740 research outputs found
Problems on Polytopes, Their Groups, and Realizations
The paper gives a collection of open problems on abstract polytopes that were
either presented at the Polytopes Day in Calgary or motivated by discussions at
the preceding Workshop on Convex and Abstract Polytopes at the Banff
International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete
Geometry, to appear
Constructing highly regular expanders from hyperbolic Coxeter groups
A graph is defined inductively to be -regular if
is -regular and for every vertex of , the sphere of radius
around is an -regular graph. Such a graph is said
to be highly regular (HR) of level if . Chapman, Linial and
Peled studied HR-graphs of level 2 and provided several methods to construct
families of graphs which are expanders "globally and locally". They ask whether
such HR-graphs of level 3 exist.
In this paper we show how the theory of Coxeter groups, and abstract regular
polytopes and their generalisations, can lead to such graphs. Given a Coxeter
system and a subset of , we construct highly regular quotients
of the 1-skeleton of the associated Wythoffian polytope ,
which form an infinite family of expander graphs when is indefinite and
has finite vertex links. The regularity of the graphs in
this family can be deduced from the Coxeter diagram of . The expansion
stems from applying superapproximation to the congruence subgroups of the
linear group .
This machinery gives a rich collection of families of HR-graphs, with various
interesting properties, and in particular answers affirmatively the question
asked by Chapman, Linial and Peled.Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest
Vinber
Convex Polytopes and Quasilattices from the Symplectic Viewpoint
We construct, for each convex polytope, possibly nonrational and nonsimple, a
family of compact spaces that are stratified by quasifolds, i.e. each of these
spaces is a collection of quasifolds glued together in an suitable way. A
quasifold is a space locally modelled on modulo the action of a
discrete, possibly infinite, group. The way strata are glued to each other also
involves the action of an (infinite) discrete group. Each stratified space is
endowed with a symplectic structure and a moment mapping having the property
that its image gives the original polytope back. These spaces may be viewed as
a natural generalization of symplectic toric varieties to the nonrational
setting.Comment: LaTeX, 29 pages. Revised version: TITLE changed, reorganization of
notations and exposition, added remarks and reference
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