250 research outputs found
Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds
Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new
homological quantum error correcting codes. They are LDPC codes with linear
rate and distance . Their rate is evaluated via Euler
characteristic arguments and their distance using -systolic
geometry. This construction answers a queston of Z\'emor, who asked whether
homological codes with such parameters could exist at all.Comment: 21 page
Can one hear the shape of the Universe?
It is shown that the recent observations of NASA's explorer mission
"Wilkinson Microwave Anisotropy Probe" (WMAP) hint that our Universe may
possess a non-trivial topology. As an example we discuss the Picard space which
is stretched out into an infinitely long horn but with finite volume.Comment: 4 page
Are Small Hyperbolic Universes Observationally Detectable?
Using recent observational constraints on cosmological density parameters,
together with recent mathematical results concerning small volume hyperbolic
manifolds, we argue that, by employing pattern repetitions, the topology of
nearly flat small hyperbolic universes can be observationally undetectable.
This is important in view of the facts that quantum cosmology may favour
hyperbolic universes with small volumes, and from the expectation coming from
inflationary scenarios, that is likely to be very close to one.Comment: 5 pages, 1 figure, LaTeX2e. A reference and two footnotes added. To
appear in Class. Quantum Grav. 18 (2001) in the present for
The Quantized Sigma Model Has No Continuum Limit in Four Dimensions. I. Theoretical Framework
The nonlinear sigma model for which the field takes its values in the coset
space is similar to quantum gravity in being
perturbatively nonrenormalizable and having a noncompact curved configuration
space. It is therefore a good model for testing nonperturbative methods that
may be useful in quantum gravity, especially methods based on lattice field
theory. In this paper we develop the theoretical framework necessary for
recognizing and studying a consistent nonperturbative quantum field theory of
the model. We describe the action, the geometry of the
configuration space, the conserved Noether currents, and the current algebra,
and we construct a version of the Ward-Slavnov identity that makes it easy to
switch from a given field to a nonlinearly related one. Renormalization of the
model is defined via the effective action and via current algebra. The two
definitions are shown to be equivalent. In a companion paper we develop a
lattice formulation of the theory that is particularly well suited to the sigma
model, and we report the results of Monte Carlo simulations of this lattice
model. These simulations indicate that as the lattice cutoff is removed the
theory becomes that of a pair of massless free fields. Because the geometry and
symmetries of these fields differ from those of the original model we conclude
that a continuum limit of the model which preserves
these properties does not exist.Comment: 25 pages, no figure
Contracting automorphisms and L^p-cohomology in degree one
We characterize those Lie groups, and algebraic groups over a local field of
characteristic zero, whose first reduced L^p-cohomology is zero for all p>1,
extending a result of Pansu. As an application, we obtain a description of
Gromov-hyperbolic groups among those groups. In particular we prove that any
non-elementary Gromov-hyperbolic algebraic group over a non-Archimedean local
field of zero characteristic is quasi-isometric to a 3-regular tree. We also
extend the study to semidirect products of a general locally compact group by a
cyclic group acting by contracting automorphisms.Comment: 27 pages, no figur
Automorphism groups of polycyclic-by-finite groups and arithmetic groups
We show that the outer automorphism group of a polycyclic-by-finite group is
an arithmetic group. This result follows from a detailed structural analysis of
the automorphism groups of such groups. We use an extended version of the
theory of the algebraic hull functor initiated by Mostow. We thus make
applicable refined methods from the theory of algebraic and arithmetic groups.
We also construct examples of polycyclic-by-finite groups which have an
automorphism group which does not contain an arithmetic group of finite index.
Finally we discuss applications of our results to the groups of homotopy
self-equivalences of K(\Gamma, 1)-spaces and obtain an extension of
arithmeticity results of Sullivan in rational homotopy theory
Shimura varieties in the Torelli locus via Galois coverings of elliptic curves
We study Shimura subvarieties of obtained from families of
Galois coverings where is a smooth complex
projective curve of genus and . We give the complete list
of all such families that satisfy a simple sufficient condition that ensures
that the closure of the image of the family via the Torelli map yields a
Shimura subvariety of for and for all and
for and . In a previous work of the first and second author
together with A. Ghigi [FGP] similar computations were done in the case .
Here we find 6 families of Galois coverings, all with and
and we show that these are the only families with satisfying this
sufficient condition. We show that among these examples two families yield new
Shimura subvarieties of , while the other examples arise from
certain Shimura subvarieties of already obtained as families of
Galois coverings of in [FGP]. Finally we prove that if a family
satisfies this sufficient condition with , then .Comment: 18 pages, to appear in Geometriae Dedicat
Compact Hyperbolic Extra Dimensions: Branes, Kaluza-Klein Modes and Cosmology
We reconsider theories with low gravitational (or string) scale M_* where
Newton's constant is generated via new large-volume spatial dimensions, while
Standard Model states are localized to a 3-brane. Utilizing compact hyperbolic
manifolds (CHM's) we show that the spectrum of Kaluza-Klein (KK) modes is
radically altered. This allows an early universe cosmology with normal
evolution up to substantial temperatures, and completely negates the
constraints on M_* arising from astrophysics. Furthermore, an exponential
hierarchy between the usual Planck scale and the true fundamental scale of
physics can emerge with only order unity coefficients. The linear size of the
internal space remains small. The proposal has striking testable signatures.Comment: 4 pages, no figure
Arithmeticity vs. non-linearity for irreducible lattices
We establish an arithmeticity vs. non-linearity alternative for irreducible
lattices in suitable product groups, such as for instance products of
topologically simple groups. This applies notably to a (large class of)
Kac-Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as
we follow Margulis' reduction of arithmeticity to superrigidity.Comment: 11 page
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