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16261 results in Geometry and topology

Page 170 of 814

  • COM volume 154 Issue 3 Cover and Back matter

  • Journal:
    Compositio Mathematica / Volume 154 / Issue 3 / March 2018
    Published online by Cambridge University Press:
    12 March 2018, pp. b1-b2
    Print publication:
    March 2018
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  • Virtual 1-domination of 3-manifolds

  • Part of
  • Yi Liu, Hongbin Sun
  • Journal:
    Compositio Mathematica / Volume 154 / Issue 3 / March 2018
    Published online by Cambridge University Press:
    27 February 2018, pp. 621-639
    Print publication:
    March 2018
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  • It is shown in this paper that given any closed oriented hyperbolic 3-manifold, every closed oriented 3-manifold is mapped onto by a finite cover of that manifold via a map of degree 1, or in other words, virtually 1-dominated by that manifold. This improves a known result of virtual 2-domination. The proof invokes a recently developed enhanced version of the connection principle in good pants constructions.

  • AN EQUIDISTRIBUTION THEOREM FOR HOLOMORPHIC SIEGEL MODULAR FORMS FOR $\mathit{GSp}_{4}$ AND ITS APPLICATIONS

  • Part of
  • Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 2 / March 2020
    Published online by Cambridge University Press:
    22 February 2018, pp. 351-419
    Print publication:
    March 2020

  • We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $\mathit{GSp}_{4}/\mathbb{Q}$ in various aspects. A main tool is Arthur’s invariant trace formula. While Shin [Automorphic Plancherel density theorem, Israel J. Math.192(1) (2012), 83–120] and Shin–Templier [Sato–Tate theorem for families and low-lying zeros of automorphic $L$-functions, Invent. Math.203(1) (2016) 1–177] used Euler–Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms $A,B_{1}$ in Theorem 1.1 which have not been studied and a mysterious second term $B_{2}$ also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato–Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor $L$-functions and degree 5 standard $L$-functions of holomorphic Siegel cusp forms.

  • Generating functions on covering groups

  • Part of
  • David Ginzburg
  • Journal:
    Compositio Mathematica / Volume 154 / Issue 4 / April 2018
    Published online by Cambridge University Press:
    20 February 2018, pp. 671-684
    Print publication:
    April 2018
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  • In this paper we prove a conjecture relating the Whittaker function of a certain generating function with the Whittaker function of the theta representation $\unicode[STIX]{x1D6E9}_{n}^{(n)}$. This enables us to establish that a certain global integral is factorizable and hence deduce the meromorphic continuation of the standard partial $L$ function $L^{S}(s,\unicode[STIX]{x1D70B}^{(n)})$. In fact we prove that this partial $L$ function has at most a simple pole at $s=1$. Here, $\unicode[STIX]{x1D70B}^{(n)}$ is a genuine irreducible cuspidal representation of the group $\text{GL}_{r}^{(n)}(\mathbf{A})$.

  • Nearby Lagrangian fibers and Whitney sphere links

  • Part of
  • Tobias Ekholm, Ivan Smith
  • Journal:
    Compositio Mathematica / Volume 154 / Issue 4 / April 2018
    Published online by Cambridge University Press:
    20 February 2018, pp. 685-718
    Print publication:
    April 2018
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  • Let $n>3$, and let $L$ be a Lagrangian embedding of $\mathbb{R}^{n}$ into the cotangent bundle $T^{\ast }\mathbb{R}^{n}$ of $\mathbb{R}^{n}$ that agrees with the cotangent fiber $T_{x}^{\ast }\mathbb{R}^{n}$ over a point $x\neq 0$ outside a compact set. Assume that $L$ is disjoint from the cotangent fiber at the origin. The projection of $L$ to the base extends to a map of the $n$-sphere $S^{n}$ into $\mathbb{R}^{n}\setminus \{0\}$. We show that this map is homotopically trivial, answering a question of Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in $T^{\ast }\mathbb{R}^{n}$, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1–59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195–240] with symplectic field theory.

  • EQUIVALENT NORMS WITH AN EXTREMELY NONLINEABLE SET OF NORM ATTAINING FUNCTIONALS

  • Part of
  • Vladimir Kadets, Ginés López, Miguel Martín, Dirk Werner
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 1 / January 2020
    Published online by Cambridge University Press:
    13 February 2018, pp. 259-279
    Print publication:
    January 2020

  • We present a construction that enables one to find Banach spaces $X$ whose sets $\operatorname{NA}(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2, Israel J. Math. (to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces, J. Funct. Anal. 272 (2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space $X$ where the set $\operatorname{NA}(X)$ for the original norm is not “too large”. The construction can be applied to every Banach space containing $c_{0}$ and having a countable system of norming functionals, in particular, to separable Banach spaces containing $c_{0}$. We also provide some geometric properties of the norms we have constructed.

  • On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

  • CHRISTIAN BONATTI, JINHUA ZHANG
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 39 / Issue 11 / November 2019
    Published online by Cambridge University Press:
    12 February 2018, pp. 2932-2967
    Print publication:
    November 2019

  • Gorodetski et al. [Nonremovability of zero Lyapunov exponents. Funktsional. Anal. i Prilozhen.39(1) (2005), 27–38 (in Russian); Engl. Transl. Funct. Anal. Appl.39(1) (2005), 21–30] and Bochi et al. [Robust criterion for the existence of nonhyperbolic ergodic measures. Comm. Math. Phys.344(3) (2016), 751–795] propose two very different ways for building non-hyperbolic measures, Gorodetski et al. (2005) building such a measure as the limit of periodic measures and Bochi et al. (2016) as the $\unicode[STIX]{x1D714}$-limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in Gorodetski et al. (2005) has been used in a generic setting in Bonatti et al. [Non-hyperbolic ergodic measures with large support. Nonlinearity23(3) (2010), 687–705] and Díaz and Gorodetski [Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes. Ergod. Th. & Dynam. Sys.29(5) (2009), 1479–1513], as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in Bochi et al. (2016) are accumulated by periodic measures. In this paper we use a shadowing lemma from Gan [A generalized shadowing lemma. Discrete Contin. Dyn. Syst.8(3) (2002), 527–632]:

    • ∙ for getting the periodic orbits in Gorodetski et al. (2005) without perturbing the dynamics;

    • ∙ for recovering the compact set in Bochi et al. (2016) with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits.

    As a consequence, we prove that there exists an open and dense subset ${\mathcal{U}}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in {\mathcal{U}}$, there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset ${\mathcal{V}}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in {\mathcal{V}}$, there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures.

  • Foreword

  • Edited by Pierre-Emmanuel Caprace, Nicolas Monod, École Polytechnique Fédérale de Lausanne
  • Book:
    New Directions in Locally Compact Groups
    Published online:
    05 February 2018
    Print publication:
    08 February 2018, pp vii-viii

  • Summary

    Totally disconnected locally compact (tdlc) groups are of interest for two reasons: on one hand, important classes of tdlc groups arise in combinatorial geometry, number theory and algebra and, on the other, an essential part of the task of describing the structure of general locally compact groups is understanding the totally disconnected case. Interest in these groups is currently very high because of the rapid progress being made with the general theory.

    Advances in our understanding of the structure of tdlc groups are being made through three loosely related approaches:

  • • the scale, a positive integer-valued function defined on automorphisms of tdlc groups that relates to eigenvalues in algebraic representations of these groups and to translation distance in geometric representations;

  • • the structure lattice of locally normal subgroups of a tdlc group, which gives rise to a local theory underpinning a typology of simple tdlc groups; and

  • • a decomposition theory for tdlc groups that exploits methods for gauging their size and breaks a given group into smaller, and often simple, pieces.

    • These approaches are developing a conceptual framework that promises to support a comprehensive description of tdlc groups. At the same time, more examples filling out this framework are being found. There is still some way to go before this description could be regarded as complete however.

    The current interest perhaps prompted the Oberwolfach Forschungsinstitut to ask P.-E. Caprace and N. Monod to organise the Arbeitsgemeinschaft Totally Disconnected Groups in October, 2014, with the aim of bringing together these approaches and the researchers and students involved. Lectures surveyed the background for the study of tdlc groups and introduced the main ideas and most recent developments in the three approaches described above. These notes, which cover the lectures as well as including a couple of other invited surveys, thus provide a valuable review of the current state of knowledge. It is to be hoped that they will serve as a reference for further work that goes toward completing the description of totally disconnected, locally compact groups.

Page 170 of 814