3,418 research outputs found
An algebraic model for rational SO(3)-spectra
Greenlees established an equivalence of categories between the homotopy
category of rational SO(3)-spectra and the derived category DA(SO(3)) of a
certain abelian category. In this paper we lift this equivalence of homotopy
categories to the level of Quillen equivalences of model categories. Methods
used in this paper provide the first step towards obtaining an algebraic model
for the toral part of rational G-spectra, for any compact Lie group G
A generalisation of the deformation variety
Given an ideal triangulation of a connected 3-manifold with non-empty
boundary consisting of a disjoint union of tori, a point of the deformation
variety is an assignment of complex numbers to the dihedral angles of the
tetrahedra subject to Thurston's gluing equations. From this, one can recover a
representation of the fundamental group of the manifold into the isometries of
3-dimensional hyperbolic space. However, the deformation variety depends
crucially on the triangulation: there may be entire components of the
representation variety which can be obtained from the deformation variety with
one triangulation but not another. We introduce a generalisation of the
deformation variety, which again consists of assignments of complex variables
to certain dihedral angles subject to polynomial equations, but together with
some extra combinatorial data concerning degenerate tetrahedra. This "extended
deformation variety" deals with many situations that the deformation variety
cannot. In particular we show that for any ideal triangulation of a small
orientable 3-manifold with a single torus boundary component, we can recover
all of the irreducible non-dihedral representations from the associated
extended deformation variety. More generally, we give an algorithm to produce a
triangulation of a given orientable 3-manifold with torus boundary components
for which the same result holds. As an application, we show that this extended
deformation variety detects all factors of the PSL(2,C) A-polynomial associated
to the components consisting of the representations it recovers.Comment: 47 pages, 26 figures. Rewrote introduction and added motivation
section based on referee's comments. Rewrote the section on retriangulation,
and added new result on small manifolds with a single cus
On the quasi-isometric rigidity of graphs of surface groups
Let be a word hyperbolic group with a cyclic JSJ decomposition that
has only rigid vertex groups, which are all fundamental groups of closed
surface groups. We show that any group quasi-isometric to is
abstractly commensurable with .Comment: 54 pages, 10 figures, comments welcom
Group-theoretic models of the inversion process in bacterial genomes
The variation in genome arrangements among bacterial taxa is largely due to
the process of inversion. Recent studies indicate that not all inversions are
equally probable, suggesting, for instance, that shorter inversions are more
frequent than longer, and those that move the terminus of replication are less
probable than those that do not. Current methods for establishing the inversion
distance between two bacterial genomes are unable to incorporate such
information. In this paper we suggest a group-theoretic framework that in
principle can take these constraints into account. In particular, we show that
by lifting the problem from circular permutations to the affine symmetric
group, the inversion distance can be found in polynomial time for a model in
which inversions are restricted to acting on two regions. This requires the
proof of new results in group theory, and suggests a vein of new combinatorial
problems concerning permutation groups on which group theorists will be needed
to collaborate with biologists. We apply the new method to inferring distances
and phylogenies for published Yersinia pestis data.Comment: 19 pages, 7 figures, in Press, Journal of Mathematical Biolog
Symmetric hyperbolic monopoles
Hyperbolic monopole solutions can be obtained from circle-invariant ADHM data
if the curvature of hyperbolic space is suitably tuned. Here we give explicit
ADHM data corresponding to axial hyperbolic monopoles in a simple, tractable
form, as well as expressions for the axial monopole fields. The data is
deformed into new 1-parameter families preserving dihedral and twisted-line
symmetries. In many cases explicit expressions are presented for their spectral
curves and rational maps of both Donaldson and Jarvis type.Comment: 20 pages, 1 figur
Hyperbolic Dehn filling in dimension four
We introduce and study some deformations of complete finite-volume hyperbolic
four-manifolds that may be interpreted as four-dimensional analogues of
Thurston's hyperbolic Dehn filling.
We construct in particular an analytic path of complete, finite-volume cone
four-manifolds that interpolates between two hyperbolic four-manifolds
and with the same volume . The deformation looks
like the familiar hyperbolic Dehn filling paths that occur in dimension three,
where the cone angle of a core simple closed geodesic varies monotonically from
to . Here, the singularity of is an immersed geodesic surface
whose cone angles also vary monotonically from to . When a cone angle
tends to a small core surface (a torus or Klein bottle) is drilled
producing a new cusp.
We show that various instances of hyperbolic Dehn fillings may arise,
including one case where a degeneration occurs when the cone angles tend to
, like in the famous figure-eight knot complement example.
The construction makes an essential use of a family of four-dimensional
deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.Comment: 60 pages, 23 figures. Final versio
- …