88 research outputs found
Cayley graphs of order kp are hamiltonian for k < 48
We provide a computer-assisted proof that if G is any finite group of order
kp, where k < 48 and p is prime, then every connected Cayley graph on G is
hamiltonian (unless kp = 2). As part of the proof, it is verified that every
connected Cayley graph of order less than 48 is either hamiltonian connected or
hamiltonian laceable (or has valence less than three).Comment: 16 pages. GAP source code is available in the ancillary file
Finite and infinite quotients of discrete and indiscrete groups
These notes are devoted to lattices in products of trees and related topics.
They provide an introduction to the construction, by M. Burger and S. Mozes, of
examples of such lattices that are simple as abstract groups. Two features of
that construction are emphasized: the relevance of non-discrete locally compact
groups, and the two-step strategy in the proof of simplicity, addressing
separately, and with completely different methods, the existence of finite and
infinite quotients. A brief history of the quest for finitely generated and
finitely presented infinite simple groups is also sketched. A comparison with
Margulis' proof of Kneser's simplicity conjecture is discussed, and the
relevance of the Classification of the Finite Simple Groups is pointed out. A
final chapter is devoted to finite and infinite quotients of hyperbolic groups
and their relation to the asymptotic properties of the finite simple groups.
Numerous open problems are discussed along the way.Comment: Revised according to referee's report; definition of BMW-groups
updated; more examples added in Section 4; new Proposition 5.1
Spectral and Combinatorial Aspects of Cayley-Crystals
Owing to their interesting spectral properties, the synthetic crystals over
lattices other than regular Euclidean lattices, such as hyperbolic and fractal
ones, have attracted renewed attention, especially from materials and
meta-materials research communities. They can be studied under the umbrella of
quantum dynamics over Cayley graphs of finitely generated groups. In this work,
we investigate numerical aspects related to the quantum dynamics over such
Cayley graphs. Using an algebraic formulation of the "periodic boundary
condition" due to Lueck [Geom. Funct. Anal. 4, 455-481 (1994)], we devise a
practical and converging numerical method that resolves the true bulk spectrum
of the Hamiltonians. Exact results on the matrix elements of the resolvent,
derived from the combinatorics of the Cayley graphs, give us the means to
validate our algorithms and also to obtain new combinatorial statements. Our
results open the systematic research of quantum dynamics over Cayley graphs of
a very large family of finitely generated groups, which includes the free and
Fuchsian groups.Comment: converging periodic bc for hyperbolic and fractal crystals, tested
against exact result
A walk in the noncommutative garden
This text is written for the volume of the school/conference "Noncommutative
Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in
noncommutative geometry, based on a discussion of significant examples of
noncommutative spaces in geometry, number theory, and physics. The paper also
contains an outline (the ``Tehran program'') of ongoing joint work with Consani
on the noncommutative geometry of the adeles class space and its relation to
number theoretic questions.Comment: 106 pages, LaTeX, 23 figure
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Combinatorics, Probability and Computing
One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly effective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b
Classical Algebraic Geometry
Progress in algebraic geometry usually comes through the introduction of new tools and ideas to tackle the classical problems of the field. Examples include new invariants that capture some aspect of geometry in a novel way, such as Voisin’s “existence of decomposition of the diagonal”, and the extension of the class of geometric objects considered to allow constructions not previously possible, such as stacks, tropical geometry, and log structures. Many famous old problems and outstanding conjectures have been resolved in this way over the last 50 years. While the new theories are sometimes studied for their own sake, they are in the end best understood in the context of the classical questions they illuminate. The goal of the workshop was to study new developments in algebraic geometry, in the context of their application to the classical problems
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
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