An application to local index theory

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Compact Manifold ◽  
Index Theory ◽  
Heat Kernels ◽  
Index Formula ◽  
Torsion Free Group ◽  
Local Index ◽  
Orbital Integrals ◽  
Index Formulas ◽  
Torsion Free

This chapter verifies the compatibility of the formula for the orbital integrals of heat kernels introduced in the previous chapter to the index formula of Atiyah-Singer, to the fixed point formulas of Atiyah-Bott, and to the index formula for orbifolds of Kawasaki. Given that the McKean-Singer formula expresses the index of a Dirac operator over a compact manifold Z as the supertrace of a heat kernel, if Z is the quotient of X by a cocompact torsion free group, this supertrace can be evaluated explicitly by the formulas provided in the previous chapter. This chapter directly checks these formulas to be compatible with the index formulas.

Introduction

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Riemannian Manifold ◽  
Trace Formula ◽  
Index Theory ◽  
Heat Kernels ◽  
Previous Literature ◽  
Orbital Integrals ◽  
Short Overview ◽  
History Of

This introductory chapter reviews the hypoelliptic Laplacian. It first explains how to mathematically conceive of a union between index theory and the trace formula through the Lefschetz fixed point formulas. The chapter then embarks on a brief history of the hypoelliptic Laplacian, hereafter turning to the construction of the hypoelliptic Laplacian that is carried out in this volume. Moreover, it discusses the analysis of the hypoelliptic orbital integrals, and its overlap with the analysis of the hypoelliptic Laplacian in previous literature, in which the Riemannian manifold X was assumed to be compact, and genuine traces or supertraces were considered. Here in this chapter X is noncompact, and the orbital integrals that appear are defined using explicit properties of the corresponding heat kernels. After this review, the chapter gives a short overview on the following chapters.

scholarly journals Spectral Flow for Nonunital Spectral Triples

2015 ◽  
Vol 67 (4) ◽  
pp. 759-794 ◽  
Author(s):  
A. L. Carey ◽  
V. Gayral ◽  
J. Phillips ◽  
A. Rennie ◽  
F. A. Sukochev
Keyword(s):  
Index Theory ◽  
Spectral Triple ◽  
Index Formula ◽  
Spectral Flow ◽  
Spectral Triples ◽  
Local Index ◽  
C Algebra ◽  
Odd Index ◽  
Definition Of ◽  
Special Case

AbstractWe prove two results about nonunital index theory left open in a previous paper. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisûes the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting, we are able to connect with earlier approaches to the analytic definition of spectral flow

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

scholarly journals A just-nonsolvable torsion-free group defined on the binary tree

1999 ◽  
Vol 211 (1) ◽  
pp. 99-114 ◽  
Author(s):  
A.M. Brunner ◽  
Said Sidki ◽  
Ana Cristina Vieira
Keyword(s):  
Free Group ◽  
Binary Tree ◽  
Torsion Free Group ◽  
Torsion Free

scholarly journals On a Nonsingular Equation of Length 9 Over Torsion Free Groups

2019 ◽  
Vol 12 (2) ◽  
pp. 590-604
Author(s):  
M. Fazeel Anwar ◽  
Mairaj Bibi ◽  
Muhammad Saeed Akram
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Torsion Free

In \cite{levin}, Levin conjectured that every equation is solvable over a torsion free group. In this paper we consider a nonsingular equation $g_{1}tg_{2}t g_{3}t g_{4} t g_{5} t g_{6} t^{-1} g_{7} t g_{8}t \\ g_{9}t^{-1} = 1$ of length $9$ and show that it is solvable over torsion free groups modulo some exceptional cases.

scholarly journals Abelian groups with small cotorsion images

1991 ◽  
Vol 50 (2) ◽  
pp. 243-247 ◽  
Author(s):  
Rüdiger Göbel
Keyword(s):  
Homomorphic Image ◽  
Free Group ◽  
Abelian Groups ◽  
Torsion Free Group ◽  
Compact Abelian Groups ◽  
Torsion Free

AbstractEpimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

scholarly journals ON THE ASPHERICITY OF LENGTH-6 RELATIVE PRESENTATIONS WITH TORSION-FREE COEFFICIENTS

2008 ◽  
Vol 51 (1) ◽  
pp. 201-214
Author(s):  
Seong Kun Kim
Keyword(s):  
Free Group ◽  
Zero Divisor ◽  
Point Of View ◽  
Torsion Free Group ◽  
Torsion Free

AbstractAn interesting result of Ivanov implies that a non-aspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture. In this point of view, we prove the asphericity of the length-6 relative presentation $\langle H,x: xh_1xh_2xh_3xh_4xh_5xh_6\rangle$, provided that each coefficient is torsion free.

scholarly journals The local index formula in semifinite Von Neumann algebras I: Spectral flow

2006 ◽  
Vol 202 (2) ◽  
pp. 451-516 ◽  
Author(s):  
Alan L. Carey ◽  
John Phillips ◽  
Adam Rennie ◽  
Fyodor A. Sukochev
Keyword(s):  
Von Neumann Algebras ◽  
Index Formula ◽  
Spectral Flow ◽  
Local Index ◽  
Von Neumann
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