802 research outputs found
Braids, posets and orthoschemes
In this article we study the curvature properties of the order complex of a
graded poset under a metric that we call the ``orthoscheme metric''. In
addition to other results, we characterize which rank 4 posets have CAT(0)
orthoscheme complexes and by applying this theorem to standard posets and
complexes associated with four-generator Artin groups, we are able to show that
the 5-string braid group is the fundamental group of a compact nonpositively
curved space.Comment: 33 pages, 16 figure
Polynomial-Sized Topological Approximations Using The Permutahedron
Classical methods to model topological properties of point clouds, such as
the Vietoris-Rips complex, suffer from the combinatorial explosion of complex
sizes. We propose a novel technique to approximate a multi-scale filtration of
the Rips complex with improved bounds for size: precisely, for points in
, we obtain a -approximation with at most simplices of dimension or lower. In conjunction with dimension
reduction techniques, our approach yields a -approximation of size for Rips filtrations on arbitrary metric
spaces. This result stems from high-dimensional lattice geometry and exploits
properties of the permutahedral lattice, a well-studied structure in discrete
geometry.
Building on the same geometric concept, we also present a lower bound result
on the size of an approximate filtration: we construct a point set for which
every -approximation of the \v{C}ech filtration has to contain
features, provided that for .Comment: 24 pages, 1 figur
Geometrically constructed bases for homology of partition lattices of types A, B and D
We use the theory of hyperplane arrangements to construct natural bases for
the homology of partition lattices of types A, B and D. This extends and
explains the "splitting basis" for the homology of the partition lattice given
in [Wa96], thus answering a question asked by R. Stanley. More explicitly, the
following general technique is presented and utilized. Let A be a central and
essential hyperplane arrangement in R^d. Let R_1,...,R_k be the bounded regions
of a generic hyperplane section of A. We show that there are induced polytopal
cycles \rho_{R_i} in the homology of the proper part \bar{L_A} of the
intersection lattice such that {\rho_{R_i}}_{i=1,...,k} is a basis for \tilde
H_{d-2}(\bar{L_A}). This geometric method for constructing combinatorial
homology bases is applied to the Coxeter arrangements of types A, B and D, and
to some interpolating arrangements.Comment: 29 pages, 4 figure
Abels's groups revisited
We generalize a class of groups introduced by Herbert Abels to produce
examples of virtually torsion free groups that have Bredon-finiteness length
m-1 and classical finiteness length n-1 for all 0 < m <= n.
The proof illustrates how Bredon-finiteness properties can be verified using
geometric methods and a version of Brown's criterion due to Martin Fluch and
the author.Comment: 17 pages, 2 figures, v2 more detaile
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