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20
L 2 determinant class and approximation of L 2 Betti numbers
 Trans. Amer. Math. Soc
"... A standing conjecture in L 2cohomology is that every finite CWcomplex X is of L 2determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually am ..."
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Cited by 16 (3 self)
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A standing conjecture in L 2cohomology is that every finite CWcomplex X is of L 2determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L 2acyclic, we also prove that the L 2determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L 2Betti numbers.
Approximating spectral invariants of Harper operators on graphs
 J. Functional Analysis
"... 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]. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete ..."
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Cited by 15 (2 self)
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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]. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in [MY]. The result is generalized to other self adjoint operators with finite propagation. 1.
Rank gradient, cost of groups and the rank versus Heegard genus problem”, preprint
"... We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a byproduct, we show that the ‘Rank vs. Heegaard genus ’ conjecture on hyperbolic 3manifolds is incompatible with the ‘Fixed Price problem ’ in topological dynamics. 1 ..."
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Cited by 13 (1 self)
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We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a byproduct, we show that the ‘Rank vs. Heegaard genus ’ conjecture on hyperbolic 3manifolds is incompatible with the ‘Fixed Price problem ’ in topological dynamics. 1
INTEGRATED DENSITY OF STATES AND WEGNER ESTIMATES FOR RANDOM SCHRÖDINGER OPERATORS
, 2003
"... We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators ..."
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Cited by 12 (2 self)
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We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators on manifolds. Subsequently we study more specific models in Euclidean space, namely of alloy type, and concentrate on the regularity properties of the IDS. We discuss the role of the integrated density of states and its regularity properties for the spectral analysis of random Schrödinger operators, particularly in relation to localisation. Proofs of the central results are given in detail. Whenever there are alternative proofs, the different approaches are compared.
A trace on fractal graphs AND THE IHARA ZETA FUNCTION
, 2008
"... 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 MokhtariSharghi have studied zeta functions for infinite graphs acted u ..."
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Cited by 9 (4 self)
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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 MokhtariSharghi have 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 selfsimilarity property. We define 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. The Ihara zeta function, originally associated to certain groups and then combinatorially
Noncommutative Riemann integration and NovikovShubin invariants for Open Manifolds
, 2001
"... Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [2 ..."
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Cited by 6 (3 self)
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Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [26], and show that A R is a C ∗algebra, and τ extends to a semicontinuous semifinite trace on A R. 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 A R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a τa.e. bimodule on A R, denoted by A R, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on A R.
Residual amenability and the approximation of L 2 –invariants
 Michigan Math. J
, 1999
"... Abstract. We generalize Lück’s Theorem to show that the L 2Betti numbers of a residually amenable covering space are the limit of the L 2Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of a finite simplicial complex is of determinant cla ..."
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Cited by 6 (0 self)
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Abstract. We generalize Lück’s Theorem to show that the L 2Betti numbers of a residually amenable covering space are the limit of the L 2Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of a finite simplicial complex is of determinant class, and that the L 2 torsion is a homotopy invariant for such spaces. We give examples of residually amenable groups, including the BaumslagSolitar groups.
Approximating L 2 torsion on amenable covering spaces. math.DG/0008211 on arxiv.org, see also [12
"... Abstract. In this paper, we prove that the L 2 combinatorial torsion of an amenable covering space can be approximated by the combinatorial torsions of a regular exhaustion. An ancillary theorem shows the L 2 spectral density function of the combinatorial Laplacian on L 2cochains on the covering sp ..."
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Cited by 3 (0 self)
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Abstract. In this paper, we prove that the L 2 combinatorial torsion of an amenable covering space can be approximated by the combinatorial torsions of a regular exhaustion. An ancillary theorem shows the L 2 spectral density function of the combinatorial Laplacian on L 2cochains on the covering space is approximated by the average spectral density functions of the combinatorial Laplacian on the cochains of the regular exhaustion, with either Dirichlet or Neumann boundary conditions, extending one of the main results in [DM]. The technique used incorporates some results of algebraic number theory.
Amenable groups, topological entropy and Betti numbers
, 1999
"... Abstract. We investigate an analogue of the L 2Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy. Partially supported by OTKA grant T 25004 and the Bolyai Fellowship ..."
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Cited by 3 (0 self)
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Abstract. We investigate an analogue of the L 2Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy. Partially supported by OTKA grant T 25004 and the Bolyai Fellowship