288 research outputs found

    Small eigenvalues of surfaces of finite type

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    Index theorems on manifolds with straight ends

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    We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume

    On eta-functions for nilmanifolds

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    Equivariant discretizations of diffusions, random walks, and harmonic functions

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    For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons–Sullivan discretizations of the processes and the associated function theory

    Bottom of spectra and coverings of orbifolds

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    We discuss the behaviour of the bottom of the spectrum of scalar Schrödinger operators under Riemannian coverings of orbifolds. We apply our results to geometrically finite and to conformally compact orbifolds

    Rank rigidity for CAT(0) cube complexes

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    We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rank one isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits Alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.Comment: 39 pages, 4 figures. Revised version according to referee repor

    Bottom of spectra and amenability of coverings

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    For a Riemannian covering π ⁣:M1M0\pi\colon M_1\to M_0, the bottoms of the spectra of M0M_0 and M1M_1 coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci curvature, we obtain a converse under a natural condition on the spectrum of M0M_0

    Symmetric spaces of higher rank do not admit differentiable compactifications

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    Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answer negatively.Comment: 13 pages, to appear in Mathematische Annale
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