891 research outputs found
Hierarchically hyperbolic spaces I: curve complexes for cubical groups
In the context of CAT(0) cubical groups, we develop an analogue of the theory
of curve complexes and subsurface projections. The role of the subsurfaces is
played by a collection of convex subcomplexes called a \emph{factor system},
and the role of the curve graph is played by the \emph{contact graph}. There
are a number of close parallels between the contact graph and the curve graph,
including hyperbolicity, acylindricity of the action, the existence of
hierarchy paths, and a Masur--Minsky-style distance formula.
We then define a \emph{hierarchically hyperbolic space}; the class of such
spaces includes a wide class of cubical groups (including all virtually compact
special groups) as well as mapping class groups and Teichm\"{u}ller space with
any of the standard metrics. We deduce a number of results about these spaces,
all of which are new for cubical or mapping class groups, and most of which are
new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent
Lie group into a hierarchically hyperbolic space lies close to a product of
hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic
spaces; this generalizes results of Behrstock--Minsky, Eskin--Masur--Rafi,
Hamenst\"{a}dt, and Kleiner. We finally prove that each hierarchically
hyperbolic group admits an acylindrical action on a hyperbolic space. This
acylindricity result is new for cubical groups, in which case the hyperbolic
space admitting the action is the contact graph; in the case of the mapping
class group, this provides a new proof of a theorem of Bowditch.Comment: To appear in "Geometry and Topology". This version incorporates the
referee's comment
Coxeter groups and random groups
For every dimension d, there is an infinite family of convex co-compact
reflection groups of isometries of hyperbolic d-space --- the superideal
(simplicial and cubical) reflection groups --- with the property that a random
group at any density less than a half (or in the few relators model) contains
quasiconvex subgroups commensurable with some member of the family, with
overwhelming probability.Comment: 18 pages, 14 figures; version 2 incorporates referee's correction
Wulff construction in statistical mechanics and in combinatorics
We present the geometric solutions to some variational problems of
statistical mechanics and combinatorics. Together with the Wulff construction,
which predicts the shape of the crystals, we discuss the construction which
exhibits the shape of a typical Young diagram and of a typical skyscraper.Comment: A revie
Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes
This article exhibits a 4-dimensional combinatorial polytope that has no
antiprism, answering a question posed by Bernt Lindst\"om. As a consequence,
any realization of this combinatorial polytope has a face that it cannot rest
upon without toppling over. To this end, we provide a general method for
solving a broad class of realizability problems. Specifically, we show that for
any semialgebraic property that faces inherit, the given property holds for
some realization of every combinatorial polytope if and only if the property
holds from some projective copy of every polytope. The proof uses the following
result by Below. Given any polytope with vertices having algebraic coordinates,
there is a combinatorial "stamp" polytope with a specified face that is
projectively equivalent to the given polytope in all realizations. Here we
construct a new stamp polytope that is closely related to Richter-Gebert's
proof of universality for 4-dimensional polytopes, and we generalize several
tools from that proof
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