187,909 research outputs found
Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids
We continue our investigations of the generalised braid groups appearing in
gauge theory, as fundamental groups of spaces of admissible deformation
parameters ("times") for the irregular isomonodromy connections. Here we study
the local wild mapping class groups in the twisted setting for arbitrary formal
structure in type . General configuration spaces will be defined and shown
to admit product decompositions, via a suitable construction of fission trees.
Moreover the fission trees will be shown to parameterise admissible deformation
classes and used to visualise the configuration spaces. Simple examples give
the braid groups of the complex reflection groups known as the generalised
symmetric groups, thereby showing how they arise naturally in gauge theory
(i.e. the theory of meromorphic connections on vector bundles on curves). This
enables us to write down the dimensions of the (global) moduli spaces of rank
, trace-free wild Riemann surfaces for any , a generalisation of
"Riemann's count".Comment: 32 pages, Comments welcome, v2: minor change
Exponential formulas for models of complex reflection groups
In this paper we find some exponential formulas for the Betti numbers of the
De Concini-Procesi minimal wonderful models Y_{G(r,p,n)} associated to the
complex reflection groups G(r,p,n). Our formulas are different from the ones
already known in the literature: they are obtained by a new combinatorial
encoding of the elements of a basis of the cohomology by means of set
partitions with weights and exponents.
We also point out that a similar combinatorial encoding can be used to
describe the faces of the real spherical wonderful models of type
A_{n-1}=G(1,1,n), B_n=G(2,1,n) and D_n=G(2,2,n). This provides exponential
formulas for the f-vectors of the associated nestohedra: the Stasheff's
associahedra (in this case closed formulas are well known) and the graph
associahedra of type D_n.Comment: with respect to v.1: misprint corrected in Example 3.
Affine actions on non-archimedean trees
We initiate the study of affine actions of groups on -trees for a
general ordered abelian group ; these are actions by dilations rather
than isometries. This gives a common generalisation of isometric action on a
-tree, and affine action on an -tree as studied by I. Liousse. The
duality between based length functions and actions on -trees is
generalised to this setting. We are led to consider a new class of groups:
those that admit a free affine action on a -tree for some .
Examples of such groups are presented, including soluble Baumslag-Solitar
groups and the discrete Heisenberg group.Comment: 27 pages. Section 1.4 expanded, typos corrected from previous versio
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
Symmetric isostatic frameworks with or distance constraints
Combinatorial characterisations of minimal rigidity are obtained for
symmetric 2-dimensional bar-joint frameworks with either or
distance constraints. The characterisations are expressed in
terms of symmetric tree packings and the number of edges fixed by the symmetry
operations. The proof uses new Henneberg-type inductive construction schemes.Comment: 20 pages. Main theorem extended. Construction schemes refined. New
titl
- …