93,493 research outputs found
On conformally flat circle bundles over surfaces
We study surface groups in , which is the group of Mobius
tranformations of , and also the group of isometries of . We
consider such so that its limit set is a quasi-circle
in , and so that the quotient is a
circle bundle over a surface. This circle bundle is said to be conformally
flat, and our main goal is to discover how twisted such bundle may be by
establishing a bound on its Euler number. By combinatorial approaches, we have
two soft bounds in this direction on certain types of nice structures. In this
article we also construct new examples, a "grafting" type path in the space of
surface group representations into : starting inside the
quasi-Fuschsian locus, going through non-discrete territory and back.Comment: 28 pages, 7 figures. Updated from Thesis version: more correct bound
of (3/2)n^2, updated exposition in section 3.
Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups
In this paper, we construct a Lagrangian submanifold of the moduli space
associated to the fundamental group of a punctured Riemann surface (the space
of representations of this fundamental group into a compact connected Lie
group). This Lagrangian submanifold is obtained as the fixed-point set of an
anti-symplectic involution defined on the moduli space. The notion of
decomposable representation provides a geometric interpretation of this
Lagrangian submanifold
Anyons in Geometric Models of Matter
We show that the "geometric models of matter" approach proposed by the first
author can be used to construct models of anyon quasiparticles with fractional
quantum numbers, using 4-dimensional edge-cone orbifold geometries with
orbifold singularities along embedded 2-dimensional surfaces. The anyon states
arise through the braid representation of surface braids wrapped around the
orbifold singularities, coming from multisections of the orbifold normal bundle
of the embedded surface. We show that the resulting braid representations can
give rise to a universal quantum computer.Comment: 22 pages LaTe
Poisson varieties from Riemann surfaces
Short survey based on talk at the Poisson 2012 conference. The main aim is to
describe and give some examples of wild character varieties (naturally
generalising the character varieties of Riemann surfaces by allowing more
complicated behaviour at the boundary), their Poisson/symplectic structures
(generalising both the Atiyah-Bott approach and the quasi-Hamiltonian
approach), and the wild mapping class groups.Comment: 33 pages, 3 figure
Deformations and stability in complex hyperbolic geometry
This paper concerns with deformations of noncompact complex hyperbolic
manifolds (with locally Bergman metric), varieties of discrete representations
of their fundamental groups into and the problem of (quasiconformal)
stability of deformations of such groups and manifolds in the sense of L.Bers
and D.Sullivan.
Despite Goldman-Millson-Yue rigidity results for such complex manifolds of
infinite volume, we present different classes of such manifolds that allow
non-trivial (quasi-Fuchsian) deformations and point out that such flexible
manifolds have a common feature being Stein spaces. While deformations of
complex surfaces from our first class are induced by quasiconformal
homeomorphisms, non-rigid complex surfaces (homotopy equivalent to their
complex analytic submanifolds) from another class are quasiconformally
unstable, but nevertheless their deformations are induced by homeomorphisms
- …