a feynman kac ito formula for magnetic schrodinger operators on graphs in this paper we prove a feynman kac ito formula for magnetic schrodinger operators on arbitrary weighted graphs to do so we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation on the operator side we identify a very general class of potentials that allows the definition of magnetic schrodinger operators on the probabilistic side we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials apart from linking the world of discrete magnetic operators with the probabilistic world through the feynman kac ito formula the insights from this paper gained on both sides should be of an independent interest as applications of the feynman kac ito formula we prove a kato inequality a golden thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials

hamiltonians on discrete structures jumps of the integrated density of states and uniform convergence we study equivariant families of discrete hamiltonians on amenable geometries and their integrated density of states ids we prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support the size of a jump of the ids is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions from this we deduce uniform convergence w r t the spectral parameter of the finite volume approximants of the ids our framework includes quasiperiodic operators on delone sets periodic and random operators on quasi transitive graphs and operators on percolation graphs

finite group extensions and the atiyah conjecture the atiyah conjecture for a discrete group g states that the l 2 betti numbers of a finite cw complex with fundamental group g are integers if g is torsion free and in general that they are rational numbers with denominators determined by the finite subgroups of g here we establish conditions under which the atiyah conjecture for a torsion free group g implies the atiyah conjecture for every finite extension of g the most important requirement is that h g z p is isomorphic to the cohomology of the p adic completion of g for every prime p 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 primitive positive one relator groups and products of these therefore every finite in fact every elementary amenable extension of these groups satisfies the atiyah conjecture as a consequence if such an extension h is torsion free then the group ring ch contains no non trivial zero divisors i e h fulfills the zero divisor conjecture in the course of the proof we prove that if these extensions are torsionfree 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 www grouptheory org our methods also apply to the baum connes conjecture this is discussed in 47 where for example the baum connes conjecture is proved for the full braid groups msc 55n25 homology with local coecients 16s34 group rings laurent rings 57m25 knots and links

the spectrum of an infinite graph abstract the spectrum of a locally finite countable graph is defined some theorems known from the theory of spectra of finite graphs are extended to infinite graphs

sofic groups and diophantine approximation we prove the algebraic eigenvalue conjecture of j dodziuk p linnell v mathai t schick and s yates see 2 for sofic groups moreover we give restrictions on the spectral measure of elements in the integral group ring finally we define integer operators and prove a quantization of the operator norm below 2 to the knowledge of the author there is no group known that is not sofic

the strong atiyah and luck approximation conjectures for one relator groups it is shown that the strong atiyah conjecture and the luck 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

a c algebra of geometric operators on self similar cw complexes novikov shubin and l2 betti numbers abstract a class of cw complexes called self similar complexes is introduced together with c algebras a j of operators endowed with a finite trace acting on square summable cellular j chains since the laplacian j belongs to a j l 2 betti numbers and novikov shubin numbers are defined for such complexes in terms of the trace in particular a relation involving the euler poincare characteristic is proved l 2 betti and novikov shubin numbers are computed for some self similar complexes arising from self similar fractals

group cocycles and the ring of affiliated operators in this article we study cocycles of discrete countable groups with values in 2 g and the ring of affiliated operators ug we clarify properties of the first cohomology of a group g with coefficients in 2 g and answer several questions from de cornulier et al transform groups 13 1 125 147 2008 moreover we obtain strong results about the existence of free subgroups and the subgroup structure provided the group has a positive first 2 betti num ber we give numerous applications and examples of groups which satisfy our assumptions

a short proof of the approximation conjecture for amenable groups abstract we give a short proof of the approximation conjecture with complex coefficients for amenable groups g elek the strong approximation conjecture holds for amenable groups j funct anal 239 1 2006 345 355

approximating spectral invariants of harper operators on graphs abstract we study harper operators and the closely related discrete magnetic laplacians dml on a graph with a free action of a discrete group as defined by sunada sun a main result in this paper is that the spectral density function of dmls associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the dmls on a regular exhaustion with either dirichlet or neumann boundary conditions this then gives a criterion for the existence of gaps in the spectrum of the dml as well as other interesting spectral properties of such dmls the technique used incorporates some results of algebraic number theory

combinatorial heat kernels and index theorems abstract in this paper we give a local heat kernel proof for the combinatorial index theorem and the lefschetz theorem we also use this method to prove the combinatorial version of roe s index theorem on open manifolds

dixmier traces and extrapolation description of noncommutative lorentz spaces we study the connections between dixmier traces functions and traces of heat semigroups beyond the dual of the macaev ideal and in the general context of semifinite von neumann algebras we show that the correct framework for this investigation is that of operator lorentz spaces possessing an extrapolation description we demonstrate the applicability of our results to hormander weyl pseudo differential calculus on rn in this context we prove that the dixmier trace of a pseudo differential operator coincides with the dixmier integral of its symbol

ihara s zeta function for periodic graphs and its approximation in the amenable case abstract in this paper we give a more direct proof of the results by clair and mokhtari sharghi b clair s mokhtari sharghi zeta functions of discrete groups acting on trees j algebra 237 2001 591 620 on the zeta functions of periodic graphs in particular using appropriate operator algebraic techniques we establish a determinant formula in this context and examine its consequences for the ihara zeta function moreover we answer in the affirmative one of the questions raised in r i grigorchuk a uk the ihara zeta function of infinite graphs the kns spectral measure and integrable maps in v a kaimanovich et al eds proc workshop random walks and geometry vienna 2001 de gruyter berlin 2004 pp 141 180 by grigorchuk and uk accordingly we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs

the parity of the cochran harvey invariants of 3 manifolds given a finitely presented group and an epimorphism cochran and harvey defined a sequence of invariants which can be viewed as the degrees of higher order alexander polynomials cochran and harvey showed that up to a minor modification this is a never decreasing sequence of numbers if is the fundamental group of a 3 manifold with empty or toroidal boundary furthermore they showed that these invariants give lower bounds on the thurston norm using a certain cohn localization and the duality of reidemeister torsion we show that for a fundamental group of a 3 manifold any jump in the sequence is necessarily even this answers in particular a question of cochran furthermore using results of turaev we show that under a mild extra hypothesis the parity of the cochran harvey invariant agrees with the parity of the thurston norm

pointwise theorems for amenable groups in this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along folner sequences that obey some restrictions these restrictions are mild enough so that such sequences exist for all amenable groups we also prove a generalization of the shannon mcmillan breiman theorem to all discrete amenable groups

sectional curvature of polygonal complexes with planar substructures in this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature these complexes can be viewed as generalizations of 2 dimensional euclidean and hyperbolic buildings we focus on the case of non positive and negative combinatorial curvature as geometric results we obtain a hadamard cartan type theorem thinness of bigons gromov hyperbolicity and estimates for the cheeger constant we employ the latter to get spectral estimates show discreteness of the spectrum in the sense of a donnelly li type theorem and present corresponding eigenvalue asymptotics moreover we prove a unique continuation theorem for eigenfunctions and the solvability of the dirichlet problem at infinity

spectral analysis of percolation hamiltonians we study the family of hamiltonians which corresponds to the adjacency operators on a percolation graph we characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions this set of energies is a dense subset of the algebraic integers the integrated density of states has discontinuities precisely at this set of energies we show that the convergence of the integrated densities of states of finite box hamiltonians to the one on the whole space holds even at the points of discontinuity for this we use an equicontinuity from the right argument the same statements hold for the restriction of the hamiltonian to the infinite cluster in this case we prove that the integrated density of states can be constructed using local data only finally we study some mixed anderson quantum percolation models and establish results in the spirit of wegner and delyon and souillard

integrality of l 2 betti numbers the atiyah conjecture predicts that the l 2 betti numbers of a finite cw complex with torsion free fundamental group are integers we establish the atiyah conjecture under the condition that it holds for g and that h lhd g is a normal subgroup for amalgamated free products g h h rtimes f here f is a free group and h rtimes f is an arbitrary semi direct product this includes free products g f and semi direct products g rtimes f we also show that the atiyah conjecture holds with an additional technical condition for direct and inverse limits of groups for which it is true as a corollary it holds for positive 1 relator groups with torsion free abelianization putting everything together we establish a new bigger class of groups for which the atiyah conjecture holds which contains all free groups and in particular is closed under taking subgroups direct sums free products extensions with torsion free elementary amenable quotient or with free quotient and under certain direct and inverse limits

spectral properties of the laplacian on bond percolation graphs bond percolation graphs are random subgraphs of the d dimensional integer lattice generated by a standard bond percolation process the associated graph laplacians subject to dirichlet or neumann conditions at cluster boundaries represent bounded self adjoint ergodic random operators with off diagonal disorder they possess almost surely the non random spectrum 0 4d and a self averaging integrated density of states the integrated density of states is shown to exhibit lifshits tails at both spectral edges in the non percolating phase while the characteristic exponent of the lifshits tail for the dirichlet neumann laplacian at the lower upper spectral edge equals d 2 and thus depends on the spatial dimension this is not the case at the upper lower spectral edge where the exponent equals 1 2

uniqueness and regularity of proper harmonic maps introduction the purpose of this paper is to study some uniqueness and regularity properties of the dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces in general if the metrics of two complete noncompact manifolds mm and n are given locally by ds2 imj gij dxzdxj and dsy en h 3 du du respectively then the energy density function of a c1 map u m n is defined by

zeta functions of discrete groups acting on trees abstract this paper generalizes bass work on zeta functions for uniform tree lattices using the theory of von neumann algebras machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree the main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree a zeta function associated to a non uniform tree lattice with appropriate hilbert representation is defined zeta functions are defined for infinite graphs with a cocompact or finite covolume group action

heat kernels on covering spaces and topological invariants it is well known that there are relationships between the heat flow acting on differential forms on a closed oriented manifold m and the topology of m from hodge theory one can recover the betti numbers of m from the heat flow furthermore ray and singer 42 defined an analytic torsion a smooth invariant of acyclic flat bundles on m and conjectured that it equals the classical reidemeister torsion this conjecture was proved to be true independently by cheeger 7 and muller 40 the analytic torsion is nonzero only on odd dimensional manifolds and behaves in some ways as an odd dimensional counterpart of the euler characteristic 20 if m is not simply connected then there is a covering space analog of the betti numbers using the heat flow on the universal cover m one can define the l2 betti numbers of m 1 by taking the trace of the heat kernel not in the ordinary sense but as an element of a certain type ii von neumann algebra more concretely this amounts to integrating the local trace of the heat kernel over a fundamental domain in m in 11 we summarize this theory we consider the covering space analog of the analytic torsion this i analytic torsion has the same relation to the i cohomology as the ordinary analytic torsion bears to de rham cohomology in 111 we define the i analytic torsion 9 m under a technical assumption which we discuss later and prove its basic properties we show that m is a smooth invariant of manifolds whose l2 betti numbers vanish the proof is similar to that of the analogous statement for the ordinary analytic torsion but requires some care because of the possible slow decay of the heat kernels for large time in order to know if there are interesting examples of m we compute it in the case where m admits a hyperbolic metric it is clear that is proportionate to the volume of the hyperbolic metric which is

approximating l2 invariants and the atiyah conjecture let g be a torsion free discrete group and let denote the field of algebraic numbers in we prove that g fulfills the atiyah conjecture if g lies in a certain class of groups d which contains in particular all groups that are residually torsion free elementary amenable or are residually free this result implies that there are no nontrivial zero divisors in g the statement relies on new approximation results for l2 betti numbers over g which are the core of the work done in this paper another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial laplacian on l2 cochains on any normal covering space of a finite cw complex we establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the linnell class we also establish the absence of eigenvalues that are liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class 2003 wiley periodicals inc

asymptotics of the transition probabilities of the simple random walk on self similar graphs it is shown explicitly how self similar graphs can be obtained as blow up constructions of finite cell graphs c this yields a larger family of graphs than the graphs obtained by discretising continuous self similar fractals for a class of symmetrically self similar graphs we study the simple random walk on a cell graph c starting at a vertex v of the boundary of c it is proved that the expected number of returns to v before hitting another vertex in the boundary coincides with the resistance scaling factor using techniques from complex rational iteration and singularity analysis for green functions we compute the asymptotic behaviour of the n step transition probabilities of the simple random walk on the whole graph the results of grabner and woess for the sierpinski graph are generalised to the class of symmetrically self similar graphs and at the same time the error term of the asymptotic expression is improved finally we present a criterion for the occurrence of oscillating phenomena of the n step transition probabilities

fuglede kadison determinants and entropy for actions of discrete amenable groups consider a discrete group t and an element in the integral group ring zt then t acts from the left on the discrete additive group zt ztf by automorphisms of groups dualizing we obtain a left t action on the compact pontrjagin dual group xf zt ztf by continuous automorphisms of groups by definition xf is a closed subshift of s1 1 for measurable or topological actions of amenable groups a good entropy theory is available ow m thus from now on let t be a finitely generated discrete amenable group we are interested in determining the topological entropy hf of the t action on xf it agrees with the measure theoretic entropy of the t action on xf equipped with its haar probability measure in their paper fk from 1952 fuglede and kadison introduced a determinant for units in finite factor von neumann algebras more recently l ck has general ized their theory to nonunits and to group von neumann algebras which are not necessarily factors l2 he needed this to define l2 torsion for general covering spaces the von neumann algebra a t of the amenable group t is a certain completion of cr and hence we may consider the fuglede kadison l ck determinant detjvr of we refer to section 6 for the definition of a log strong f0lner sequence it is known that finitely generated virtually nilpotent groups have log strong f0lner sequences our main result is

the strong approximation conjecture holds for amenable groups abstract let g be a finitely generated group and g g 1 g 2 be normal subgroups such that k 1 g k 1 let a mat d d c g and a k mat d d c g g k be the images of a under the maps induced by the epimorphisms g g g k according to the strong form of the approximation conjecture of luck w luck l 2 invariants theory and applications to geometry and k theory ergeb math grenzgeb 3 vol 44 springer verlag berlin 2002 dim g ker a lim k dim g g k ker a k where dim g denotes the von neumann dimension in j dodziuk p linnell v mathai t schick s yates approximating l 2 invariants and the atiyah conjecture comm pure appl math 56 7 2003 839 873 dodziuk et al proved the conjecture for torsion free elementary amenable groups in this paper we extend their result for all amenable groups using the quasi tilings of ornstein and weiss d s ornstein b weiss entropy and isomorphism theorems for actions of amenable groups j anal math 48 1987 1 141

approximatingl2invariants of amenable covering spaces a combinatorial approach abstract in this paper we prove that the l 2 betti numbers of an amenable covering space can be approximated by the average betti numbers of a regular exhaustion proving a conjecture in dm we also prove that an arbitrary amenable covering space of a finite simplicial complex is of determinant class

asymptotic localization in the bose hubbard model we consider the bose hubbard model our focus is on many body localization which was described by many authors in such models even in the absence of disorder since our work is rigorous and since we believe that the localization in this type of models is not strictly valid in the infinite time limit we necessarily restrict our study to asymptotic localization also known as quasi localization we prove that transport and thermalization are small beyond perturbation theory in the limit of large particle density our theorem takes the form of a many body nekhoroshev estimate an interesting and new aspect of this model is the following the localization cannot be inferred from a lack of hybridization between zero hopping eigenstates naively speaking all these eigenstates appear resonant and one has to move to a dressed basis to see the absence of resonances that are responsible for quasi localization

linear wegner estimate for alloy type schrodinger operators on metric graphs we study spectra of alloy type random schrodinger operators on metric graphs for finite edge subsets we prove a wegner estimate which is linear in the volume i e the total length of the edges and the length of the energy interval the single site potential needs to have fixed sign the metric graph does not need to have a periodic structure a further result is the existence of the integrated density of states for ergodic random hamiltonians on metric graphs with a z structure for certain models the two above results together imply the lipschitz continuity of the integrated density of states

spectral gaps and discrete magnetic laplacians abstract the aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete laplacian on periodic graphs for proving this we analyse the discrete magnetic laplacian dml on the finite quotient and interpret the vector potential as a floquet parameter we develop a procedure of virtualising edges and vertices that produces matrices whose eigenvalues written in ascending order and counting multiplicities specify the bracketing intervals where the spectrum of the laplacian is localised we prove higuchi shirai s conjecture for z periodic trees and apply our technique in several examples like the polypropylene or the polyacetylene to show the existence of spectral gaps

a sears type self adjointness result for discrete magnetic schrodinger operators abstract in the context of a weighted graph with vertex set v and bounded vertex degree we give a sufficient condition for the essential self adjointness of the operator w where is the magnetic laplacian and w v r is a function satisfying w x q x for all x v with q v 1 the condition is expressed in terms of completeness of a metric that depends on q and the weights of the graph the main result is a discrete analogue of the results of i oleinik and m a shubin in the setting of non compact riemannian manifolds

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

on the spectrum of a finite volume negatively curved manifold we show that a noncompact manifold with bounded sectional curvature whose ends are sufficiently gromov hausdorff close to rays has a finite dimensional space of square integrable harmonic forms in the special case of a finite volume manifold with pinched negative sectional curvature we show that the essential spectrum of the p iorm laplacian is the union of the essential spectra of a collection of ordinary differential operators associated to the ends we give examples of such manifolds with curvature pinched arbitrarily close to 1 and with an infinite number of gaps in the spectrum of the function laplacian 1 introduction in this paper we consider riemannian manifolds of finite volume and pinched negative sectional curvature we give results about the kernel of the differential form laplacian and about its essential spectrum our first result is the finite dimensionality of the space of square integrable harmonic forms for a more general class of riemannian manifolds which can be roughly characterized as those with bounded sectional curvature and with ends that are sufficiently gromov hausdorff close to rays let m be a complete connected rc dimensional riemannian manifold with a basepoint m let br m denote the distance ball around m and let sr m dbr m be the distance sphere around m put

log holder continuity of the integrated density of states for stochastic jacobi matrices we consider the integrated density of states k e of a general operator on l2 v of the formh h0 v where h 0 u n sum limits left i right 1 u n i and vu n v n u n wherev is a general bounded ergodic stationary process on v we show that k e k e c log e e 1 when e e 1 2 the key is a thouless formula for the strip

uniform existence of the integrated density of states for random schrodinger operators on metric graphs over zd abstract we consider ergodic random schrodinger operators on the metric graph z d with random potentials and random boundary conditions taking values in a finite set we show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable this limit the integrated density of states can be expressed by a closed shubin pastur type trace formula it supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions among other examples we discuss random magnetic fields and percolation models

on the spectrum of lamplighter groups and percolation clusters let mathfrak g be a finitely generated group and x its cayley graph with respect to a finite symmetric generating set s furthermore let mathfrak h be a finite group and mathfrak h wr mathfrak g the lamplighter group wreath product over mathfrak g with group of lamps mathfrak h we show that the spectral measure plancherel measure of any symmetric switch walk switch random walk on mathfrak h wr mathfrak g coincides with the expected spectral measure integrated density of states of the random walk with absorbing boundary on the cluster of the group identity for bernoulli site percolation on x with parameter mathsf p 1 mathfrak h the return probabilities of the lamplighter random walk coincide with the expected annealed return probabilities on the percolation cluster in particular if the clusters of percolation with parameter mathsf p are almost surely finite then the spectrum of the lamplighter group is pure point this generalizes results of grigorchuk and uk resp dicks and schick regarding the case when mathfrak g is infinite cyclic analogous results relate bond percolation with another lamplighter random walk in general the integrated density of states of site or bond percolation with arbitrary parameter mathsf p is always related with the plancherel measure of a convolution operator by a signed measure on mathfrak h wr mathfrak g where mathfrak h mathbb z or another suitable group

l2 spectral invariants and convergent sequences of finite graphs abstract using the spectral theory of weakly convergent sequences of finite graphs we prove the uniform existence of the integrated density of states for a large class of infinite graphs

characters l 2 betti numbers and an equivariant approximation theorem let g be a group with a finite subgroup h we define the l 2 multiplicity of an irreducible representation of h in the l 2 homology of a proper g cw complex these invariants generalize the l 2 betti numbers our main results are approximation theorems for l 2 multiplicities which extend the approximation theorems for l 2 betti numbers of luck farber and elek szabo respectively the main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups we discuss applications to the cohomology of arithmetic groups

arithmetic properties of eigenvalues of generalized harper operators on graphs let open image in new window denote the field of algebraic numbers in open image in new window a discrete group g is said to have the multiplier algebraic eigenvalue property if for every matrix a m d open image in new window g regarded as an operator on l 2 g d the eigenvalues of a are algebraic numbers where z 2 g open image in new window is an algebraic multiplier and open image in new window denotes the unitary elements of open image in new window such operators include the harper operator and the discrete magnetic laplacian that occur in solid state physics we prove that any finitely generated amenable free or surface group has this property for any algebraic multiplier in the special case when is rational n 1 for some positive integer n this property holds for a larger class of groups open image in new window containing free groups and amenable groups and closed under taking directed unions and extensions with amenable quotients included in the paper are proofs of other spectral properties of such operators

group ring elements with large spectral density given delta 0 we construct a group g and a group ring element s in mathbb z g such that the spectral measure mu of s fulfils mu 0 varepsilon frac c log varepsilon 1 delta for small varepsilon in particular the novikov shubin invariant of any such s is 0 the constructed examples show that the best known upper bounds on mu 0 varepsilon are not far from being optimal

a trace on fractal graphs and the ihara zeta function starting with ihara s work in 1968 there has been a growing interest in the study of zeta functions of finite graphs by sunada hashimoto bass stark and terras mizuno and sato to name just a few authors then clair and mokhtari sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms the main formula in all these treatments establishes a connection between the zeta function originally defined as an infinite product and the laplacian of the graph in this article we consider a different class of infinite graphs they are fractal graphs i e they enjoy a self similarity property we de ne a zeta function for these graphs and using the machinery of operator algebras we prove a determinant formula which relates the zeta function with the laplacian of the graph we also prove functional equations and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs

on turing dynamical systems and the atiyah problem main theorems of the article concern the problem of atiyah on possible values of l 2 betti numbers it is shown that all non negative real numbers are l 2 betti numbers and that many for example all non negative algebraic real numbers are l 2 betti numbers of simply connected manifolds with respect to a free cocompact action also an explicit example is constructed which leads to a simply connected manifold with a transcendental l 2 betti number with respect to an action of the threefold direct product of the lamplighter group mathbb z 2 mathbb z wr mathbb z the main new idea is embedding turing machines into integral group rings the main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids

isomorphism conjectures in algebraic k theory duff in this paper we are concerned with the four functors y y3 if and which map from the category of topological spaces x to the category diff of q spectra the functor 3 or if maps the space x to the qspectrum of stable topological or smooth pseudoisotopies of x the spaces diff dif 3i x id9 x i 0 are hatcher s deloopings of 970 x yd x cf 29 the functor j maps a path connected space x to the algebraic ktheoretic s2 spectrum for the integral group ring z7r1x the spaces x x i 0 are the gersten wagoner deloopings of z o x cf 27 52 5 x is also defined on a nonpath connected space x to be x1 where xi i e i are the path components of x and ii xi indicates the direct limit of all finite products of the i xi i e i if x is path connected then the functor maps x to the l surgery classifying spaces for oriented surgery problems with fundamental group r i1x identified by their fourfold periodicity k 90 x k 4j x cf 35 39 44 47 in addition if x is not path connected then we set y77 x tiei5 fj xi where xi i e i are the path components of x results obtained by the authors over the past five years cf 15 25 and in particular 19 16 a 1 a 18 together with many earlier results and conjectures which are reviewed in 1 6 1 1 6 6 suggest that for any connected cwcomplex x and for 5 denoting any of the above functors the q spectrum x should be computable in a simple way from the values taken by 9 on covering spaces of x which have very simple fundamental groups in more detail we let 9 x denote the collection of all subgroups h c ir ix which are either finite or virtually infinite cyclic a group h is virtually infinite cyclic if there is a short exact sequence of groups 1 z h f 1 with f equal to a finite group and z equal to the infinite cyclic group for each such group we let xh x denote the covering space projection corresponding to h then the q spectrum 5 x should be computable in a simple way from the q spectra 5 xh h e 9 x the purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any x

hyperlinearity essentially free actions and l2 invariants the sofic property we prove that connes embedding conjecture holds for the von neumann algebras of sofic groups that is sofic groups are hyperlinear hence we provide some new examples of hyperlinearity we also show that the determinant conjecture holds for sofic groups as well we introduce the notion of essentially free actions and amenable actions and study their properties

discontinuities of the integrated density of states for random operators on delone sets despite all the analogies with usual random models tight binding operators for quasicrystals exhibit a feature that clearly distinguishes them from the former the integrated density of states may be discontinuous this phenomenon is identified as a local effect due to the occurrence of eigenfunctions with bounded support

an index theorem on open manifolds ii on decrit et on applique un nouveau theoreme d indice pour des operateurs elliptiques sur certaines varietes non compactes on calcule un indice a valeur reelle pour des operateurs de type dirac sur des varietes non compactes de geometrie bornee

spectral preorder and perturbations of discrete weighted graphs in this article we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential the first preorder is expressed through the existence of a graph homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights the second preorder refers to the spectrum of the associated laplacian of the magnetic weighted graph these relations give a quantitative control of the effect of elementary and composite perturbations of the graph deleting edges contracting vertices etc on the spectrum of the corresponding laplacians generalising interlacing of eigenvalues we give several applications of the preorders we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal d clique moreover we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic cheeger constants with respect to the geometric preorder finally we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph

quantum hall effect on the hyperbolic plane in this paper we study both the continuous model and the discrete model of the quantum hall effect qhe on the hyperbolic plane the hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables we define a twisted analogue of the kasparov map which enables us to use the pairing between k theory and cyclic cohomology theory to identify this geometric invariant with a topological index thereby proving the integrality of the hall conductivity in this case

hyperpolygon spaces and moduli spaces of parabolic higgs bundles abstract given an n tuple of positive real numbers we consider the hyperpolygon space x the hyperkahler quotient analogue to the kahler moduli space of polygons in r 3 we prove the existence of an isomorphism between hyperpolygon spaces and moduli spaces of stable rank 2 holomorphically trivial parabolic higgs bundles over c p 1 with fixed determinant and trace free higgs field this isomorphism allows us to prove that hyperpolygon spaces x undergo an elementary transformation in the sense of mukai as crosses a wall in the space of its admissible values we describe the changes in the core of x as a result of this transformation as well as the changes in the nilpotent cone of the corresponding moduli spaces of parabolic higgs bundles moreover we study the intersection rings of the core components of x in particular we find generators of these rings prove a recursion relation in n for their intersection numbers and use it to obtain explicit formulas for the computation of these numbers using our isomorphism we obtain similar formulas for each connected component of the nilpotent cone of the corresponding moduli spaces of parabolic higgs bundles thus determining their intersection rings as a final application of our isomorphism we describe the cohomology ring structure of these moduli spaces of parabolic higgs bundles and of the components of their nilpotent cone

fourier duality for fractal measures with affine scales for a family of fractal measures wend an explicit fourier d uality the measures in the pair have compact support in r d and they both have the same matrix scaling but the two use dierent translation vectors one by a subset b in r d and the other by a related subset l among other things we show that there is then a pair of innite disc rete sets l and b in r d such that the l fourier exponentials are orthogonal in l 2 b and the b fourier exponentials are orthogonal in l 2 l these sets of orthogonal frequencies are typically lac unary and they will be obtained by scaling in the large the nature of our duality is explored below both in higher dimensions and for examples on the real line our duality pairs do not always yield orthonormal fourier bases in the respective l 2 hilbert spaces but depending on the geometry of certainnite orbit s we show that they do in some cases we further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these ane fractal measures

all the git quotients at once let g be an algebraic torus acting on a smooth variety v we study the relationship between the various git quotients of v and the symplectic quotient of the cotangent bundle of v

noncommutative riemann integration and novikov shubin invariants for open manifolds given a c algebra a with a semicontinuous semifinite trace acting on the hilbert space h we define the family ar of bounded riemann measurable elements w r t as a suitable closure a 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 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 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 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

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

entropy determinants and l 2 torsion we show that for any amenable group and any z module m of type fl with vanishing euler characteristic the entropy of the natural action on the pontryagin dual of m is equal to the l 2 torsion of m as a particular case the entropy of the principal algebraic action associated with the module z z f is equal to the logarithm of the fuglede kadison determinant of f whenever f is a non zero divisor in z this conrms a conjecture of deninger as a key step in the proof we provide a general szeg o type approximation theorem for the fuglede kadison determinant on the group von neumann algebra of an amenable group as a consequence of the equality between l 2 torsion and entropy we show that the l 2 torsion of a non trivial amenable group with nite classifying space vanishes this was conjectured by luck finally we establish a milnor turaev formula for the l 2 torsion of a nite acyclic chain complex

remark on the continuity of the density of states of ergodic finite difference operators we give an elementary proof that for a large class ofd dimensional finite difference operators including tight binding models for electron propagation and models for harmonic phonons with random masses or couplings the integrated density of states is a continuous function of the energy

discrete magnetic laplacian we consider a 2 dimensional discrete operator which we call the discrete magnetic laplacian dml it is an analogue of the magnetic schrodinger operator it follows from well known arguments that dml has the same spectrum as a subset inr as the almost mathieu operator am they also have the same integrated density of states ids which is known to be continuous we show that dml is an element in a ii1 factor and its ids can be expressed through the trace in the ii1 factor it follows that dml never has anyl2 eigenfunctions i e has no point spectrum then we formulate a natural algebraic conjecture which implies that the spectrum of dml hence the spectrum of am is a cantor set

integrated density of states for random metrics on manifolds we study ergodic random schrodinger operators on a covering manifold where the randomness enters both via the potential and the metric we prove measurability of the random operators almost sure constancy of their spectral properties the existence of a self averaging integrated density of states and a pastur ubin type trace formula

erratum to integrality of l2 betti numbers a gap has been found in the proof of proposition 3 of 3 consequently this result and the consequences drawn in the paper have to be considered as conjectures in particular it is not proved that the atiyah conjecture is stable under extensions with free quotient groups and under free products with free groups

hyperpolygon spaces and their cores given an n tuple of positive real numbers 1 an konno 2000 defines the hyperpolygon space x a a hyperkahler analogue of the kahler variety m a parametrizing polygons in r 3 with edge lengths 1 an the polygon space m can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector from this point of view x a is the hyperkahler quiver variety defined by nakajima a quiver variety admits a natural c action and the union of the precompact orbits is called the core we study the components of the core of x interpreting each one as a moduli space of pairs of polygons in r 3 with certain properties konno gives a presentation of the cohomology ring of x a we extend this result by computing the c equivariant cohomology ring as well as the ordinary and equivariant cohomology rings of the core components

singular traces on semifinite von neumann algebras abstract singular traces are constructed on a general semifinite von neumann algebra thus generalizing the result of dixmier c r acad sci paris 262 1966 moreover our technique produces singular traces on type ii 1 factors such traces though vanishing on all bounded operators are non trivial on the algebra of affiliated unbounded operators on a semifinite factor we show that all traces are given by a dilation invariant functional on the cone of positive decreasing functions on 0 and we prove that the existence of a singular trace which is non trivial on a given operator is equivalent to an eccentricity condition on the singular values function a result which generalizes the theorem given in s albeverio d guido a ponosov and s scarlatti j funct anal to appear for b h

on the density of state for the quantum percolation problem for the site dilution model on the hypercubic lattice zd d or 2 the authors examine the density of states for the tight binding hamiltonian projected onto the infinite cluster it is shown that with probability one the corresponding integrated density of states is discontinuous on a set of energies which is dense in the band this result is proved by constructing states supported on finite regions of the infinite cluster analogous to the kirkpatrick and eggarter zero energy molecular state