234 research outputs found
Realizations of self branched coverings of the 2-sphere
For a degree d self branched covering of the 2-sphere, a notable
combinatorial invariant is an integer partition of 2d -- 2, consisting of the
multiplicities of the critical points. A finer invariant is the so called
Hurwitz passport. The realization problem of Hurwitz passports remain largely
open till today. In this article, we introduce two different types of finer
invariants: a bipartite map and an incident matrix. We then settle completely
their realization problem by showing that a map, or a matrix, is realized by a
branched covering if and only if it satisfies a certain balanced condition. A
variant of the bipartite map approach was initiated by W. Thurston. Our results
shed some new lights to the Hurwitz passport problem
On balanced planar graphs, following W. Thurston
Let be an orientation-preserving branched covering map of
degree , and let be an oriented Jordan curve passing through
the critical values of . Then is an oriented graph
on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced
balanced planar graphs and showed that they combinatorially characterize all
such , where has distinct critical values. We give a
detailed account of this discussion, along with some examples and an appendix
about Hurwitz numbers.Comment: 17 page
Discrete approaches to quantum gravity in four dimensions
The construction of a consistent theory of quantum gravity is a problem in
theoretical physics that has so far defied all attempts at resolution. One
ansatz to try to obtain a non-trivial quantum theory proceeds via a
discretization of space-time and the Einstein action. I review here three major
areas of research: gauge-theoretic approaches, both in a path-integral and a
Hamiltonian formulation, quantum Regge calculus, and the method of dynamical
triangulations, confining attention to work that is strictly four-dimensional,
strictly discrete, and strictly quantum in nature.Comment: 33 pages, invited contribution to Living Reviews in Relativity; the
author welcomes any comments and suggestion
Entropy of random coverings and 4D quantum gravity
We discuss the counting of minimal geodesic ball coverings of -dimensional
riemannian manifolds of bounded geometry, fixed Euler characteristic and
Reidemeister torsion in a given representation of the fundamental group. This
counting bears relevance to the analysis of the continuum limit of discrete
models of quantum gravity. We establish the conditions under which the number
of coverings grows exponentially with the volume, thus allowing for the search
of a continuum limit of the corresponding discretized models. The resulting
entropy estimates depend on representations of the fundamental group of the
manifold through the corresponding Reidemeister torsion. We discuss the sum
over inequivalent representations both in the two-dimensional and in the
four-dimensional case. Explicit entropy functions as well as significant bounds
on the associated critical exponents are obtained in both cases.Comment: 54 pages, latex, no figure
Combinatorial and Algebraic Enumeration: a survey of the work of Ian P. Goulden and David M. Jackson
Finite covers of random 3-manifolds
A 3-manifold is Haken if it contains a topologically essential surface. The
Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite
fundamental group has a finite cover which is Haken. In this paper, we study
random 3-manifolds and their finite covers in an attempt to shed light on this
difficult question. In particular, we consider random Heegaard splittings by
gluing two handlebodies by the result of a random walk in the mapping class
group of a surface. For this model of random 3-manifold, we are able to compute
the probabilities that the resulting manifolds have finite covers of particular
kinds. Our results contrast with the analogous probabilities for groups coming
from random balanced presentations, giving quantitative theorems to the effect
that 3-manifold groups have many more finite quotients than random groups. The
next natural question is whether these covers have positive betti number. For
abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show
that the probability of positive betti number is 0.
In fact, many of these questions boil down to questions about the mapping
class group. We are lead to consider the action of mapping class group of a
surface S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show
that if the genus of S is large, then this action is very mixing. In
particular, the action factors through the alternating group of each orbit.
This is analogous to Goldman's theorem that the action of the mapping class
group on the SU(2) character variety is ergodic.Comment: 60 pages; v2: minor changes. v3: minor changes; final versio
Unified bijections for planar hypermaps with general cycle-length constraints
We present a general bijective approach to planar hypermaps with two main
results. First we obtain unified bijections for all classes of maps or
hypermaps defined by face-degree constraints and girth constraints. To any such
class we associate bijectively a class of plane trees characterized by local
constraints. This unifies and greatly generalizes several bijections for maps
and hypermaps. Second, we present yet another level of generalization of the
bijective approach by considering classes of maps with non-uniform girth
constraints. More precisely, we consider "well-charged maps", which are maps
with an assignment of "charges" (real numbers) on vertices and faces, with the
constraints that the length of any cycle of the map is at least equal to the
sum of the charges of the vertices and faces enclosed by the cycle. We obtain a
bijection between charged hypermaps and a class of plane trees characterized by
local constraints
Generic method for bijections between blossoming trees and planar maps
This article presents a unified bijective scheme between planar maps and
blossoming trees, where a blossoming tree is defined as a spanning tree of the
map decorated with some dangling half-edges that enable to reconstruct its
faces. Our method generalizes a previous construction of Bernardi by loosening
its conditions of applications so as to include annular maps, that is maps
embedded in the plane with a root face different from the outer face.
The bijective construction presented here relies deeply on the theory of
\alpha-orientations introduced by Felsner, and in particular on the existence
of minimal and accessible orientations. Since most of the families of maps can
be characterized by such orientations, our generic bijective method is proved
to capture as special cases all previously known bijections involving
blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable
maps and simple triangulations and quadrangulations of a k-gon. Moreover, it
also permits to obtain new bijective constructions for bipolar orientations and
d-angulations of girth d of a k-gon.
As for applications, each specialization of the construction translates into
enumerative by-products, either via a closed formula or via a recursive
computational scheme. Besides, for every family of maps described in the paper,
the construction can be implemented in linear time. It yields thus an effective
way to encode and generate planar maps.
In a recent work, Bernardi and Fusy introduced another unified bijective
scheme, we adopt here a different strategy which allows us to capture different
bijections. These two approaches should be seen as two complementary ways of
unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom
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