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.

LOCALLY INVARIANT ORDERS ON GROUPS

2006 ◽  
Vol 16 (06) ◽  
pp. 1161-1179 ◽  
Author(s):  
I. M. CHISWELL
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Recent Work ◽  
Analogous Result ◽  
Hyperbolic Manifold ◽  
Free Groups ◽  
Free Action ◽  
Detailed Proof ◽  
Unique Product ◽  
Invariant Order

Recent work by T. Delzant and S. Hair shows that certain groups are unique product groups. In effect, they show that the groups have a locally invariant order, an idea introduced by D. Promislow in the early eighties. Having a locally invariant order implies the group is a unique product group, and a strict left (or right) ordering on a group is a locally invariant order. We study properties of the class of LIO groups, that is, groups having a locally invariant order. The main result gives conditions under which the fundamental group of a graph of LIO groups is LIO. In particular, the free product of two LIO groups is LIO. There is an analogous result for a graph of right orderable groups. We also study tree-free groups (those having a free action without inversions on a Λ-tree, for some ordered abelian group Λ). In particular, a detailed proof that tree-free groups are LIO is given. There is also a detailed proof of an observation made by Hair, that the fundamental group of a compact hyperbolic manifold is virtually LIO.

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 The geometry of k-free hyperbolic 3-manifolds

2019 ◽  
Vol 12 (04) ◽  
pp. 1195-1212
Author(s):  
R. K. Guzman ◽  
P. B. Shalen
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Geometry ◽  
Hyperbolic Manifold ◽  
Free Groups ◽  
Kleinian Groups ◽  
Fundamental Groups ◽  
Group Topology ◽  
Topological Properties ◽  
Cyclic Subgroups ◽  
Differential Geom

We investigate the geometry of closed, orientable, hyperbolic 3-manifolds whose fundamental groups are [Formula: see text]-free for a given integer [Formula: see text]. We show that any such manifold [Formula: see text] contains a point [Formula: see text] with the following property: If [Formula: see text] is the set of maximal cyclic subgroups of [Formula: see text] that contain non-trivial elements represented by loops of [Formula: see text], then for every subset [Formula: see text], we have rank [Formula: see text]. This generalizes to all [Formula: see text] results proved in [J. W. Anderson, R. D. Canary, M. Culler and P. B. Shalen, Free Kleinian groups and volumes of hyperbolic 3-manifolds, J. Differential Geom. 43 (1996) 738–782; M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156] for [Formula: see text]. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136; R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156].

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

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)).

On the SL(2, C)-representation rings of free abelian groups

2018 ◽  
Vol 167 (02) ◽  
pp. 229-247
Author(s):  
TAKAO SATOH
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Groups ◽  
The Other ◽  
Subsequent Research ◽  
Representation Rings ◽  
Free Abelian Groups

AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

scholarly journals Biorder-preserving coextensions of fundamental semigroups

1988 ◽  
Vol 31 (3) ◽  
pp. 463-467 ◽  
Author(s):  
David Easdown
Keyword(s):  
Group Theory ◽  
Semigroup Theory ◽  
Building Blocks ◽  
Inverse Semigroups ◽  
Extension Theory ◽  
Precise Definition ◽  
Simple Groups ◽  
Seminal Work ◽  
Fundamental Semigroups

In any extension theory for semigroups one must determine the basic building blocks and then discover how they fit together to create more complicated semigroups. For example, in group theory the basic building blocks are simple groups. In semigroup theory however there are several natural choices. One that has received considerable attention, particularly since the seminal work on inverse semigroups by Munn ([14, 15]), is the notion of a fundamental semigroup. A semigroup is called fundamental if it cannot be [shrunk] homomorphically without collapsing some of its idempotents (see below for a precise definition).

scholarly journals Assembling homology classes in automorphism groups of free groups

2016 ◽  
Vol 91 (4) ◽  
pp. 751-806 ◽  
Author(s):  
James Conant ◽  
Allen Hatcher ◽  
Martin Kassabov ◽  
Karen Vogtmann
Keyword(s):  
Automorphism Groups ◽  
Free Groups

scholarly journals Computing the fundamental group of a higher-rank graph

2021 ◽  
pp. 1-12
Author(s):  
Sooran Kang ◽  
David Pask ◽  
Samuel B.G. Webster
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Klein Bottle ◽  
Fundamental Groups ◽  
The Third ◽  
Higher Rank ◽  
The One ◽  
Standard Presentation

Abstract We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about higher-rank graphs when we show that the abelianization of the fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

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