On certain Subgroups of the fundamental group of a closed surface

1970 ◽  
Vol 67 (1) ◽  
pp. 17-18 ◽  
Author(s):  
William Jaco
Keyword(s):  
Fundamental Group ◽  
Closed Surface ◽  
The Other

1. Introduction. Although Corollaries 4 and 5 to Theorem 1 below appear elsewhere in the literature (1, 2), the proofs given seem to use rather long and involved arguments and refer to other results in the literature for their completeness. The proofs given below are brief and follow quite naturally in sequence with the other corollaries from Theorem 1. The arguments presented are independent of references to the literature except for the reference in the proof of Theorem 1 to Lemma 2·1 of (3), which is well known to topologists. For our purposes this theorem may be stated as: For each open 2-manifold M (non-compact and without boundary), there is a subcomplex L made up of edges of some triangulation of M such that the open simplicial neighbourhood of L is piecewise linearly homeomorphic to M.

scholarly journals Non-injective representations of a closed surface group into PSL(2, )

2007 ◽  
Vol 142 (2) ◽  
pp. 289-304 ◽  
Author(s):  
LOUIS FUNAR ◽  
MAXIME WOLFF
Keyword(s):  
Fundamental Group ◽  
Euler Class ◽  
Discrete Subgroup ◽  
Surface Group ◽  
Closed Surface ◽  
The Other ◽  
Connected Components ◽  
Discrete Representation ◽  
Discrete Representations

AbstractLet e denote the Euler class on the space ${\text{\rm Hom}(\Gamma_g,{PSL(2,{\mathbb R})}}$ of representations of the fundamental group Γg of the closed surface Σg of genus g. Goldman showed that the connected components of ${\text{\rm Hom}(\Gamma_g,{PSL(2,{\mathbb R})}}$ are precisely the inverse images e−1(k), for 2−2g≤ k≤ 2g−2, and that the components of Euler class 2−2g and 2g−2 consist of the injective representations whose image is a discrete subgroup of ${PSL(2,{\mathbb R})}$. We prove that non-faithful representations are dense in all the other components. We show that the image of a discrete representation essentially determines its Euler class. Moreover, we show that for every genus and possible corresponding Euler class, there exist discrete representations.

scholarly journals A Lemma on Continuous Functions

1960 ◽  
Vol 3 (2) ◽  
pp. 186-187
Author(s):  
J. Lipman
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Closed Interval ◽  
Continuous Functions ◽  
The Other ◽  
Special Consideration ◽  
Boundary Lines

The point of this note is to get a lemma which is useful in treating homotopy between paths in a topological space [1].As explained in the reference, two paths joining a given pair of points in a space E are homotopic if there exists a mapping F: I x I →E (I being the closed interval [0,1] ) which deforms one path continuously into the other. In practice, when two paths are homotopic and the mapping F is constructed, then the verification of all its required properties, with the possible exception of continuity, is trivial. The snag occurs when F is a combination of two or three functions on different subsets of I x I. Then the boundary lines between these subsets have to be given special consideration, and although the problems resulting are routine their disposal can involve some tedious calculation and repetition. In the development [l] of the fundamental group of a space, for example, this sort of situation comes up four or five times.

Various Generalizations and Deformations of PSL(2,ℝ) Surface Group Representations and their Higgs Bundles

2018 ◽  
pp. 471-502
Author(s):  
Brian Collier
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Surface Group ◽  
Closed Surface ◽  
Connected Components ◽  
Character Variety ◽  
Higgs Bundles ◽  
Group Representations ◽  
Symmetric Powers ◽  
Higher Rank

The goal of this chapter is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus g into PSL(2, R) and their associated Higgs bundles generalize to the higher-rank groups PSL(n, R), PSp(2n, R), SO0(2, n), SO0(n,n+1) and PU(n, n). For the SO0(n,n+1)-character variety, it parameterises n(2g−2) new connected components as the total spaces of vector bundles over appropriate symmetric powers of the surface, and shows how these components deform in the character variety. This generalizes results of Hitchin for PSL(2, R).

Curvature-torsion quasitensor of Laptev fundamental-group connection

2020 ◽  
pp. 155-169
Author(s):  
Yu. I. Shevchenko
Keyword(s):  
Fundamental Group ◽  
Curvature Tensor ◽  
Principal Bundle ◽  
Torsion Tensor ◽  
Structural Equations ◽  
The Other ◽  
Cartan Connection ◽  
Cartan Connections ◽  
Special Cases ◽  
The Given

We consider a space with Laptev's fundamental group connection generalizing spaces with Cartan connections. Laptev structural equations are reduced to a simpler form. The continuation of the given structural equations made it possible to find differential comparisons for the coeffi­cients in these equations. It is proved that one part of these coefficients forms a tensor, and the other part forms is quasitensor, which justifies the name quasitensor of torsion-curvature for the entire set. From differential congruences for the components of this quasitensor, congruences are ob­tained for the components of the Laptev curvature-torsion tensor, which contains 9 subtensors included in the unreduced structural equations. In two special cases, a space with a fundamental connection is a spa­ce with a Cartan connection, having a quasitensor of torsion-curvature, which contains a quasitensor of torsion. In the reductive case, the space of the Cartan connection is turned into such a principal bundle with connec­tion that has not only a curvature tensor, but also a torsion tensor.

scholarly journals SPHERICAL TRIANGLES AND THE TWO COMPONENTS OF THE SO(3)-CHARACTER SPACE OF THE FUNDAMENTAL GROUP OF A CLOSED SURFACE OF GENUS 2

2011 ◽  
Vol 22 (09) ◽  
pp. 1261-1364 ◽  
Author(s):  
SUHYOUNG CHOI
Keyword(s):  
Fundamental Group ◽  
Closed Surface ◽  
Branch Locus ◽  
Character Space ◽  
Branch Covering ◽  
Higher Genus ◽  
Genus 2 ◽  
Spherical Triangles ◽  
Geometric Techniques ◽  
Future Work

We use geometric techniques to explicitly find the topological structure of the space of SO (3)-representations of the fundamental group of a closed surface of genus two quotient by the conjugation action by SO (3). There are two components of the space. We will describe the topology of both components and describe the corresponding SU (2)-character spaces by parametrizing them by spherical triangles. There is the sixteen to one branch-covering for each component, and the branch locus is a union of two-spheres or two-tori. Along the way, we also describe the topology of both spaces. We will later relate this result to future work into higher-genus cases and the SL (3, ℝ)-representations.

scholarly journals A 3-manifold with a non-subgroup-separable fundamental group

1997 ◽  
Vol 55 (2) ◽  
pp. 261-279
Author(s):  
Saburo Matsumoto
Keyword(s):  
Fundamental Group ◽  
The Other ◽  
Covering Space ◽  
Finite Degree ◽  
Incompressible Surfaces ◽  
Other Hand ◽  
Graph Manifold

We examine a 3-manifold Γ whose fundamental group is known to be non-subgroup-separable (non-LERF). We show that this manifold Γ is a graph manifold and that the subgroup known to be non-separable is not geometric. On the other hand, there are incompressible surfaces immersed in the manifold which do not lift to embeddings in any finite-degree covering space. We then prove that these bad incompressible surfaces must have non-empty boundary.

Classification of one-dimensional attractors of diffeomorphisms of surfaces by means of pseudo-anosov homeomorphisms

2019 ◽  
Vol 485 (2) ◽  
pp. 135-138 ◽  
Author(s):  
V. Z. Grines ◽  
E. D. Kurenkov
Keyword(s):  
Fundamental Group ◽  
The Other ◽  
One Dimensional ◽  
The Given ◽  
Nonwandering Set

In the present paper axiom diffeomorphisms of closed 2-manifolds of genus whose nonwandering set contains perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of the paper is the following. Two diffeomorphisms from the given class are topologically conjugate on attractors if and only if corresponding pseudo-Anosov homeomorphisms are topologically conjugate by means of homeomorphism that maps a certain subset of one pseudo-Anosov map onto the certain subset of the other pseudo-Anosov map.

Complete systems of surfaces in 3-manifolds

1997 ◽  
Vol 122 (1) ◽  
pp. 185-191 ◽  
Author(s):  
FENGCHUN LEI
Keyword(s):  
Finite Number ◽  
Boundary Component ◽  
Pairwise Disjoint ◽  
Complete System ◽  
Closed Surface ◽  
The Other ◽  
Closed Curves ◽  
Orientable Surfaces

A complete system (CS) [Jscr ]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr ] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr ] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr ] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr ] is a CS of surfaces for M, then any CS equivalent to [Jscr ] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds:

scholarly journals $$\pi _1$$ of Miranda moduli spaces of elliptic surfaces

2021 ◽  
Author(s):  
Michael Lönne
Keyword(s):  
Fundamental Group ◽  
Moduli Spaces ◽  
Classical Result ◽  
Projective Line ◽  
The Other ◽  
Fundamental Groups ◽  
Elliptic Surfaces ◽  
Generators And Relations ◽  
Moduli Stacks ◽  
Finite Presentations

AbstractWe give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over $${\mathbf {P}}^1$$ P 1 which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of $$SL_2{{\mathbb {Z}}}$$ S L 2 Z presented in terms of $$\Bigg (\begin{array}{ll} 1&{}1\\ 0&{}1\end{array} \Bigg )$$ ( 1 1 0 1 ) , $$\Bigg (\begin{array}{ll} 1&{}0\\ {-1}&{}1\end{array} \Bigg )$$ ( 1 0 - 1 1 ) on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other.Our approach exploits the natural $${\mathbb {Z}}_2$$ Z 2 -action on Weierstrass curves and the identification of $${\mathbb {Z}}_2$$ Z 2 -fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over $${{\mathbf {P}}}^1$$ P 1 . The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition.

scholarly journals MAPS FROM THE MINIMAL GROPE TO AN ARBITRARY GROPE

2013 ◽  
Vol 23 (03) ◽  
pp. 503-519 ◽  
Author(s):  
MATIJA CENCELJ ◽  
KATSUYA EDA ◽  
ALEŠ VAVPETIČ
Keyword(s):  
Group Theory ◽  
Fundamental Group ◽  
The Other ◽  
Classifying Space ◽  
Continuous Map

We consider open infinite gropes and prove that every continuous map from the minimal grope to another grope is nulhomotopic unless the other grope has a "branch" which is a copy of the minimal grope. Since every grope is the classifying space of its fundamental group, the problem is translated to group theory and a suitable block cancellation of words is used to obtain the result.

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