14 research outputs found
Graph curves
AbstractWe study a family of stable curves defined combinatorially from a trivalent graph. Most of our results are related to the conjecture of Green which relates the Clifford index of a smooth curve, an important intrinsic invariant measuring the “specialness” of the geometry of the curve, to the “resolution Clifford index,” a projective invariant defined from the canonical embedding. Thus we study the canonical linear series and its powers and identify them in terms of combinatorial data on the graph; we given combinatorial criteria for the canonical series to be base point free or very ample; we prove the analogue of Noether's theorem on the projective normality of smooth canonical curves; we define a combinatorial invariant of a graph which we conjecture to be equal to the resolution Clifford index of the associated graph curve, at least for “most” graphs; and we prove our conjecture for planar graphs and for graphs of Clifford index 0. Along the way we prove a result of some independent interest on the canonical sheaves of (not necessarily arithmetically Cohen-Macaulay) face varieties. The Appendix establishes a formula connecting the combinatorics of a trivalent graph G and the minimal degree of an admissible covering of a curve of arithmetic genus 0 by the corresponding graph curve
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
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