An Introduction to Combinatorial Group Theory and the Word Problem

1997 ◽  
Vol 70 (1) ◽  
pp. 3
Author(s):  
David Peifer
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Combinatorial Group Theory ◽  
Combinatorial Group

scholarly journals A Descriptive View of Combinatorial Group Theory

2011 ◽  
Vol 17 (2) ◽  
pp. 252-264
Author(s):  
Simon Thomas
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Embedding Theorem ◽  
Combinatorial Group Theory ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Combinatorial Group

AbstractIn this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman–Neumann–Neumann Embedding Theorem.

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.

COMBINATORIAL GROUP THEORY, INVERSE MONOIDS, AUTOMATA, AND GLOBAL SEMIGROUP THEORY

2002 ◽  
Vol 12 (01n02) ◽  
pp. 179-211 ◽  
Author(s):  
MANUEL DELGADO ◽  
STUART MARGOLIS ◽  
BENJAMIN STEINBERG
Keyword(s):  
Group Theory ◽  
Formal Language ◽  
Finite Semigroup ◽  
Semigroup Theory ◽  
Nilpotent Groups ◽  
Language Theory ◽  
Combinatorial Group Theory ◽  
Inverse Monoid ◽  
Group Presentation ◽  
Combinatorial Group

This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = <A|R> be a group presentation and ℬA, R its standard 2-complex. Suppose X is a 2-complex with a morphism to ℬA, R which restricts to an immersion on the 1-skeleton. Then we associate an inverse monoid to X which algebraically encodes topological properties of the morphism. Applications are given to separability properties of groups. We also associate an inverse monoid M(A, R) to the presentation <A|R> with the property that pointed subgraphs of covers of ℬA, R are classified by closed inverse submonoids of M(A, R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to be quasiconvex allowing semigroup theoretic variants on the usual proofs that the intersection of such subgroups is quasiconvex and that such subgroups are finitely generated. Generalizations are given to non-geodesic combings. We also obtain a formal language theoretic equivalence to quasiconvexity which holds even for groups which are not hyperbolic. Finally, we illustrate some applications of separability properties of relatively free groups to finite semigroup theory. In particular, we can deduce the decidability of various semidirect and Mal/cev products of pseudovarieties of monoids with equational pseudovarieties of nilpotent groups and with the pseudovariety of metabelian groups.

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