266 research outputs found
On hierarchical hyperbolicity of cubical groups
Let X be a proper CAT(0) cube complex admitting a proper cocompact action by
a group G. We give three conditions on the action, any one of which ensures
that X has a factor system in the sense of [BHS14]. We also prove that one of
these conditions is necessary. This combines with results of
Behrstock--Hagen--Sisto to show that is a hierarchically hyperbolic group;
this partially answers questions raised by those authors. Under any of these
conditions, our results also affirm a conjecture of BehrstockHagen on
boundaries of cube complexes, which implies that X cannot contain a convex
staircase. The conditions on the action are all strictly weaker than virtual
cospecialness, and we are not aware of a cocompactly cubulated group that does
not satisfy at least one of the conditions.Comment: Minor changes in response to referee report. Streamlined the proof of
Lemma 5.2, and added an examples of non-rotational action
Throughput constrained parallelism reduction in cyclo-static dataflow applications
International audienceThis paper deals with semantics-preserving parallelism reduction methods for cyclo-static dataflow applications. Parallelism reduction is the process of equivalent actors fusioning. The principal objectives of parallelism reduction are to decrease the memory footprint of an application and to increase its execution performance. We focus on parallelism reduction methodologies constrained by application throughput. A generic parallelism reduction methodology is introduced. Experimental results are provided for asserting the performance of the proposed method
Spectral morphisms, K-theory, and stable ranks
We give a brief account of the interplay between spectral morphisms,
K-theory, and stable ranks in the context of Banach algebras.Comment: 12 pages; to appear in the Proceedings of the Workshop on
Noncommutative Geometry (Fields Institute, Toronto 2008
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
Branch Rings, Thinned Rings, Tree Enveloping Rings
We develop the theory of ``branch algebras'', which are infinite-dimensional
associative algebras that are isomorphic, up to taking subrings of finite
codimension, to a matrix ring over themselves. The main examples come from
groups acting on trees.
In particular, for every field k we construct a k-algebra K which (1) is
finitely generated and infinite-dimensional, but has only finite-dimensional
quotients;
(2) has a subalgebra of finite codimension, isomorphic to ;
(3) is prime;
(4) has quadratic growth, and therefore Gelfand-Kirillov dimension 2;
(5) is recursively presented;
(6) satisfies no identity;
(7) contains a transcendental, invertible element;
(8) is semiprimitive if k has characteristic ;
(9) is graded if k has characteristic 2;
(10) is primitive if k is a non-algebraic extension of GF(2);
(11) is graded nil and Jacobson radical if k is an algebraic extension of
GF(2).Comment: 35 pages; small changes wrt previous versio
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