A class of non-Kählerian manifolds

1986 ◽  
Vol 100 (3) ◽  
pp. 519-521 ◽  
Author(s):  
F. E. A. Johnson
Keyword(s):  
Group Theory ◽  
Fundamental Groups ◽  
Genus 2 ◽  
Kählerian Manifolds

Let S+ (resp. S−) denote the class of fundamental groups of closed orientable (resp. non-orientable) 2-manifolds of genus ≥ 2, and let surface = S+ ∪ S−. In the list of problems raised at the 1977 Durham Conference on Homological Group Theory occurs the following([7], p. 391, (G. 3)).

scholarly journals Residually finite rationally p groups

2019 ◽  
Vol 22 (03) ◽  
pp. 1950016
Author(s):  
Thomas Koberda ◽  
Alexander I. Suciu
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Large Class ◽  
Algebraic Curves ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residual Property ◽  
Residually Finite ◽  
Torsion Freeness ◽  
Do So

In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so.

scholarly journals Applications of L systems to group theory

2018 ◽  
Vol 28 (02) ◽  
pp. 309-329 ◽  
Author(s):  
Laura Ciobanu ◽  
Murray Elder ◽  
Michal Ferov
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Normal Forms ◽  
Free Groups ◽  
Fundamental Groups ◽  
Context Sensitive ◽  
Rank 2 ◽  
L Systems ◽  
Primitive Sets ◽  
Context Free

L systems generalize context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper, we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives and bases in the free group on two generators, the Bridson–Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk’s group can be generated by L systems. To complement the result on primitives in rank 2 free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L languages of intermediate growth.

scholarly journals A Note on Regular Coverings of Closed Orientable Surfaces

1961 ◽  
Vol 5 (2) ◽  
pp. 49-66 ◽  
Author(s):  
Jens Mennicke
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Finite Index ◽  
Orientable Surface ◽  
Fundamental Groups ◽  
Closed Orientable Surface ◽  
Regular Covering ◽  
Genus 2 ◽  
Image Position ◽  
Orientable Surfaces

The object of this note is to study the regular coverings of the closed orientable surface of genus 2.Let the closed orientable surfaceFhof genushbe a covering ofF2and letand f be the fundamental groups respectively. Thenis a subgroup of f of indexn = h − 1. A covering is called regular ifis normal in f.Conversely, letbe a normal subgroup of f of finite index. Then there is a uniquely determined regular coveringFhsuch thatis the fundamental group ofFh. The coveringFhis an orientable surface. Since the indexnofin f is supposed to be finite,Fhis closed, and its genus is given byn=h − 1.The fundamental group f can be defined by.

scholarly journals The Farrell-Jones isomorphism conjecture for 3-manifold groups

2007 ◽  
Vol 1 (1) ◽  
pp. 49-82 ◽  
Author(s):  
S.K. Roushon
Keyword(s):  
Fundamental Group ◽  
Large Class ◽  
Partial Solution ◽  
Fundamental Groups ◽  
Problem List ◽  
Index Subgroup ◽  
Main Aspect ◽  
Graph Of Groups ◽  
Genus 2 ◽  
Finite Index Subgroup

AbstractWe show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for fundamental groups of a large class of 3-manifolds. We also prove that if the FIC is true for irreducible 3-manifold groups then it is true for all 3-manifold groups. In fact, this follows from a more general result we prove, namely we show that if the FIC is true for each vertex group of a graph of groups with trivial edge groups then the FIC is true for the fundamental group of the graph of groups. This result is part of a program to prove the FIC for the fundamental group of a graph of groups where all the vertex and edge groups satisfy the FIC. A consequence of the first result gives a partial solution to a problem in the problem list of R. Kirby. We also deduce that the FIC is true for a class of virtually P D3-groups.Another main aspect of this article is to prove the FIC for all Haken 3-manifold groups assuming that the FIC is true for B-groups. By definition a B-group contains a finite index subgroup isomorphic to the fundamental group of a compact irreducible 3-manifold with incompressible nonempty boundary so that each boundary component is of genus ≥ 2. We also prove the FIC for a large class of B-groups and moreover, using a recent result of L.E. Jones we show that the surjective part of the FIC is true for any B-group.

scholarly journals Characterizations of Schützenberger graphs in terms of their automorphism groups and fundamental groups

1993 ◽  
Vol 35 (3) ◽  
pp. 275-291 ◽  
Author(s):  
David Cowan ◽  
Norman R. Reilly
Keyword(s):  
Group Theory ◽  
Fundamental Group ◽  
Recent Work ◽  
Word Problems ◽  
Automorphism Groups ◽  
Free Groups ◽  
Inverse Semigroups ◽  
Fundamental Groups ◽  
Graph Theoretic

AbstractThe importance of the fundamental group of a graph in group theory has been well known for many years. The recent work of Meakin, Margolis and Stephen has shown how effective graph theoretic techniques can be in the study of word problems in inverse semigroups. Our goal here is to characterize those deterministic inverse word graphs that are Schutzenberger graphs and consider how deterministic inverse word graphs and Schutzenberger graphs can be constructed from subgroups of free groups.

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.

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