Free groups, the 𝑇-property, and 𝐼𝐼₁-factors with different countable fundamental groups

1986 ◽  
pp. 71-74
Author(s):  
V. Ya. Golodets ◽  
N. I. Nessonov
Keyword(s):  
Free Groups ◽  
Fundamental Groups

Classification of epimorphisms of fundamental groups of surfaces onto free groups

1990 ◽  
Vol 48 (2) ◽  
pp. 736-742
Author(s):  
R. I. Grigorchuk ◽  
P. F. Kurchanov
Keyword(s):  
Free Groups ◽  
Fundamental Groups

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.

Hierarchies of Computable groups and the word problem

1966 ◽  
Vol 31 (3) ◽  
pp. 376-392 ◽  
Author(s):  
Frank B. Cannonito
Keyword(s):  
Word Problem ◽  
Free Groups ◽  
Orientable Surface ◽  
Fundamental Groups ◽  
Church’S Thesis ◽  
Free Products ◽  
Closed Orientable Surface ◽  
Products Of Groups ◽  
Solvable Word Problem ◽  
Solvable Word

The word problem for groups was first formulated by M. Dehn [1], who gave a solution for the fundamental groups of a closed orientable surface of genus g ≧ 2. In the following years solutions were given, for example, for groups with one defining relator [2], free groups, free products of groups with a solvable word problem and, in certain cases, free products of groups with amalgamated subgroups [3], [4], [5]. During the period 1953–1957, it was shown independently by Novikov and Boone that the word problem for groups is recursively undecidable [6], [7]; granting Church's Thesis [8], their work implies that the word problem for groups is effectively undecidable.

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 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].

scholarly journals Folding-like techniques for CAT(0) cube complexes

2021 ◽  
pp. 1-12
Author(s):  
MICHAEL BEN–ZVI ◽  
ROBERT KROPHOLLER ◽  
RYLEE ALANZA LYMAN
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Coxeter Groups ◽  
Free Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Seminal Paper ◽  
Finite Graphs ◽  
Cube Complexes ◽  
Positively Curved

Abstract In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

scholarly journals On model-theoretic connected components in some group extensions

2015 ◽  
Vol 15 (02) ◽  
pp. 1550009 ◽  
Author(s):  
Jakub Gismatullin ◽  
Krzysztof Krupiński
Keyword(s):  
Free Groups ◽  
Invariant Subgroup ◽  
Universal Cover ◽  
Connected Components ◽  
Fundamental Groups ◽  
Bounded Cohomology ◽  
Central Extensions ◽  
Finite Image ◽  
Genus Formula ◽  
Orientable Surfaces

We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We characterize, in terms of these 2-cocycles, when the smallest type-definable subgroup of the corresponding extension differs from the smallest invariant subgroup. In some situations, we also describe the quotient of these two connected components. Using our general results about extensions of groups together with Matsumoto–Moore theory or various quasi-characters considered in bounded cohomology, we obtain new classes of examples of groups whose smallest type-definable subgroup of bounded index differs from the smallest invariant subgroup of bounded index. This includes the first known example of a group with this property found by Conversano and Pillay, namely the universal cover of [Formula: see text] (interpreted in a monster model), as well as various examples of different nature, e.g. some central extensions of free groups or of fundamental groups of closed orientable surfaces. As a corollary, we get that both non-abelian free groups and fundamental groups of closed orientable surfaces of genus [Formula: see text], expanded by predicates for all subsets, have this property, too. We also obtain a variant of the example of Conversano and Pillay for [Formula: see text] instead of [Formula: see text], which (as most of our examples) was not accessible by the previously known methods.

scholarly journals Free quotients of fundamental groups of smooth quasi-projective varieties

2021 ◽  
pp. 1-23
Author(s):  
Jose I. Cogolludo ◽  
Anatoly Libgober
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Fundamental Groups ◽  
Projective Varieties ◽  
Large Rank ◽  
Small Rank ◽  
Irreducible Components ◽  
Simply Connected

Abstract We study the fundamental groups of the complements to curves on simply connected surfaces, admitting non-abelian free groups as their quotients. We show that given a subset of the Néron–Severi group of such a surface, there are only finitely many classes of equisingular isotopy of curves with irreducible components belonging to this subset for which the fundamental groups of the complement admit surjections onto a free group of a given sufficiently large rank. Examples of subsets of the Néron–Severi group are given with infinitely many isotopy classes of curves with irreducible components from such a subset and fundamental groups of the complements admitting surjections on a free group only of a small rank.

scholarly journals An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements

2003 ◽  
Vol 55 (4) ◽  
pp. 822-838 ◽  
Author(s):  
Djun Maximilian Kim ◽  
Dale Rolfsen
Keyword(s):  
Fibre Type ◽  
Free Groups ◽  
Braid Groups ◽  
Hyperplane Arrangements ◽  
The Other ◽  
Fundamental Groups ◽  
Magnus Expansion ◽  
Direct Generalization ◽  
Complex Hyperplane

AbstractWe define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic.By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.

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.

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