Combinatorial Group Theory, Discrete Groups, and Number Theory

2006 ◽  
Keyword(s):  
Number Theory ◽  
Group Theory ◽  
Combinatorial Group Theory ◽  
Discrete Groups ◽  
Combinatorial Group

scholarly journals Negative curvature and combinatorial group theory

1973 ◽  
Vol 16 (1) ◽  
pp. 14-15
Author(s):  
Joseph A. Wolf
Keyword(s):  
Number Theory ◽  
Group Theory ◽  
Analytic Number Theory ◽  
Combinatorial Group Theory ◽  
Fundamental Groups ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Analytic Number ◽  
Topology Of Manifolds ◽  
Combinatorial Group

Combinatorial group theory has roots in Poincaré's work on the topology of manifolds, which in turn was based on problems in differential equations and analytic number theory. Thus the Fuchsian groups, which are the fundamental (first homotopy) groups of oriented negatively curved compact surfaces, served as important models in their day. In the last few years there have been advances in the understanding of the structure of fundamental groups of negatively curved manifolds, some of them based on examples from analytic number theory. Here I describe one of these developments and pose a few difficult combinatorial questions.

The History, Classes and Presentations of Groups

2012 ◽  
Vol 430-432 ◽  
pp. 834-837
Author(s):  
Xiao Qiang Guo ◽  
Zheng Jun He
Keyword(s):  
Group Theory ◽  
Algebraic Groups ◽  
Transformation Groups ◽  
Combinatorial Group Theory ◽  
Algebraic Equations ◽  
Historical Sources ◽  
Matrix Groups ◽  
History Of ◽  
History Classes ◽  
Combinatorial Group

First we introduce the history of group theory. Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. Secondly, we give the main classes of groups: permutation groups, matrix groups, transformation groups, abstract groups and topological and algebraic groups. Finally, we give two different presentations of a group: combinatorial group theory and geometric group theory.

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