15,344 research outputs found
Computability of probability measures and Martin-Lof randomness over metric spaces
In this paper we investigate algorithmic randomness on more general spaces
than the Cantor space, namely computable metric spaces. To do this, we first
develop a unified framework allowing computations with probability measures. We
show that any computable metric space with a computable probability measure is
isomorphic to the Cantor space in a computable and measure-theoretic sense. We
show that any computable metric space admits a universal uniform randomness
test (without further assumption).Comment: 29 page
Bihomogeneity of solenoids
Solenoids are inverse limit spaces over regular covering maps of closed
manifolds. M.C. McCord has shown that solenoids are topologically homogeneous
and that they are principal bundles with a profinite structure group. We show
that if a solenoid is bihomogeneous, then its structure group contains an open
abelian subgroup. This leads to new examples of homogeneous continua that are
not bihomogeneous.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-1.abs.htm
Augmented Homotopical Algebraic Geometry
We develop the framework for augmented homotopical algebraic geometry. This
is an extension of homotopical algebraic geometry, which itself is a
homotopification of classical algebraic geometry. To do so, we define the
notion of augmentation categories, which are a special class of generalised
Reedy categories. For an augmentation category, we prove the existence of a
closed Quillen model structure on the presheaf category which is compatible
with the Kan-Quillen model structure on simplicial sets. Moreover, we use the
concept of augmented hypercovers to define a local model structure on the
category of augmented presheaves. We prove that crossed simplicial groups, and
the planar rooted tree category are examples of augmentation categories.
Finally, we introduce a method for generating new examples from old via a
categorical pushout construction.Comment: 36 pages, comments welcom
Couplings of Uniform Spanniing Forests
We prove the existence of an automorphism-invariant coupling for the wired
and the free uniform spanning forests on Cayley graphs of finitely generated
residually amenable groups.Comment: 7 page
Topological rigidity and H_1-negative involutions on tori
We prove there is only one involution (up to conjugacy) on the n-torus which
acts as on the first homology group when is of the form
, is of the form , or is less than . In all other cases we prove
there are infinitely many such involutions up to conjugacy, but each of them
has exactly fixed points and is conjugate to a smooth involution. The key
technical point is that we completely compute the equivariant structure set for
the corresponding crystallographic group action on in terms of
the Cappell -groups arising from its infinite dihedral
subgroups. We give a complete analysis of equivariant topological rigidity for
this family of groups.Comment: 50 pages, to appear in Geometry & Topolog
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