scholarly journals The geometry and fundamental groups of solenoid complements

2015 ◽  
Vol 24 (14) ◽  
pp. 1550069 ◽  
Author(s):  
Gregory R. Conner ◽  
Mark Meilstrup ◽  
Dušan Repovš
Keyword(s):  
Fundamental Group ◽  
Inverse Limit ◽  
Abelian Groups ◽  
Inverse Limits ◽  
Fundamental Groups ◽  
Three Space

A solenoid is an inverse limit of circles. When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with nonhomeomorphic complements.

Profinite Modules

1973 ◽  
Vol 16 (3) ◽  
pp. 405-415
Author(s):  
Gerard Elie Cohen
Keyword(s):  
Finite Group ◽  
General Theory ◽  
Finite Length ◽  
Finite Groups ◽  
Inverse Limit ◽  
Abelian Groups ◽  
Point Of View ◽  
Inverse Limits ◽  
Profinite Groups ◽  
Inverse Systems

An inverse limit of finite groups has been called in the literature a pro-finite group and we have extensive studies of profinite groups from the cohomological point of view by J. P. Serre. The general theory of non-abelian modules has not yet been developed and therefore we consider a generalization of profinite abelian groups. We study inverse systems of discrete finite length R-modules. Profinite modules are inverse limits of discrete finite length R-modules with the inverse limit topology.

scholarly journals Topological realizations and fundamental groups of higher-rank graphs

2015 ◽  
Vol 59 (1) ◽  
pp. 143-168 ◽  
Author(s):  
S. Kaliszewski ◽  
Alex Kumjian ◽  
John Quigg ◽  
Aidan Sims
Keyword(s):  
Fundamental Group ◽  
Abelian Groups ◽  
Fundamental Groups ◽  
Crossed Products ◽  
Finitely Generated ◽  
Topological Realization ◽  
Category Equivalence ◽  
Free Abelian Groups ◽  
Higher Rank ◽  
Projective Limits

AbstractWe investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graphΛ, this functor determines a category equivalence between the category of coverings ofΛand the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions fork-graphs: projective limits and crossed products by finitely generated free abelian 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.

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.

scholarly journals Fundamental Group of Simple C*-algebras with Unique Trace III

2012 ◽  
Vol 64 (3) ◽  
pp. 573-587 ◽  
Author(s):  
Norio Nawata
Keyword(s):  
Fundamental Group ◽  
Lower Semicontinuous ◽  
Classification Theorem ◽  
Fundamental Groups ◽  
C Algebras ◽  
Scalar Multiple ◽  
Unique Trace ◽  
Densely Defined

Abstract We introduce the fundamental group ℱ(A) of a simple σ-inital C*-algebra A with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple C*-algebras with Unique Trace I and II by Nawata andWatatani. Our definition in this paper makes sense for stably projectionless C*-algebras. We show that there exist separable stably projectionless C*-algebras such that their fundamental groups are equal to ℝ×+ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

Elementary amenable groups and 4-manifolds with Euler characteristic 0

1991 ◽  
Vol 50 (1) ◽  
pp. 160-170 ◽  
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Euler Characteristic ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Euler Characteristics ◽  
Amenable Groups ◽  
Amenable Subgroup ◽  
Finite Subgroups ◽  
Torsion Free

AbstractWe extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

Finiteness of Isomorphism Classes of Hyperbolic Polycurves with Prescribed Fundamental Groups

2020 ◽  
Author(s):  
Koichiro Sawada
Keyword(s):  
Fundamental Group ◽  
Profinite Group ◽  
Fundamental Groups ◽  
Isomorphism Classes

Abstract In the present paper, we show that there are at most finitely many isomorphism classes of hyperbolic polycurves (i.e., successive extensions of families of hyperbolic curves) over certain types of fields whose étale fundamental group is isomorphic to a prescribed profinite group.

scholarly journals On l-Adic Iterated Integrals, III Galois Actions on Fundamental Groups

2005 ◽  
Vol 178 ◽  
pp. 1-36 ◽  
Author(s):  
Zdzisław Wojtkowiak
Keyword(s):  
Fundamental Group ◽  
Galois Representations ◽  
Fundamental Groups ◽  
Iterated Integrals ◽  
Galois Actions

We continue to study l-adic iterated integrals introduced in the first part. We shall calculate explicitly l-adic logarithm and l-adic polylogarithms. Next we shall use these results to study Galois representations on the fundamental group of .

scholarly journals On Gromov’s conjecture for totally non-spin manifolds

2016 ◽  
Vol 08 (04) ◽  
pp. 571-587
Author(s):  
Dmitry Bolotov ◽  
Alexander Dranishnikov
Keyword(s):  
Fundamental Group ◽  
Scalar Curvature ◽  
Universal Covering ◽  
Positive Scalar Curvature ◽  
Fundamental Groups ◽  
Closed Formula ◽  
Duality Group ◽  
Positive Scalar ◽  
Spin Manifolds ◽  
Stability Conjecture

Gromov’s conjecture states that for a closed [Formula: see text]-manifold [Formula: see text] with positive scalar curvature, the macroscopic dimension of its universal covering [Formula: see text] satisfies the inequality [Formula: see text] [9]. We prove that for totally non-spin [Formula: see text]-manifolds, the inequality [Formula: see text] implies the inequality [Formula: see text]. This implication together with the main result of [6] allows us to prove Gromov’s conjecture for totally non-spin [Formula: see text]-manifolds whose fundamental group is a virtual duality group with [Formula: see text]. In the case of virtually abelian groups, we reduce Gromov’s conjecture for totally non-spin manifolds to the problem whether [Formula: see text]. This problem can be further reduced to the [Formula: see text]-stability conjecture for manifolds with free abelian fundamental groups.

Sign in / Sign up

Export Citation Format

Share Document