The difference Laplacian on a square lattice in Rn has been studied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient.
Standard Subdivision of a Complex 3 Approximation Theorem 4 Inner Product in Cochain Spaces. Combinatorial and Continuous Hodge Theories 5 Eigenvalues of the Laplacian Acting on Functions Acknowledgments References 0. Introduction. Let K be a finite simplicial complex. Eckmann (see [3]) observed that any inner product in real cochain spaces of K gives rise to a combinatorial Hodge theory. We show that if K is a smooth triangulation of a compact oriented, Riemannian manifold X, then the combinatorial Hodge theory (for a suitable choice of inner product) is an approximation of the Hodge theory of forms on X. Before giving a more detailed description of our results we introduce some notation and formulas. Thus let X be a compact, oriented, C 00 Riemannian manifold of dimension N, whose boundary consists of two disjoint closed submanifolds M1 and M 2. We do not exclude the possibility that M1, M 2 or both are empty. The Riemannian metric provides the space A= �A q of C 00 differen-Manuscript received June 26, 1973. *Columbia University Ph.D. Thesis.
Let G be a torsion-free discrete group, and let Q denote the field of algebraic numbers in C. We prove that QG 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 CG. The statement relies on new approximation results for L 2 -Betti numbers over QG, 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 L 2 -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 C. 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 G.
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