2,448 research outputs found
On tube-log Riemann surfaces and primitives of rational functions
For a generic class of rational functions, we give an explicit description of
the flat structure on the Riemann sphere induced by a meromorphic 1-form
R(z)dz, where R is a rational function. The rational functions in the generic
class we consider have only simple poles. We show that the flat structure may
be obtained by pasting isometrically flat half-cylinders to a 'log-polygon',
which is a domain bounded by straight line segments in a simply connected
finite sheeted branched cover of C
Distortion of surfaces in graph manifolds
Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We
show that the fundamental group of the surface S is quadratically distorted
whenever the surface is virtually embedded (i.e., separable) and is
exponentially distorted when the surface is not virtually embedded.Comment: 29 pages, 2 figure
Rigidity of geodesic completeness in the Brinkmann class of gravitational wave spacetimes
We consider restrictions placed by geodesic completeness on spacetimes
possessing a null parallel vector field, the so-called Brinkmann spacetimes.
This class of spacetimes includes important idealized gravitational wave models
in General Relativity, namely the plane-fronted waves with parallel rays, or
pp-waves, which in turn have been intensely and fruitfully studied in the
mathematical and physical literatures for over half a century. More concretely,
we prove a restricted version of a conjectural analogue for Brinkmann
spacetimes of a rigidity result obtained by M.T. Anderson for stationary
spacetimes. We also highlight its relation with a long-standing 1962 conjecture
by Ehlers and Kundt. Indeed, it turns out that the subclass of Brinkmann
spacetimes we consider in our main theorem is enough to settle an important
special case of the Ehlers-Kundt conjecture in terms of the well known class of
Cahen-Wallach spaces.Comment: Second version including new references, some extra motivation on the
Introduction and Propositions 2.2, 2.4 and Remark 2.5. 17 page
Completions of normed algebras of differentiable functions
In this paper we look at normed spaces of differentiable functions on compact
plane sets, including the spaces of infinitely differentiable functions
originally considered by Dales and Davie. For many compact plane sets the
classical definitions give rise to incomplete spaces. We introduce an
alternative definition of differentiability which allows us to describe the
completions of these spaces. We also consider some associated problems of
polynomial and rational approximation.Comment: 18 pages LaTeX, one figur
Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
Let G be a group acting geometrically on a CAT(0) cube complex X. We prove
first that G is hyperbolic relative to the collection P of subgroups if and
only if the simplicial boundary of X is the disjoint union of a nonempty
discrete set, together with a pairwise-disjoint collection of subcomplexes
corresponding, in the appropriate sense, to elements of P. As a special case of
this result is a new proof, in the cubical case, of a Theorem of
Hruska--Kleiner regarding Tits boundaries of relatively hyperbolic CAT(0)
spaces. Second, we relate the existence of cut-points in asymptotic cones of a
cube complex X to boundedness of the 1-skeleton of the boundary of X. We deduce
characterizations of thickness and strong algebraic thickness of a group G
acting properly and cocompactly on the CAT(0) cube complex X in terms of the
structure of, and nature of the G-action on, the boundary of X. Finally, we
construct, for each n,k, infinitely many quasi-isometry types of group G such
that G is strongly algebraically thick of order n, has polynomial divergence of
order n+1, and acts properly and cocompactly on a k-dimensional CAT(0) cube
complex.Comment: Corrections according to referee report. Fixed proof of Theorem 4.3.
To appear in "Groups, Geometry, and Dynamics
Cell decompositions of moduli space, lattice points and Hurwitz problems
In this article we describe cell decompositions of the moduli space of
Riemann surfaces and their relationship to a Hurwitz problem. The cells possess
natural linear structures and with respect to this they can be described as
rational convex polytopes which come equipped with natural integer points and a
volume form. We show how to effectively calculate the number of lattice points
and the volumes over all the cells and that these calculations contain deep
information about the moduli space.Comment: 38 pages, To appear in Handbook of Modul
Divergence in lattices in semisimple Lie groups and graphs of groups
Divergence functions of a metric space estimate the length of a path
connecting two points , at distance avoiding a large enough ball
around a third point . We characterize groups with non-linear divergence
functions as groups having cut-points in their asymptotic cones. By
Olshanskii-Osin-Sapir, that property is weaker than the property of having
Morse (rank 1) quasi-geodesics. Using our characterization of Morse
quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur
that states that mapping class groups cannot contain copies of irreducible
lattices in semi-simple Lie groups of higher ranks. It also gives a
generalization of the result of Birman-Lubotzky-McCarthy about solvable
subgroups of mapping class groups not covered by the Tits alternative of Ivanov
and McCarthy.
We show that any group acting acylindrically on a simplicial tree or a
locally compact hyperbolic graph always has "many" periodic Morse
quasi-geodesics (i.e. Morse elements), so its divergence functions are never
linear. We also show that the same result holds in many cases when the
hyperbolic graph satisfies Bowditch's properties that are weaker than local
compactness. This gives a new proof of Behrstock's result that every
pseudo-Anosov element in a mapping class group is Morse.
On the other hand, we conjecture that lattices in semi-simple Lie groups of
higher rank always have linear divergence. We prove it in the case when the
-rank is 1 and when the lattice is where
, is a finite set of valuations of a number field including all
infinite valuations, and is the corresponding ring of
-integers.Comment: v1: 34 pages; v2: implemented referee's comments/ The paper is
accepted in Tr.AMS v3: more small changes are made, especially in Section 3,
v6: added an erratum correcting Proposition 3.24 and Theorems 4.4, 4.9; v7:
fixed the proof of Theorem 7.
Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori
We consider the relationship between hyperbolic cone-manifold structures on
surfaces, and algebraic representations of the fundamental group into a group
of isometries. A hyperbolic cone-manifold structure on a surface, with all
interior cone angles being integer multiples of , determines a holonomy
representation of the fundamental group. We ask, conversely, when a
representation of the fundamental group is the holonomy of a hyperbolic
cone-manifold structure. In this paper we prove results for the punctured
torus; in the sequel, for higher genus surfaces. We show that a representation
of the fundamental group of a punctured torus is a holonomy representation of a
hyperbolic cone-manifold structure with no interior cone points and a single
corner point if and only if it is not virtually abelian. We construct a
pentagonal fundamental domain for hyperbolic structures, from the geometry of a
representation. Our techniques involve the universal covering group of the
group of orientation-preserving isometries of the hyperbolic plane, and Markoff
moves arising from the action of the mapping class group on the character
variety.Comment: v.2: 41 pages, 42 figures. Improvements in graphics and exposition,
incorporating referee comments. To appear in Geometriae Dedicat
Algebraic curve in the unit ball in C^2 passing through the center, all whose boundary components are arbitrarily short
We prove that curves indicated in the title exist. This results answers to a
question posed by A.G.Vitushkin about 30 years ago. We also discuss the minimal
number of boundary components of a curve in the unit ball passing through the
center, under the condition that all these components are shorter than a given
number. More precisely, we discuss the order of growth of the number of the
components as their maximal length tends to zero
On Geometry of Flat Complete Strictly Causal Lorentzian Manifolds
A flat complete causal Lorentzian manifold is called {\it strictly causal} if
the past and the future of each its point are closed near this point. We
consider strictly causal manifolds with unipotent holonomy groups and assign to
a manifold of this type four nonnegative integers (a signature) and a parabola
in the cone of positive definite matrices. Two manifolds are equivalent if and
only if their signatures coincides and the corresponding parabolas are equal
(up to a suitable automorphism of the cone and an affine change of variable).
Also, we give necessary and sufficient conditions, which distinguish parabolas
of this type among all parabolas in the cone.Comment: The exposition is revised (no essential change in the content). The
paper is publishe
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