Piotr Przytycki
Bipolar Coxeter groups.
Tue, 02/23/2010 - 16:01 — SomNambulAuthors: Pierre-Emmanuel Caprace, Piotr Przytycki
Subjects: Group Theory
AbstractWe consider the class of those Coxeter groups for which removing from the
Cayley graph any tubular neighbourhood of any wall leaves exactly two connected
components. We call these Coxeter groups bipolar. They include both the
virtually Poincare duality Coxeter groups and the infinite irreducible
2-spherical ones. We show in a geometric way that a bipolar Coxeter group
admits a unique conjugacy class of Coxeter generating sets. Moreover, we
provide a characterisation of bipolar Coxeter groups in terms of the associated
Coxeter diagram.Twist-rigid Coxeter groups.
Tue, 11/03/2009 - 16:01 — SomNambulAuthors: Pierre-Emmanuel Caprace, Piotr Przytycki
Subjects: Group Theory
AbstractWe prove that two angle-compatible Coxeter generating sets of a given
finitely generated Coxeter group are conjugate provided one of them does not
admit any elementary twist. This confirms a basic case of a general conjecture
which describes a potential solution to the isomorphism problem for Coxeter
groups.Acute triangulations of polyhedra and R^n.
Tue, 09/22/2009 - 14:09 — SomNambulAuthors: Eryk Kopczyński, Igor Pak, Piotr Przytycki
Subjects: Metric Geometry
AbstractWe study the problem of acute triangulations of convex polyhedra and the
space R^n. Here an acute triangulation is a triangulation into simplices whose
dihedral angles are acute. We prove that acute triangulations of the n-cube do
not exist for n>=4. Further, we prove that acute triangulations of the space
R^n do not exist for n>= 5. In the opposite direction, in R^3, we present a
construction of an acute triangulation of the cube, the regular octahedron and
a non-trivial acute triangulation of the regular tetrahedron.Acute triangulations of polyhedra and R^n.
Tue, 09/22/2009 - 14:09 — SomNambulAuthors: Eryk Kopczyński, Igor Pak, Piotr Przytycki
Subjects: Metric Geometry
AbstractWe study the problem of acute triangulations of convex polyhedra and the
space R^n. Here an acute triangulation is a triangulation into simplices whose
dihedral angles are acute. We prove that acute triangulations of the n-cube do
not exist for n>=4. Further, we prove that acute triangulations of the space
R^n do not exist for n>= 5. In the opposite direction, in R^3, we present a
construction of an acute triangulation of the cube, the regular octahedron and
a non-trivial acute triangulation of the regular tetrahedron.
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