1,053 research outputs found
A locally minimal, but not globally minimal bridge position of a knot
We give a locally minimal, but not globally minimal bridge position of a
knot, that is, an unstabilized, nonminimal bridge position of a knot. It
implies that a bridge position cannot always be simplified so that the bridge
number monotonically decreases to the minimal.Comment: 27 pages, 12 figures, v3: minor corrections throughout the pape
Cannon-Thurston Maps for Kleinian Groups
We show that Cannon-Thurston maps exist for degenerate free groups without
parabolics, i.e. for handlebody groups. Combining these techniques with earlier
work proving the existence of Cannon-Thurston maps for surface groups, we show
that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian
groups without parabolics, proving conjectures of Thurston and McMullen. We
also show that point pre-images under Cannon-Thurston maps for degenerate free
groups without parabolics correspond to end-points of leaves of an ending
lamination in the Masur domain, whenever a point has more than one pre-image.
This proves a conjecture of Otal. We also prove a similar result for point
pre-images under Cannon-Thurston maps for arbitrary finitely generated Kleinian
groups without parabolics.Comment: 39 pgs 1 fig. Final version incorporating referee comments. To appear
in Forum of Mathematics, P
Non-minimal bridge positions of torus knots are stabilized
We show that any non-minimal bridge decomposition of a torus knot is
stabilized and that -bridge decompositions of a torus knot are unique for
any integer . This implies that a knot in a bridge position is a torus knot
if and only if there exists a torus containing the knot such that it intersects
the bridge sphere in two essential loops.Comment: 11 pages, 4 figure
Chimneys, leopard spots, and the identities of Basmajian and Bridgeman
We give a simple geometric argument to derive in a common manner
orthospectrum identities of Basmajian and Bridgeman. Our method also
considerably simplifies the determination of the summands in these identities.
For example, for every odd integer n, there is a rational function q_n of
degree 2(n-2) so that if M is a compact hyperbolic manifold of dimension n with
totally geodesic boundary S, there is an identity \chi(S) = \sum_i q_n(e^{l_i})
where the sum is taken over the orthospectrum of M. When n=3, this has the
explicit form \sum_i 1/(e^{2l_i}-1) = -\chi(S)/4.Comment: 6 pages; version 2 incorporates referee's comment
Mutation and SL(2,C)-Reidemeister torsion for hyperbolic knots
Given a hyperbolic knot, we prove that the Reidemeister torsion of any lift
of the holonomy to SL(2,C) is invariant under mutation along a Conway sphere.Comment: 19 pages, 1 figur
Invariant solutions to the Strominger system and the heterotic equations of motion
We construct many new invariant solutions to the Strominger system with
respect to a 2-parameter family of metric connections
in the anomaly cancellation equation. The ansatz
is a natural extension of the canonical 1-parameter
family of Hermitian connections found by Gauduchon, as one recovers the Chern
connection for , and the Bismut
connection for . In particular,
explicit invariant solutions to the Strominger system with respect to the Chern
connection, with non-flat instanton and positive are obtained.
Furthermore, we give invariant solutions to the heterotic equations of motion
with respect to the Bismut connection. Our solutions live on three different
compact non-K\"ahler homogeneous spaces, obtained as the quotient by a lattice
of maximal rank of a nilpotent Lie group, the semisimple group
SL(2,) and a solvable Lie group. To our knowledge, these are the
only known invariant solutions to the heterotic equations of motion, and we
conjecture that there is no other such homogeneous space admitting an invariant
solution to the heterotic equations of motion with respect to a connection in
the ansatz .Comment: 27 pages, 3 figure
Six dimensional solvmanifolds with holomorphically trivial canonical bundle
We determine the 6-dimensional solvmanifolds admitting an invariant complex
structure with holomorphically trivial canonical bundle. Such complex
structures are classified up to isomorphism, and the existence of strong
K\"ahler with torsion (SKT), generalized Gauduchon, balanced and strongly
Gauduchon metrics is studied. As an application we construct a holomorphic
family of compact complex manifolds such that satisfies the
-Lemma and admits a balanced metric for any ,
but the central limit neither satisfies the -Lemma nor
admits balanced metrics.Comment: 32 pages; to appear in IMR
On the influence of transitively normal subgroups on the structure of some infinite groups
A transitively normal subgroup of a group G is one that is normal in every subgroup in which it is subnormal. This concept is obviously related to the transitivity of normality because the latter holds in every subgroup of a group G if and only if every subgroup of G is transitively normal. In this paper we describe the structure of a group whose cyclic subgroups (or a part of them) are transitively normal
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