82,912 research outputs found
The Patterson-Sullivan embedding and minimal volume entropy for outer space
Motivated by Bonahon's result for hyperbolic surfaces, we construct an
analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann
outer space into the space of projectivized geodesic currents on a
free group. We prove that this map is a topological embedding. We also prove
that for every the minimum of the volume entropy of the universal
covers of finite connected volume-one metric graphs with fundamental group of
rank and without degree-one vertices is equal to and that
this minimum is realized by trivalent graphs with all edges of equal lengths,
and only by such graphs.Comment: An updated versio
Strings from Feynman Graph counting : without large N
A well-known connection between n strings winding around a circle and
permutations of n objects plays a fundamental role in the string theory of
large N two dimensional Yang Mills theory and elsewhere in topological and
physical string theories. Basic questions in the enumeration of Feynman graphs
can be expressed elegantly in terms of permutation groups. We show that these
permutation techniques for Feynman graph enumeration, along with the Burnside
counting lemma, lead to equalities between counting problems of Feynman graphs
in scalar field theories and Quantum Electrodynamics with the counting of
amplitudes in a string theory with torus or cylinder target space. This string
theory arises in the large N expansion of two dimensional Yang Mills and is
closely related to lattice gauge theory with S_n gauge group. We collect and
extend results on generating functions for Feynman graph counting, which
connect directly with the string picture. We propose that the connection
between string combinatorics and permutations has implications for QFT-string
dualities, beyond the framework of large N gauge theory.Comment: 55 pages + 10 pages Appendices, 23 figures ; version 2 - typos
correcte
Modular Theory, Non-Commutative Geometry and Quantum Gravity
This paper contains the first written exposition of some ideas (announced in
a previous survey) on an approach to quantum gravity based on Tomita-Takesaki
modular theory and A. Connes non-commutative geometry aiming at the
reconstruction of spectral geometries from an operational formalism of states
and categories of observables in a covariant theory. Care has been taken to
provide a coverage of the relevant background on modular theory, its
applications in non-commutative geometry and physics and to the detailed
discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields
A geometric approach to (semi)-groups defined by automata via dual transducers
We give a geometric approach to groups defined by automata via the notion of
enriched dual of an inverse transducer. Using this geometric correspondence we
first provide some finiteness results, then we consider groups generated by the
dual of Cayley type of machines. Lastly, we address the problem of the study of
the action of these groups in the boundary. We show that examples of groups
having essentially free actions without critical points lie in the class of
groups defined by the transducers whose enriched dual generate a torsion-free
semigroup. Finally, we provide necessary and sufficient conditions to have
finite Schreier graphs on the boundary yielding to the decidability of the
algorithmic problem of checking the existence of Schreier graphs on the
boundary whose cardinalities are upper bounded by some fixed integer
From self-similar groups to self-similar sets and spectra
The survey presents developments in the theory of self-similar groups leading
to applications to the study of fractal sets and graphs, and their associated
spectra
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