48 research outputs found
Pseudo-Riemannian geodesic foliations by circles
We investigate under which assumptions an orientable pseudo-Riemannian
geodesic foliations by circles is generated by an -action. We construct
examples showing that, contrary to the Riemannian case, it is not always true.
However, we prove that such an action always exists when the foliation does not
contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley's Theorem. As an
application, we show that every Lorentzian surface all of whose
spacelike/timelike geodesics are closed, is finitely covered by .
It follows that every Lorentzian surface contains a non-closed geodesic.Comment: 14 page
Aperiodic invariant continua for surface homeomorphisms
We prove that if a homeomorphism of a closed orientable surface S has no
wandering points and leaves invariant a compact, connected set K which contains
no periodic points, then either K=S and S is a torus, or is the
intersection of a decreasing sequence of annuli. A version for non-orientable
surfaces is given.Comment: 8 pages, to appear in Mathematische Zeitschrif
Automatic structures, rational growth and geometrically finite hyperbolic groups
We show that the set of equivalence classes of synchronously
automatic structures on a geometrically finite hyperbolic group is dense in
the product of the sets over all maximal parabolic subgroups . The
set of equivalence classes of biautomatic structures on is
isomorphic to the product of the sets over the cusps (conjugacy
classes of maximal parabolic subgroups) of . Each maximal parabolic is a
virtually abelian group, so and were computed in ``Equivalent
automatic structures and their boundaries'' by M.Shapiro and W.Neumann, Intern.
J. of Alg. Comp. 2 (1992) We show that any geometrically finite hyperbolic
group has a generating set for which the full language of geodesics for is
regular. Moreover, the growth function of with respect to this generating
set is rational. We also determine which automatic structures on such a group
are equivalent to geodesic ones. Not all are, though all biautomatic structures
are.Comment: Plain Tex, 26 pages, no figure
Measurable versions of the LS category on laminations
We give two new versions of the LS category for the set-up of measurable
laminations defined by Berm\'udez. Both of these versions must be considered as
"tangential categories". The first one, simply called (LS) category, is the
direct analogue for measurable laminations of the tangential category of
(topological) laminations introduced by Colman Vale and Mac\'ias Virg\'os. For
the measurable lamination that underlies any lamination, our measurable
tangential category is a lower bound of the tangential category. The second
version, called the measured category, depends on the choice of a transverse
invariant measure. We show that both of these "tangential categories" satisfy
appropriate versions of some well known properties of the classical category:
the homotopy invariance, a dimensional upper bound, a cohomological lower bound
(cup length), and an upper bound given by the critical points of a smooth
function.Comment: 22 page
On three-manifolds dominated by circle bundles
We determine which three-manifolds are dominated by products. The result is
that a closed, oriented, connected three-manifold is dominated by a product if
and only if it is finitely covered either by a product or by a connected sum of
copies of the product of the two-sphere and the circle. This characterization
can also be formulated in terms of Thurston geometries, or in terms of purely
algebraic properties of the fundamental group. We also determine which
three-manifolds are dominated by non-trivial circle bundles, and which
three-manifold groups are presentable by products.Comment: 12 pages; to appear in Math. Zeitschrift; ISSN 1103-467
Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group
We study Sobolev-type metrics of fractional order on the group
\Diff_c(M) of compactly supported diffeomorphisms of a manifold . We show
that for the important special case the geodesic distance on
\Diff_c(S^1) vanishes if and only if . For other manifolds we
obtain a partial characterization: the geodesic distance on \Diff_c(M)
vanishes for and for ,
with being a compact Riemannian manifold. On the other hand the geodesic
distance on \Diff_c(M) is positive for and
.
For we discuss the geodesic equations for these metrics. For
we obtain some well known PDEs of hydrodynamics: Burgers' equation for ,
the modified Constantin-Lax-Majda equation for and the
Camassa-Holm equation for .Comment: 16 pages. Final versio
Fuchsian convex bodies: basics of Brunn--Minkowski theory
The hyperbolic space \H^d can be defined as a pseudo-sphere in the
Minkowski space-time. In this paper, a Fuchsian group is a group of
linear isometries of the Minkowski space such that \H^d/\Gamma is a compact
manifold. We introduce Fuchsian convex bodies, which are closed convex sets in
Minkowski space, globally invariant for the action of a Fuchsian group. A
volume can be associated to each Fuchsian convex body, and, if the group is
fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be
studied in the same manner as convex bodies of Euclidean space in the classical
Brunn--Minkowski theory. For example, support functions can be defined, as
functions on a compact hyperbolic manifold instead of the sphere.
The main result is the convexity of the associated volume (it is log concave
in the classical setting). This implies analogs of Alexandrov--Fenchel and
Brunn--Minkowski inequalities. Here the inequalities are reversed
Reversible maps and composites of involutions in groups of piecewise linear homeomorphisms of the real line
An element of a group is reversible if it is conjugate to its own inverse, and it is strongly reversible if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be expressed as a composite of two involutions. In this paper the reversible maps, the strongly reversible maps, and those maps that can be expressed as a composite of involutions are determined in certain groups of piecewise linear homeomorphisms of the real line
Tight Beltrami fields with symmetry
Let be a compact orientable Seifered fibered 3-manifold without a
boundary, and an -invariant contact form on . In a suitable
adapted Riemannian metric to , we provide a bound for the volume
and the curvature, which implies the universal tightness of the
contact structure .Comment: 26 page