55 research outputs found

    On the Regularization of the Kepler Problem

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    We show that for the Kepler problem the canonical Ligon-Schaaf regularization map can be understood in a straightforward manner as an adaptation of the Moser regularization. In turn this explains the hidden symmetry in a geometric way.Comment: 12 page

    Simple Lie groups without the Approximation Property II

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    We prove that the universal covering group Sp~(2,R)\widetilde{\mathrm{Sp}}(2,\mathbb{R}) of Sp(2,R)\mathrm{Sp}(2,\mathbb{R}) does not have the Approximation Property (AP). Together with the fact that SL(3,R)\mathrm{SL}(3,\mathbb{R}) does not have the AP, which was proved by Lafforgue and de la Salle, and the fact that Sp(2,R)\mathrm{Sp}(2,\mathbb{R}) does not have the AP, which was proved by the authors of this article, this finishes the description of the AP for connected simple Lie groups. Indeed, it follows that a connected simple Lie group has the AP if and only if its real rank is zero or one. By an adaptation of the methods we use to study the AP, we obtain results on approximation properties for noncommutative LpL^p-spaces associated with lattices in Sp~(2,R)\widetilde{\mathrm{Sp}}(2,\mathbb{R}). Combining this with earlier results of Lafforgue and de la Salle and results of the second named author of this article, this gives rise to results on approximation properties of noncommutative LpL^p-spaces associated with lattices in any connected simple Lie group.Comment: Final version. Continuation of the work in 1201.1250 and 1208.593

    Simple Lie groups without the Approximation Property

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    For a locally compact group G, let A(G) denote its Fourier algebra, and let M_0A(G) denote the space of completely bounded Fourier multipliers on G. The group G is said to have the Approximation Property (AP) if the constant function 1 can be approximated by a net in A(G) in the weak-* topology on the space M_0A(G). Recently, Lafforgue and de la Salle proved that SL(3,R) does not have the AP, implying the first example of an exact discrete group without it, namely SL(3,Z). In this paper we prove that Sp(2,R) does not have the AP. It follows that all connected simple Lie groups with finite center and real rank greater than or equal to two do not have the AP. This naturally gives rise to many examples of exact discrete groups without the AP.Comment: Version 4, 29 pages. Minor correction

    Superexpanders from group actions on compact manifolds

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    It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the corresponding warped cone does not coarsely embed into this Banach space. Combining this with non-embeddability results for warped cones by Nowak and Sawicki, which relate the non-embeddability of a warped cone to a spectral gap property of the underlying action, we provide new examples of expanders that do not coarsely embed into any Banach space with nontrivial type. Moreover, we prove that these expanders are not coarsely equivalent to a Lafforgue expander. In particular, we provide infinitely many coarsely distinct superexpanders that are not Lafforgue expanders. In addition, we prove a quasi-isometric rigidity result for warped cones.Comment: 16 pages, to appear in Geometriae Dedicat

    Strong property (T) for higher rank simple Lie groups

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    We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces E10\mathcal{E}_{10} containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that SL(3,R)\mathrm{SL}(3,\mathbb{R}) has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that SL(3,R)\mathrm{SL}(3,\mathbb{R}) has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank groups, it is sufficient to prove strong property (T) for Sp(2,R)\mathrm{Sp}(2,\mathbb{R}) and its universal covering group. As consequences of our main result, it follows that for X∈E10X \in \mathcal{E}_{10}, connected higher rank simple Lie groups and their lattices have property (FX_X) of Bader, Furman, Gelander and Monod, and that the expanders contructed from a lattice in a connected higher rank simple Lie group do not admit a coarse embedding into XX.Comment: 33 pages, 1 figur

    A complete characterization of connected Lie groups with the Approximation Property

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    We give a complete characterization of connected Lie groups with the Approximation Property for groups (AP). To this end, we introduce a strengthening of property (T), that we call property (T*), which is a natural obstruction to the AP. In order to define property (T*), we first prove that for every locally compact group G, there exists a unique left invariant mean on the space of completely bounded Fourier multipliers of G. A locally compact group G is said to have property (T*) if this mean is a weak* continuous functional. After proving that the groups SL(3,R), Sp(2,R), and the universal covering of Sp(2,R) have property (T*), we address the question which connected Lie groups have the AP. A technical problem that arises when considering this question from the point of view of the AP is that the semisimple part of the global Levi decomposition of a connected Lie group need not be closed. Because of an important permanence property of property (T*), this problem vanishes. It follows that a connected Lie group has the AP if and only if all simple factors in the semisimple part of its Levi decomposition have real rank 0 or 1. Finally, we are able to establish property (T*) for all connected simple higher rank Lie groups with finite center.Comment: 18 pages, more details were included in Sections 5 and 6 and some additional minor changes were mad

    On strong property (T) and fixed point properties for Lie groups

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    We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional subspaces is bounded above by a power of k strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank depending on X has strong property (T) of Lafforgue with respect to X. As a consequence, we obtain that every continuous affine isometric action of such a high rank group (or a lattice in such a group) on X has a fixed point. This result corroborates a conjecture of Bader, Furman, Gelander and Monod. For the special linear Lie groups, we also present a more direct approach to fixed point properties, or, more precisely, to the boundedness of quasi-cocycles. Without appealing to strong property (T), we prove that given a Banach space X as above, every special linear group of sufficiently large rank satisfies the following property: every quasi-1-cocycle with values in an isometric representation on X is bounded.Comment: 26 pages. v2: correction in Proposition 2.1 and other small change

    Exotic group C∗C^*-algebras of simple Lie groups with real rank one

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    Exotic group C∗C^*-algebras are C∗C^*-algebras that lie between the universal and the reduced group C∗C^*-algebra of a locally compact group. We consider simple Lie groups GG with real rank one and investigate their exotic group C∗C^{*}-algebras CLp+∗(G)C^*_{L^{p+}}(G), which are defined through LpL^p-integrability properties of matrix coefficients of unitary representations. First, we show that the subset of equivalence classes of irreducible unitary Lp+L^{p+}-representations forms a closed ideal of the unitary dual of these groups. This result holds more generally for groups with the Kunze-Stein property. Second, for every classical simple Lie group GG with real rank one and every 2≤q<p≤∞2 \leq q < p \leq \infty, we determine whether the canonical quotient map CLp+∗(G)↠CLq+∗(G)C^*_{L^{p+}}(G) \twoheadrightarrow C^*_{L^{q+}}(G) has non-trivial kernel. Our results generalize, with different methods, recent results of Samei and Wiersma on exotic group C∗C^*-algebras of SO0(n,1)\mathrm{SO}_{0}(n,1) and SU(n,1)\mathrm{SU}(n,1). In particular, our approach also works for groups with property (T).Comment: 17 pages, minor improvement
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