360 research outputs found

    Twisted limit formula for torsion and cyclic base change

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    Let GG be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. G= \SL_2 (\C) \times \SL_2 (\C) or \SL_3 (\C). Then the fundamental rank of GG is 2,2, and according to the conjecture made in \cite{BV}, lattices in GG should have 'little' --- in the very weak sense of 'subexponential in the co-volume' --- torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the \emph{square root} of the volume. This is deduced from a general theorem that compares twisted and untwisted L2L^2-torsions in the general base-change situation. This also makes uses of a precise equivariant 'Cheeger-M\"uller Theorem' proved by the second author \cite{Lip1}.Comment: 23 page

    Tetrahedra of flags, volume and homology of SL(3)

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    In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes. We describe when this volume belongs to the Bloch group. In doing so, we recover and generalize results of Neumann-Zagier, Neumann, and Kabaya. Our approach is very related to the work of Fock and Goncharov.Comment: 45 pages, 14 figures. The first version of the paper contained a mistake which is correct here. Hopefully the relation between the works of Neumann-Zagier on one side and Fock-Goncharov on the other side is now much cleare

    Eigenfunctions and Random Waves in the Benjamini-Schramm limit

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    We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we present a mathematically precise formulation of Berry's conjecture for a compact negatively curved manifold and formulate a Berry-type conjecture for sequences of locally symmetric spaces. We prove some weak versions of these conjectures. Using ergodic theory, we also analyze the connections of these conjectures to Quantum Unique Ergodicity.Comment: 40 page

    The Relative Lie Algebra Cohomology of the Weil Representation of SO(n,1)

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    In Part 1 of this paper we construct a spectral sequence converging to the relative Lie algebra cohomology associated to the action of any subgroup GG of the symplectic group on the polynomial Fock model of the Weil representation, see Section 7. These relative Lie algebra cohomology groups are of interest because they map to the cohomology of suitable arithmetic quotients of the symmetric space G/KG/K of GG. We apply this spectral sequence to the case G=SO0(n,1)G = \mathrm{SO}_0(n,1) in Sections 8, 9, and 10 to compute the relative Lie algebra cohomology groups H∙(so(n,1),SO(n);P(Vk))H^{\bullet} \big(\mathfrak{so}(n,1), \mathrm{SO}(n); \mathcal{P}(V^k) \big). Here V=Rn,1V = \mathbb{R}^{n,1} is Minkowski space and P(Vk)\mathcal{P}(V^k) is the subspace of L2(Vk)L^2(V^k) consisting of all products of polynomials with the Gaussian. In Part 2 of this paper we compute the cohomology groups H∙(so(n,1),SO(n);L2(Vk))H^{\bullet}\big(\mathfrak{so}(n,1), \mathrm{SO}(n); L^2(V^k) \big) using spectral theory and representation theory. In Part 3 of this paper we compute the maps between the polynomial Fock and L2L^2 cohomology groups induced by the inclusions P(Vk)⊂L2(Vk)\mathcal{P}(V^k) \subset L^2(V^k).Comment: 64 pages, 5 figure
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