852 research outputs found
On uniform canonical bases in lattices and other metric structures
We discuss the notion of \emph{uniform canonical bases}, both in an abstract
manner and specifically for the theory of atomless lattices. We also
discuss the connection between the definability of the set of uniform canonical
bases and the existence of the theory of beautiful pairs (i.e., with the finite
cover property), and prove in particular that the set of uniform canonical
bases is definable in algebraically closed metric valued fields
On Roeckle-precompact Polish group which cannot act transitively on a complete metric space
We study when a continuous isometric action of a Polish group on a complete
metric space is, or can be, transitive. Our main results consist of showing
that certain Polish groups, namely and
, such an action can never be transitive (unless the
space acted upon is a singleton). We also point out "circumstantial evidence"
that this pathology could be related to that of Polish groups which are not
closed permutation groups and yet have discrete uniform distance, and give a
general characterisation of continuous isometric action of a Roeckle-precompact
Polish group on a complete metric space is transitive. It follows that the
morphism from a Roeckle-precompact Polish group to its Bohr compactification is
surjective
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
L2-invariants of nonuniform lattices in semisimple Lie groups
We compute L2-invariants of certain nonuniform lattices in semisimple Lie
groups by means of the Borel-Serre compactification of arithmetically defined
locally symmetric spaces. The main results give new estimates for
Novikov-Shubin numbers and vanishing L2-torsion for lattices in groups with
even deficiency. We discuss applications to Gromov's Zero-in-the-Spectrum
Conjecture as well as to a proportionality conjecture for the L2-torsion of
measure equivalent groups.Comment: 35 pages, 2 figure
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