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.

UNKNOTTING ORIENTABLE SURFACES IN THE 4-SPHERE

1995 ◽  
Vol 04 (02) ◽  
pp. 213-224 ◽  
Author(s):  
JONATHAN A. HILLMAN ◽  
AKIO KAWAUCHI
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Flat Embedding ◽  
Orientable Surfaces

We show that a topologically locally flat embedding of a closed orientable surface in the 4-sphere is isotopic to one whose image lies in the equatorial 3-sphere if and only if its exterior has an infinite cyclic fundamental group.

scholarly journals Ranks of mapping tori via the curve complex

2019 ◽  
Vol 2019 (748) ◽  
pp. 153-172 ◽  
Author(s):  
Ian Biringer ◽  
Juan Souto
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Curve Complex ◽  
Mapping Torus ◽  
Closed Orientable Surface ◽  
Mapping Tori

Abstract We show that if ϕ is a homeomorphism of a closed, orientable surface of genus g, and ϕ has large translation distance in the curve complex, then the fundamental group of the mapping torus {M_{\phi}} has rank {2g+1} .

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.

scholarly journals Surface groups within Baumslag doubles

2011 ◽  
Vol 54 (1) ◽  
pp. 91-97 ◽  
Author(s):  
Benjamin Fine ◽  
Gerhard Rosenberger
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Group ◽  
Sufficient Conditions ◽  
Surface Group ◽  
Orientable Surface ◽  
Hyperbolic Surface ◽  
Surface Groups ◽  
Word Hyperbolic Group ◽  
Genus 2

AbstractA conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V ∈ F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y2 for some X, Y ∈ F. G can contain no non-orientable surface group of smaller genus.

scholarly journals EXTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE

2019 ◽  
Vol 71 (1) ◽  
pp. 175-196
Author(s):  
Kenta Funayoshi ◽  
Yuya Koda
Keyword(s):  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Genus 2

Abstract An automorphism $f$ of a closed orientable surface $\Sigma $ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, \Sigma )$ with respect to some embedding $\Sigma \hookrightarrow S^3$. We prove that if an automorphism of a genus-2 surface $\Sigma $ is extendable over $S^3$, then $f$ extends to an automorphism of the pair $(S^3, \Sigma )$ with respect to an embedding $\Sigma \hookrightarrow S^3$ such that $\Sigma $ bounds genus-2 handlebodies on both sides. The classification of essential annuli in the exterior of genus-2 handlebodies embedded in $S^3$ due to Ozawa, and the second author plays a key role.

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.

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.

Enumeration of branched coverings of closed orientable surfaces whose branch orders coincide with multiplicity

2007 ◽  
Vol 44 (2) ◽  
pp. 215-223 ◽  
Author(s):  
Jin Kwak ◽  
Alexander Mednykh
Keyword(s):  
Euler Characteristic ◽  
Orientable Surface ◽  
Branch Points ◽  
Closed Orientable Surface ◽  
Branched Coverings ◽  
Orientable Surfaces

The number Nn , g , r of nonisomorphic n -fold branched coverings of a given closed orientable surface S of genus g with r ≧ 1 branch points of order n is determined. The result is given in terms of the Euler characteristic of the surface S with r points removed and the von Sterneck-Ramanujan function ϕ( k,n ) = Σ ( d , n )=1 exp (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{{2\pi ikd}}{n}$$ \end{document}). More precisely, if v = 2 g − 2 + r then \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N_{n,g,r} = \sum\limits_{\ell |n,\ell m = n} {(m!\ell ^m )^\nu } \sum\limits_{k = 1} {\left( {\frac{{\Phi (k,\ell )}}{n}} \right)^r } \sum\limits_{s = 0}^{m - 1} {( - 1)^{sr} \left( {\begin{array}{*{20}c} {m - 1} \\ s \\ \end{array} } \right)^{ - \nu } } .$$ \end{document}.

scholarly journals Enumeration of orientable coverings of a non-orientable manifold

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Jin Ho Kwak ◽  
Alexander Mednykh ◽  
Roman Nedela
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Finitely Generated ◽  
Closed Orientable Surface ◽  
Orientable Manifold ◽  
International Audience

International audience In this paper we solve the known V.A. Liskovets problem (1996) on the enumeration of orientable coverings over a non-orientable manifold with an arbitrary finitely generated fundamental group. As an application we obtain general formulas for the number of chiral and reflexible coverings over the manifold. As a further application, we count the chiral and reflexible maps and hypermaps on a closed orientable surface by the number of edges. Also, by this method the number of self-dual and Petri-dual maps can be determined. This will be done in forthcoming papers by authors.

scholarly journals THE CO-HOPFIAN PROPERTY OF SURFACE BRAID GROUPS

2013 ◽  
Vol 22 (10) ◽  
pp. 1350055 ◽  
Author(s):  
YOSHIKATA KIDA ◽  
SAEKO YAMAGATA
Keyword(s):  
Finite Index ◽  
Braid Group ◽  
Orientable Surface ◽  
Braid Groups ◽  
Index Subgroup ◽  
Closed Orientable Surface ◽  
Pure Braid Group ◽  
Injective Homomorphism ◽  
Finite Index Subgroup

Let g and n be integers at least 2, and let G be the pure braid group with n strands on a closed orientable surface of genus g. We describe any injective homomorphism from a finite index subgroup of G into G. As a consequence, we show that any finite index subgroup of G is co-Hopfian.

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