From Practice to New Concepts: Geometric Properties of Groups

Cet article cherche à montrer comment la pratique mathématique, particulièrement celle admettant des représentations visuelles, peut conduire à de nouveaux résultats mathématiques. L'argumentation est basée sur l'étude du cas d'un domaine des mathématiques relativement récent et prometteur: la théorie géométrique des groupes. L'article discute comment la représentation des groupes par les graphes de Cayley rendit possible la découverte de nouvelles propriétés géométriques de groupes.

Oberwolfach Reports, 2013

The overall theme of the conference was geometric group theory, interpreted quite broadly. In general, geometric group theory seeks to understand algebraic properties of groups by studying their actions on spaces with various topological and geometric properties; in particular these spaces must have enough structure-preserving symmetry to admit interesting group actions. Although traditionally geometric group theorists have focused on finitely generated (and even finitely presented) countable discrete groups, the techniques that have been developed are now applied to more general groups, such as Lie groups and Kac-Moody groups, and although metric properties of the spaces have played a key role in geometric group theory, other structure such as complex or projective structures and measure-theoretic structures are being used more and more frequently.

Contains a few results about geometric group theory. Finitely generated groups admitting a free action on a quasi-tree (a graph with a bound on the size of its simple loops) are shown to be free products of free groups and finite groups. Groups admitting thin combings are shown to be hyperbolic. Finitely presented infinite groups with a sublinear isoperimetric inequality are free. There are also some results about groups admitting tripodal combings and tripodal pairs of combings, which are a simplified form of Rips-Sela Canonical Representatives.

Logic and Logical Philosophy, 2004

Klein's Erlangen program contains the postulate to study the group of automorphisms instead of a structure itself. This postulate, taken literally, sometimes means a substantial loss of information. For example, the group of automorphisms of the field of rational numbers is trivial. However in the case of Euclidean plane geometry the situation is different. We shall prove that the plane Euclidean geometry is mutually interpretable with the elementary theory of the group of authomorphisms of its standard model. Thus both theories differ practically in the language only.

In this course, we develop the basic notions of Manifolds and Geometry, with applications in physics, and also we develop the basic notions of the theory of Lie Groups, and their applications in physics.

Contemporary Mathematics, 2015

European Journal of Pure and Applied Mathematics, 2020

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power a...

2009

The theory of non-positively curved spaces and groups is tremendously powerful and has enormous consequences for the groups and spaces involved. Nevertheless, our ability to construct examples to which the theory can be applied has been severely limited by an inability to test-in real timewhether a random finite piecewise Euclidean complex is non-positively curved. In this article I focus on the question of how to construct examples of nonpositively curved spaces and groups, highlighting in particular the boundary between what is currently do-able and what is not yet feasible. Since this is intended primarily as a survey, the key ideas are merely sketched with references pointing the interested reader to the original articles.

Proceedings of the 2008 ACM symposium on Applied computing - SAC '08, 2008

This paper shows that considering the group generated by orthogonal symmetries relatively to lines may give very short and readable proofs of geometric theorems. A short and readable proof of the fundamental Pascal's theorem is provided for illustration.

Geometriae Dedicata, 1985