Journal: Geometriae Dedicata

Abbreviation

Geom Dedicata

Publisher

Springer

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ISSN

0046-5755
1572-9168

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Publications 1 - 10 of 16
  • Lagrangian Rabinowitz Floer homology and twisted cotangent bundles
    Item type: Journal Article
    Merry, Will J. (2014)
    Geometriae Dedicata
    We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser (Acta. Math. 141(1–2):17–34, 1978). Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the ‘virtually contact’ setting. By means of an Abbondandolo–Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.
  • Convex geodesic bicombings and hyperbolicity
    Item type: Journal Article
    Descombes, Dominic; Lang, Urs (2015)
    Geometriae Dedicata
    A geodesic bicombing on a metric space selects for every pair of points a geodesic connecting them. We prove existence and uniqueness results for geodesic bicombings satisfying different convexity conditions. In combination with recent work by the second author on injective hulls, this shows that every word hyperbolic group acts geometrically on a proper, finite dimensional space 𝑋 with a unique (hence equivariant) convex geodesic bicombing of the strongest type. Furthermore, the Gromov boundary of 𝑋 is a 𝑍-set in the closure of 𝑋, and the latter is a metrizable absolute retract, in analogy with the Bestvina–Mess theorem on the Rips complex.
  • Some groups having only elementary actions on metric spaces with hyperbolic boundaries
    Item type: Journal Article
    Karlsson, Anders; Noskov, Guennadi A. (2004)
    Geometriae Dedicata
  • h-Principle for curves with prescribed curvature
    Item type: Journal Article
    Wasem, Micha (2016)
    Geometriae Dedicata
  • Ford fundamental domains in symmetric spaces of rank one
    Item type: Journal Article
    Pohl, Anke D. (2010)
    Geometriae Dedicata
    We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and domains are essentially subsumed by our work.
  • Tessellations of Homogeneous Spaces of Classical Groups of Real Rank Two
    Item type: Journal Article
    Iozzi, Alessandra; Morris, Dave Witte (2004)
    Geometriae Dedicata
  • Twisted Weyl groups of Lie groups and nonabelian cohomology
    Item type: Journal Article
    An, Jinpeng (2007)
    Geometriae Dedicata
  • Elementary abelian 2-subgroups of compact Lie groups
    Item type: Journal Article
    Yu, Jun (2013)
    Geometriae Dedicata
  • Weyl-Einstein structures on conformal solvmanifolds
    Item type: Journal Article
    del Barco, Viviana; Moroianu, Andrei; Schichl, Arthur (2023)
    Geometriae Dedicata
    A conformal Lie group is a conformal manifold (M, c) such that M has a Lie group structure and c is the conformal structure defined by a left-invariant metric g on M. We study Weyl-Einstein structures on conformal solvable Lie groups and on their compact quotients. In the compact case, we show that every conformal solvmanifold carrying a Weyl-Einstein structure is Einstein. We also show that there are no left-invariant Weyl-Einstein structures on non-abelian nilpotent conformal Lie groups, and classify them on conformal solvable Lie groups in the almost abelian case. Furthermore, we determine the precise list (up to automorphisms) of left-invariant metrics on simply connected solvable Lie groups of dimension 3 carrying left-invariant Weyl-Einstein structures.
  • Cutoff on hyperbolic surfaces
    Item type: Journal Article
    Golubev, Konstantin; Kamber, Amitay (2019)
    Geometriae Dedicata
    In this paper, we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least , then the distances on the surface are highly concentrated around the minimal possible value of the diameter, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres (Geom Funct Anal 26(4):1190–1216, 2016. https://doi.org/10.1007/s00039-016-0382-7) from the setting of graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from Sarnak and Xue (Duke Math J 64(1):207–227, 1991), we are able to show that the results apply to congruence subgroups of SL₂ (Z) and other arithmetic lattices, without relying on the well-known conjecture of Selberg (Proc Symp Pure Math 8:1–15, 1965), thus relaxing the condition on the Laplace spectrum of a surface. Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis (Proc Natl Acad Sci 93(4):1659–1664, 1996), who asked under what general phenomena cutoff exists.
Publications 1 - 10 of 16