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THE GEOMETRY OF CYCLES IN THE CAYLEY DIAGRAM OF A GROUP
, 1993
"... Dedicated to the memory of Wilhelm Magnus Abstract. A study of triangulations of cycles in the Cayley diagrams of finitely generated groups leads to a new geometric characterization of hyperbolic groups. 1. ..."
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Dedicated to the memory of Wilhelm Magnus Abstract. A study of triangulations of cycles in the Cayley diagrams of finitely generated groups leads to a new geometric characterization of hyperbolic groups. 1.
ON A DIAGRAMMATIC PROOF OF THE CAYLEYHAMILTON THEOREM
, 2009
"... This note concerns a oneline diagrammatic proof of the CayleyHamilton Theorem. We discuss the proof’s implications regarding the “core truth ” of the theorem, and provide a generalization. We review the notation of trace diagrams and exhibit explicit diagrammatic descriptions of the coefficients ..."
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This note concerns a oneline diagrammatic proof of the CayleyHamilton Theorem. We discuss the proof’s implications regarding the “core truth ” of the theorem, and provide a generalization. We review the notation of trace diagrams and exhibit explicit diagrammatic descriptions of the coefficients
Cayley Graphs, Cori Hypermaps, and Dessins d’Enfants
, 2008
"... This paper explains some facts probably known to experts and implicitely contained in the literature about dessins d’enfants but which seem to be nowhere explicitely stated. The 1skeleton of every regular Cori hypermap is the Cayley graph of its automorphism group, embedded in the underlying orient ..."
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orientable surface. Conversely, every Cayley graph of a finite twogenerator group has an embedding as the 1skeleton of a regular hypermap in the Cori representation. For nonregular hypermaps there is an analogous correspondence with Schreier coset diagrams.
CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS AND MONOMIAL IDEALS
, 2007
"... In the study of doubleloop computer networks, the diagrams known as Lshapes arise as a graphical representation of an optimal routing for every graph’s node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the correspon ..."
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of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece
DISKS IN TRIVIAL BRAID DIAGRAMS
, 2003
"... We show that every trivial 3strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin–Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a partition ..."
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We show that every trivial 3strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin–Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a
Forest diagrams for elements of Thompson’s group F
, 2003
"... We introduce forest diagrams to represent elements of Thompson’s group F. These diagrams relate to a certain action of F on the real line in the same way that tree diagrams relate to the standard action of F on the unit interval. Using forest diagrams, we give a conceptually simple length formula fo ..."
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for elements of F with respect to the {x0,x1} generating set, and we discuss the construction of minimumlength words for positive elements. Finally, we use forest diagrams and the length formula to examine the structure of the Cayley graph of F.
On groups whose word problem is solved by a nested stack automaton. arXiv:math.GR/9812028
, 1998
"... Abstract. Accessible groups whose word problems are accepted by a deterministic nested stack automaton with limited erasing are virtually free. 1. Introduction. During the past several years combinatorial group theory has received an infusion of ideas both from topology and from the theory of formal ..."
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of formal languages. The resulting interplay between groups, the geometry of their Cayley diagrams, and associated formal languages has led to several developments including
Mathematical tools for computergenerated ornamental patterns
 In Electronic Publishing, Artistic Imaging and Digital Typography, number 1375 in Lecture Notes in Computer Science
, 1998
"... Abstract. This article presents mathematical tools for computergenerated ornamental patterns, with a particular attention payed to Islamic patterns. The article shows how, starting from a photo or a sketch of an ornamental figure, the designer analyzes its structure and produces the analytical repr ..."
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symmetry groups and Cayley diagrams. A simple and intuitive stepbystep guide is provided. 1
Groups and simple languages
 Trans. Amer. Math. Soc
, 1983
"... Abstract. With any finitely generated group presentation, one can associate a formal language (called the reduced word problem) consisting of those words on the generators and their inverses which are equal to the identity but which have no proper prefix equal to the identity. We show that the reduc ..."
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that the reduced word problem is a simple language if and only if each vertex of the presentation's Cayley diagram has only a finite number of simple closed paths passing through it. Furthermore, if the reduced word problem is simple, then the group is a free product of a free group of finite rank and a
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