8,202 research outputs found
Stability inequalities and universal Schubert calculus of rank 2
The goal of the paper is to introduce a version of Schubert calculus for each
dihedral reflection group W. That is, to each "sufficiently rich'' spherical
building Y of type W we associate a certain cohomology theory and verify that,
first, it depends only on W (i.e., all such buildings are "homotopy
equivalent'') and second, the cohomology ring is the associated graded of the
coinvariant algebra of W under certain filtration. We also construct the dual
homology "pre-ring'' of Y. The convex "stability'' cones defined via these
(co)homology theories of Y are then shown to solve the problem of classifying
weighted semistable m-tuples on Y in the sense of Kapovich, Leeb and Millson
equivalently, they are cut out by the generalized triangle inequalities for
thick Euclidean buildings with the Tits boundary Y. Quite remarkably, the
cohomology ring is obtained from a certain universal algebra A by a kind of
"crystal limit'' that has been previously introduced by Belkale-Kumar for the
cohomology of flag varieties and Grassmannians. Another degeneration of A leads
to the homology theory of Y.Comment: 55 pages, 1 figur
Shapes of polyhedra and triangulations of the sphere
The space of shapes of a polyhedron with given total angles less than 2\pi at
each of its n vertices has a Kaehler metric, locally isometric to complex
hyperbolic space CH^{n-3}. The metric is not complete: collisions between
vertices take place a finite distance from a nonsingular point. The metric
completion is a complex hyperbolic cone-manifold. In some interesting special
cases, the metric completion is an orbifold. The concrete description of these
spaces of shapes gives information about the combinatorial classification of
triangulations of the sphere with no more than 6 triangles at a vertex.Comment: 39 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper25.abs.htm
Dehn filling in relatively hyperbolic groups
We introduce a number of new tools for the study of relatively hyperbolic
groups. First, given a relatively hyperbolic group G, we construct a nice
combinatorial Gromov hyperbolic model space acted on properly by G, which
reflects the relative hyperbolicity of G in many natural ways. Second, we
construct two useful bicombings on this space. The first of these, "preferred
paths", is combinatorial in nature and allows us to define the second, a
relatively hyperbolic version of a construction of Mineyev.
As an application, we prove a group-theoretic analog of the Gromov-Thurston
2\pi Theorem in the context of relatively hyperbolic groups.Comment: 83 pages. v2: An improved version of preferred paths is given, in
which preferred triangles no longer need feet. v3: Fixed several small errors
pointed out by the referee, and repaired several broken figures. v4:
corrected definition 2.38. This is very close to the published versio
Relation lifting, with an application to the many-valued cover modality
We introduce basic notions and results about relation liftings on categories
enriched in a commutative quantale. We derive two necessary and sufficient
conditions for a 2-functor T to admit a functorial relation lifting: one is the
existence of a distributive law of T over the "powerset monad" on categories,
one is the preservation by T of "exactness" of certain squares. Both
characterisations are generalisations of the "classical" results known for set
functors: the first characterisation generalises the existence of a
distributive law over the genuine powerset monad, the second generalises
preservation of weak pullbacks. The results presented in this paper enable us
to compute predicate liftings of endofunctors of, for example, generalised
(ultra)metric spaces. We illustrate this by studying the coalgebraic cover
modality in this setting.Comment: 48 pages, accepted for publication in LMC
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