Representation varieties of the fundamental groups of non-orientable surfaces

1997 ◽  
Vol 188 (7) ◽  
pp. 997-1039 ◽  
Author(s):  
V V Benyash-Krivets ◽  
V I Chernousov
Keyword(s):  
Fundamental Groups ◽  
Representation Varieties ◽  
Orientable Surfaces

scholarly journals Representation varieties of the fundamental groups of compact orientable surfaces

1996 ◽  
Vol 93 (1) ◽  
pp. 29-71 ◽  
Author(s):  
A. S. Rapinchuk ◽  
V. V. Benyash-Krivetz ◽  
V. I. Chernousov
Keyword(s):  
Fundamental Groups ◽  
Representation Varieties ◽  
Orientable Surfaces

scholarly journals RANK TWO TOPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI OF QUASI-PROJECTIVE MANIFOLDS

2018 ◽  
Vol 19 (2) ◽  
pp. 451-485 ◽  
Author(s):  
Stefan Papadima ◽  
Alexander I. Suciu
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Linear Algebraic Group ◽  
The Other ◽  
Fundamental Groups ◽  
Flat Connections ◽  
Other Hand ◽  
Projective Manifolds ◽  
Representation Varieties

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\text{SL}_{2}(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\text{SL}_{n}(\mathbb{C})$ for some $n\geqslant 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.

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.

scholarly journals Spider evaluation and representations of web groups

2019 ◽  
Vol 28 (04) ◽  
pp. 1950021
Author(s):  
Charles Frohman
Keyword(s):  
Irreducible Component ◽  
Rooted Tree ◽  
Fundamental Groups ◽  
Text Representation ◽  
Poincaré Polynomial ◽  
Representation Variety ◽  
Representation Varieties ◽  
The Web

The topology of [Formula: see text]-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to [Formula: see text] are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate to each geodesic [Formula: see text] from the root of the tree to the tip of a leaf an irreducible component [Formula: see text] of the representation variety of the web, and a graded subalgebra [Formula: see text] of [Formula: see text]. The spider evaluation of geodesic [Formula: see text] is the symmetrized Poincaré polynomial of [Formula: see text]. The spider evaluation of the web is the sum of the symmetrized Poincaré polynomials of the graded subalgebras associated to all maximal geodesics from the root of the tree to the leaves.

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 The Topology of Compact Lie Group Actions Through the Lens of Finite Models

2018 ◽  
Vol 2019 (20) ◽  
pp. 6390-6436 ◽  
Author(s):  
Stefan Papadima ◽  
Alexander I Suciu
Keyword(s):  
Lie Group ◽  
Orbit Space ◽  
Algebraic Model ◽  
Characteristic Classes ◽  
Fundamental Groups ◽  
Algebraic Models ◽  
Sasakian Manifolds ◽  
Finite Dimensional ◽  
Surface Singularities ◽  
Representation Varieties

AbstractGiven a compact, connected Lie group K, we use principal K-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let M be a compact, connected, smooth manifold, which supports an almost free K-action. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe an algebraic model for M with commensurate finiteness and partial formality properties. The existence of such a model has various implications on the structure of the cohomology jump loci of M and of the representation varieties of π1(M). As an application, we show that compact Sasakian manifolds of dimension 2n + 1 are (n − 1)-formal, and that their fundamental groups are filtered-formal. Further applications to the study of weighted-homogeneous isolated surface singularities are also given.

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.

QUALITATIVE CHANGES IN RHYTHMIC GYMNASTICS HOOP THROWS FROM THE PERSPECTIVE OF BIOMECHANICAL ANALYSIS

2020 ◽  
pp. 1-8
Author(s):  
Raluca Tanasa
Keyword(s):  
Correlation Coefficient ◽  
Video Recording ◽  
Biomechanical Analysis ◽  
Fundamental Groups ◽  
Horizontal Distance ◽  
Video Recordings ◽  
Rhythmic Gymnastics ◽  
Female Gymnasts ◽  
Execution Technique ◽  
Qualitative Changes

Throws and catches in rhythmic gymnastics represent one of the fundamental groups of apparatus actuation. They represent for the hoop actions of great showmanship, but also elements of risk. The purpose of this paper is to improve the throw execution technique through biomechanical analysis in order to increase the performance of female gymnasts in competitions. The subjects of this study were 8 gymnasts aged 9-10 years old, practiced performance Rhythmic Gymnastics. The experiment consisted in video recording and the biomechanical analysis of the element “Hoop throw, step jump and catch”. After processing the video recordings using the Simi Motion software, we have calculated and obtained values concerning: launch height, horizontal distance and throwing angle between the arm and the horizontal. Pursuant to the data obtained, we have designed a series of means to improve the execution technique for the elements comprised within the research and we have implemented them in the training process. Regarding the interpretation of the results, it may be highlighted as follows: height and horizontal distance in this element have values of the correlation coefficient of 0.438 and 0.323, thus a mean significance of 0.005. The values of the arm/horizontal angle have improved for all the gymnasts, the correlation coefficient being 0.931, with a significance of 0.01. As a general conclusion, after the results obtained, it may be stated that the means introduced in the experiment have proven their efficacy, which has led to the optimisation of the execution technique, thus confirming the research hypothesis.

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