20,381 research outputs found
Independence property and hyperbolic groups
We prove that existentially closed -groups have the independence
property. This is done by showing that there exist words having the
independence property relatively to the class of torsion-free hyperbolic
groups.Comment: v3: 10 pages (11pt), a few typos corrected, minor rearrangements
(e.g. Fact 2.3 and Lemma 2.5); v2: 8 pages (10pt), a false statement in the
proof of Fact 2.4 is replaced with a true one; v1: 8 page
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
Ampleness in the free group
We show that the theory of the free group -- and more generally the theory of
any torsion-free hyperbolic group -- is -ample for any . We give
also an explicit description of the imaginary algebraic closure in free groups
Relative Quasiconvexity using Fine Hyperbolic Graphs
We provide a new and elegant approach to relative quasiconvexity for
relatively hyperbolic groups in the context of Bowditch's approach to relative
hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach
to quasiconvexity generalizes the other definitions in the literature that
apply only for countable relatively hyperbolic groups. We also provide an
elementary and self-contained proof that relatively quasiconvex subgroups are
relatively hyperbolic.Comment: 21 pages, 6 figures. New section on fine graphs. Version to appear in
AG
Groups acting on trees with almost prescribed local action
We investigate a family of groups acting on a regular tree, defined by
prescribing the local action almost everywhere. We study lattices in these
groups and give examples of compactly generated simple groups of finite
asymptotic dimension (actually one) not containing lattices. We also obtain
examples of simple groups with simple lattices, and we prove the existence of
(infinitely many) finitely generated simple groups of asymptotic dimension one.
We also prove various properties of these groups, including the existence of a
proper action on a CAT(0) cube complex.Comment: v2: 35 pages; argument slightly modified in 4.2.2; final versio
Tsallis entropy composition and the Heisenberg group
We present an embedding of the Tsallis entropy into the 3-dimensional
Heisenberg group, in order to understand the meaning of generalized
independence as encoded in the Tsallis entropy composition property. We infer
that the Tsallis entropy composition induces fractal properties on the
underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we
justify why the underlying configuration/phase space of systems described by
the Tsallis entropy has polynomial growth for both discrete and Riemannian
cases. We provide a geometric framework that elucidates Abe's formula for the
Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian
spaces.Comment: 26 pages, No figures, LaTeX2e. To be published in Int. J. Geom.
Methods Mod. Physic
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