the delocalized L 2-combinatorial torsion.

Homological Invariants and Quasi-Isometry

Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cd R over a commutative ring R satisfies the inequality $$ \,{\text{cd}}_R (\Lambda ) \leq {\text{cd}}_R (\Gamma ) $$ if Λ embeds uniformly into Γ and $$ {\text{cd}}_R (\Lambda ) < \infty $$ holds. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants, and show their invariance under quasi-isometry.

Riemann-Roch-Grothendieck and torsion for foliations

In this article we prove a Riemann-Roch-Grothendieck theorem for the characteristic classes of a flat vector bundle over a foliation whose graph is Hausdorff. We assume that the strong foliation Novikov-Shubin invariants of the flat bundle are greater than three times the codimension of the foliation. Using transgression, we define a torsion form which in the odd acyclic case determines a Haefliger cohomology class which only depends on the foliation and the flat bundle. We construct examples where this torsion class is highly non-trivial.

$L^2$-invariants of locally symmetric spaces

Let X=G/K be a Riemannian symmetric space of the noncompact type, \Gamma\subset G a discrete, torsion-free, cocompact subgroup, and let Y=\Gamma\backslash X be the corresponding locally symmetric space. In this paper we explain how the Harish-Chandra Plancherel Theorem for L^2(G) and results on ({\mathfrak g}, K) -cohomology can be used in order to compute the L^2 -Betti numbers, the Novikov–Shubin invariants, and the L^2 -torsion of Y in a uniform way thus completing results previously obtained by Borel, Lott, Mathai, Hess and Schick, Lohoue and Mehdi. It turns out that the behaviour of these invariants is essentially determined by the fundamental rank m=\mathrm{rk}_{\Bbb C}G -\mathrm{rk}_{\Bbb C}K of G . In particular, we show the nonvanishing of the L^2 -torsion of Y whenever m=1 .

Noncommutative Riemann Integration and Novikov–Shubin Invariants for Open Manifolds

Given a C*-algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family AR of bounded Riemann measurable elements w.r.t. τ as a suitable closure, á la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that AR is a C*-algebra, and τ extends to a semicontinuous semifinite trace on AR. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A″ and can be approximated in measure by operators in AR, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a τ-a.e. bimodule on AR, denoted by AR, and such a bimodule contains the functional calculi of selfadjoint elements of AR under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on AR. Type II1 singular traces for C*-algebras can be defined on the bimodule of unbounded Riemann-measurable operators. Noncommutative Riemann integration and singular traces for C*-algebras are then used to define Novikov–Shubin numbers for amenable open manifolds, to show their invariance under quasi-isometries, and to prove that they are (noncommutative) asymptotic dimensions.

The Novikov–Shubin invariants for locally symmetric spaces

Let M be a closed oriented Riemannian smooth manifold. The Von Neumann dimension of the spectrum of the Laplace–Beltrami operator acting on differential forms of degree l on M is a homotopy invariant of M, called the lth Novikov–Shubin invariant of M. In this paper, using Representation Theory of semisimple Lie groups, we compute the Novikov–Shubin invariants of locally symmetric spaces.

Von Neumann spectra near the spectral gap

In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L 2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L 2 theta and L 2 zera functions defined by metric dependent combinatorial Laplacians acting on L 2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold (M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of π₁(M) on a finite dimensional vector space to a representation on a A-Hilbert module W of finite type where A is a finite von Neumann algebra. If (M,W) is of determinant class we prove, generalizing the Cheeger-Muller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, the L₂-analytic and L₂-Reidemeister torsions are equal.

Homological algebra of Novikov-Shubin invariants and morse inequalities

It is shown in this paper that the topological phenomenon“zero in the continuous spectrum”, discovered by S.P. Novikov and M.A. Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in a certain abelian category. This approach implies homotopy invariance of the Novikov-Shubin invariants. Its main advantage is that it allows the use of the standard homological techniques, such as spectral sequences, derived functors, universal coefficients etc., while studying the Novikov-Shubin invariants. It also leads to some new quantitative invariants, measuring the Novikov-Shubin phenomenon in a different way, which are used in the present paper in order to strengthen the Morse type inequalities of Novikov and Shubin [NSh2].

Inequalities for the Novikov-Shubin invariants

In this paper, we prove that the Novikov-Shubin invariants satisfy a sequence of inequalities and deduce some useful consequences of this result.