2,759 research outputs found
Hyperbolic towers and independent generic sets in the theory of free groups
We use hyperbolic towers to answer some model theoretic questions around the
generic type in the theory of free groups. We show that all the finitely
generated models of this theory realize the generic type , but that there
is a finitely generated model which omits . We exhibit a finitely
generated model in which there are two maximal independent sets of realizations
of the generic type which have different cardinalities. We also show that a
free product of homogeneous groups is not necessarily homogeneous.Comment: to appear in Proceedings of the conference "Recent developments in
Model Theory", Notre Dame Journal of Formal Logi
Homogeneity in the free group
We show that any non abelian free group \F is strongly
-homogeneous, i.e. that finite tuples of elements which satisfy the
same first-order properties are in the same orbit under \Aut(\F). We give a
characterization of elements in finitely generated groups which have the same
first-order properties as a primitive element of the free group. We deduce as a
consequence that most hyperbolic surface groups are not -homogeneous.Comment: 26 page
Arithmetic lattices and weak spectral geometry
This note is an expansion of three lectures given at the workshop "Topology,
Complex Analysis and Arithmetic of Hyperbolic Spaces" held at Kyoto University
in December of 2006 and will appear in the proceedings for this workshop.Comment: To appear in workshop proceedings for "Topology, Complex Analysis and
Arithmetic of Hyperbolic Spaces". Comments welcom
Inflation and topological phase transition driven by exotic smoothness
In this paper we will discuss a model which describes the cause of inflation
by a topological transition. The guiding principle is the choice of an exotic
smoothness structure for the space-time. Here we consider a space-time with
topology . In case of an exotic ,
there is a change in the spatial topology from a 3-sphere to a homology
3-sphere which can carry a hyperbolic structure. From the physical point of
view, we will discuss the path integral for the Einstein-Hilbert action with
respect to a decomposition of the space-time. The inclusion of the boundary
terms produces fermionic contributions to the partition function. The
expectation value of an area (with respect to some surface) shows an
exponential increase, i.e. we obtain inflationary behavior. We will calculate
the amount of this increase to be a topological invariant. Then we will
describe this transition by an effective model, the Starobinski or
model which is consistent with the current measurement of the Planck satellite.
The spectral index and other observables are also calculated. Finally we obtain
a realistic cosmological constant.Comment: 21 pages, no figures, iopart styla, accepted in Advances in High
Energy Physics, special issue "Experimental Tests of Quantum Gravity and
Exotic Quantum Field Theory Effects (QGEQ)
Graph towers, laminations and their invariant measures
In this paper we present a combinatorial machinery, consisting of a graph
tower and vector towers on
, which allows us to efficiently describe all invariant
measures on any given shift space over a finite
alphabet.
The new technology admits a number of direct applications, in particular
concerning invariant measures on non-primitive substitution subshifts, minimal
subshifts with many ergodic measures, or an efficient calculation of the
measure of a given cylinder. It also applies to currents on a free group ,
and in particular the set of projectively fixed currents under the action of a
(possibly reducible) endomorphism is determined, when
is represented by a train track map.Comment: 52 pages, 3 figures. This is rather a new paper than a new version of
the old one. The setting is much more general, and also closer to a symbolic
dynamics spirit. Also, some of the work from the original paper has been
removed and will be taken up in a forthcoming paper. Accepted in Journal of
London Mathematical society. To appear 201
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