scholarly journals ROOTS OF DEHN TWISTS ABOUT MULTICURVES

2017 ◽  
Vol 60 (3) ◽  
pp. 555-583 ◽  
Author(s):  
KASHYAP RAJEEVSARATHY ◽  
PRAHLAD VAIDYANATHAN
Keyword(s):  
Sufficient Conditions ◽  
Orientable Surface ◽  
Mapping Class ◽  
Finite Collection ◽  
Closed Curves ◽  
Dehn Twists ◽  
Left Handed ◽  
The Individual ◽  
Necessary And Sufficient ◽  
Combinatorial Data

AbstractA multicurve${\mathcal{C}}$ in a closed orientable surface Sg of genus g is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. A left-handed Dehn twist$t_{\mathcal{C}}$about${\mathcal{C}}$ is the product of left-handed Dehn twists about the individual curves in ${\mathcal{C}}$. In this paper, we derive necessary and sufficient conditions for the existence of a root of $t_{\mathcal{C}}$ in the mapping class group Mod(Sg). Using these conditions, we obtain combinatorial data that correspond to roots, and use it to determine upper bounds on the degree of a root. As an application of our theory, we classify all such roots up to conjugacy in Mod(S4). Finally, we establish that no such root can lie in the level m congruence subgroup of Mod(Sg), for m ≥ 3.

scholarly journals FRACTIONAL POWERS OF DEHN TWISTS ABOUT NONSEPARATING CURVES

2013 ◽  
Vol 56 (1) ◽  
pp. 197-210 ◽  
Author(s):  
KASHYAP RAJEEVSARATHY
Keyword(s):  
Sufficient Conditions ◽  
Fractional Power ◽  
Orientable Surface ◽  
Dehn Twist ◽  
Necessary And Sufficient Conditions ◽  
Fractional Powers ◽  
Closed Orientable Surface ◽  
Left Handed ◽  
Complete Listing ◽  
Necessary And Sufficient

AbstractLet Sg be a closed orientable surface of genus g ≥ 2 and C a simple closed nonseparating curve in F. Let tC denote a left-handed Dehn twist about C. A fractional power of tC of exponent ℓ//n is an h ∈ Mod(Sg) such that hn = tCℓ. Unlike a root of a tC, a fractional power h can exchange the sides of C. We derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We show in the side-preserving case that if gcd(ℓ,n) = 1, then h will be isotopic to the ℓth power of an nth root of tC and that n ≤ 2g+1. In general, we show that n ≤ 4g, and that side-preserving fractional powers of exponents 2g//2g+2 and 2g//4g always exist. For a side-exchanging fractional power of exponent ℓ//2n, we show that 2n ≥ 2g+2, and that side-exchanging fractional powers of exponent 2g+2//4g+2 and 4g+1//4g+2 always exist. We give a complete listing of certain side-preserving and side-exchanging fractional powers on S5.

Some Applications of Artamonov-Quillen-Suslin Theorems to Metabelian Inner Rank and Primitivity

1994 ◽  
Vol 46 (2) ◽  
pp. 298-307 ◽  
Author(s):  
C. K. Gupta ◽  
N. D. Gupta ◽  
G. A. Noskov
Keyword(s):  
Metabelian Group ◽  
Sufficient Conditions ◽  
Surface Group ◽  
Maximal Rank ◽  
Orientable Surface ◽  
Surface Groups ◽  
Modular Element ◽  
Free Metabelian Group ◽  
One Relator Groups ◽  
Necessary And Sufficient

AbstractFor any variety of groups, the relative inner rank of a given groupG is defined to be the maximal rank of the -free homomorphic images of G. In this paper we explore metabelian inner ranks of certain one-relator groups. Using the well-known Quillen-Suslin Theorem, in conjunction with an elegant result of Artamonov, we prove that if r is any "Δ-modular" element of the free metabelian group Mn of rank n > 2 then the metabelian inner rank of the quotient group Mn/(r) is at most [n/2]. As a corollary we deduce that the metabelian inner rank of the (orientable) surface group of genus k is precisely k. This extends the corresponding result of Zieschang about the absolute inner ranks of these surface groups. In continuation of some further applications of the Quillen-Suslin Theorem we give necessary and sufficient conditions for a system g = (g1,..., gk) of k elements of a free metabelian group Mn, k ≤ n, to be a part of a basis of Mn. This extends results of Bachmuth and Timoshenko who considered the cases k = n and k < n — 3 respectively.

Dynamic Capabilities and Where to Find Them

2017 ◽  
Vol 29 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Seidali Kurtmollaiev
Keyword(s):  
Dynamic Capabilities ◽  
Sufficient Conditions ◽  
Resource Base ◽  
Firm Level ◽  
Individual Level ◽  
The Status ◽  
The Individual ◽  
High Level ◽  
Necessary And Sufficient ◽  
Intention To Change

Despite its immense popularity, the dynamic capabilities framework faces fierce criticism because of the ambiguous and contradictory interpretations of dynamic capabilities. Especially challenging are the aspects related to the nature of dynamic capabilities and the issue of agency. In an attempt to avoid circular and overlapping definitions, I explicate dynamic capabilities as the regular actions of creating, extending, and modifying an organizational resource base. This implies that the individual’s intention to change the status quo in the organization and the individual’s high level of influence in the organization are necessary and sufficient conditions for dynamic capabilities. This approach overcomes challenges associated with current interpretations of dynamic capabilities, necessarily focusing on the actions and interactions of individuals in organizations. Following the micro-foundations movement, I present a multilevel approach for studying the individual-level causes and the firm-level effects of dynamic capabilities.

scholarly journals Classification of periodic transformations of an orientable surface of genus two

2021 ◽  
Vol 23 (2) ◽  
pp. 147-158
Author(s):  
Denis A. Baranov ◽  
Olga V. Pochinka
Keyword(s):  
Sufficient Conditions ◽  
Orientable Surface ◽  
Topological Conjugacy ◽  
Modular Surface ◽  
Periodic Surface ◽  
Genus 2 ◽  
Genus Two ◽  
Necessary And Sufficient ◽  
Genus One

Abstract. In this paper, we find all admissible topological conjugacy classes of periodic transformations of a two-dimensional surface of genus two. It is proved that there are exactly seventeen pairwise topologically non-conjugate orientation-preserving periodic pretzel transformations. The implementation of all classes by lifting the full characteristics of mappings from a modular surface to a surface of genus two is also presented. The classification results are based on Nielsen’s theory of periodic surface transformations, according to which the topological conjugacy class of any such homeomorphism is completely determined by its characteristic. The complete characteristic carries information about the genus of the modular surface, the ramified bearing surface, the periods of the ramification points and the turns around them. The necessary and sufficient conditions for the admissibility of the complete characteristic are described by Nielsen and for any surface they give a finite number of admissible collections. For surfaces of a small genus, one can compile a complete list of admissible characteristics, which was done by the authors of the work for a surface of genus 2.

scholarly journals What is the I that I am? An Enquiry into the Self

2021 ◽  
Author(s):  
◽  
Jennifer Leigh Gosnell
Keyword(s):  
Sufficient Conditions ◽  
Psychological Disorders ◽  
Necessary And Sufficient Conditions ◽  
The Self ◽  
Multiple Selves ◽  
The Individual ◽  
Necessary And Sufficient

<p>This dissertation is an elucidation of the nature of the self. It consists of two major parts. The first part is an investigation of the necessary and sufficient conditions of the self, appealing to four theses: the Conceivability Thesis, the Equilibrium Thesis, Panpsychism and the Multiple Selves Doctrine and the Locus Thesis. Proponents of these views are examined in detail, including Descartes, Avicenna, Strawson, Parfit and Dennett. The conditions of selfhood are established through an examination of the individual’s perception and how they arrange their perceptions. The second part of the dissertation discusses the influences of the outside or others’ perception of a self, and how this can influence an individual’s own impression of the self. This is considered using as examples the psychological disorders of autism and schizophrenia. The primary aim of this dissertation is to establish criteria for the presence of the self in the individual and to examine some of the ways in which the self can be expressed. Furthermore, this dissertation begins to clarify the importance of the contribution the self makes towards a person’s successful functioning within his/her selected community.</p>

scholarly journals Generating the twist subgroup by involutions

2020 ◽  
pp. 1-24
Author(s):  
Tüli̇n Altunöz ◽  
Mehmetci̇k Pamuk ◽  
Oguz Yildiz
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Index Formula ◽  
Mapping Class ◽  
New Techniques ◽  
Closed Curves ◽  
Nonorientable Surface ◽  
Dehn Twists ◽  
Generating Sets ◽  
Number Of Generators

For a nonorientable surface, the twist subgroup is an index [Formula: see text] subgroup of the mapping class group generated by Dehn twists about two-sided simple closed curves. In this paper, we consider involution generators of the twist subgroup and give generating sets of involutions with smaller number of generators than the ones known in the literature using new techniques for finding involution generators.

Mapping Class Groups

2017 ◽  
Author(s):  
Leah Childers ◽  
Dan Margalit
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Class Groups ◽  
Open Problems ◽  
Dehn Twists ◽  
Group Of Homeomorphisms ◽  
Compact Orientable Surface ◽  
Group Of Symmetries

This chapter considers the mapping class group, the group of symmetries of a surface, and some of its basic properties. It first provides an overview of surfaces and the concept of homeomorphism before giving examples of homeomorphisms and defining the mapping class group as a certain quotient of the group of homeomorphisms of a surface. It then looks at Dehn twists and describes some of the relations they satisfy. It also presents a theorem stating that the mapping class group of a compact orientable surface is generated by Dehn twists and proves it. It concludes with some projects and open problems. The discussion also includes various exercises.

scholarly journals An Appropriate Way to Switch from the Individual Risk Model to the Collective One

1993 ◽  
Vol 23 (1) ◽  
pp. 23-54 ◽  
Author(s):  
S. Kuon ◽  
M. Radtke ◽  
A. Reich
Keyword(s):  
Risk Model ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Individual Risk ◽  
Risk Models ◽  
Approximation Quality ◽  
Collective Risk Model ◽  
Collective Risk ◽  
The Individual ◽  
Necessary And Sufficient

AbstractFor some time now, the convenient and fast calculability of collective risk models using the Panjer-algorithm has been a well-known fact, and indeed practitioners almost always make use of collective risk models in their daily numerical computations. In doing so, a standard link has been preferred for relating such calculations to the underlying heterogeneous risk portfolio and to the approximation of the aggregate claims distribution function in the individual risk model. In this procedure until now, the approximation quality of the collective risk model upon which such calculations are based is unknown.It is proved that the approximation error which arises does not converge to zero if the risk portfolio in question continues to grow. Therefore, necessary and sufficient conditions are derived in order to obtain well-adjusted collective risk models which supply convergent approximations. Moreover, it is shown how in practical situations the previous natural link between the individual and the collective risk model can easily be modified to improve its calculation accuracy. A numerical example elucidates this.

scholarly journals Free products in mapping class groups generated by Dehn twists

1989 ◽  
Vol 31 (2) ◽  
pp. 213-218
Author(s):  
Stephen P. Humphries
Keyword(s):  
Class Group ◽  
Simple Closed Curve ◽  
Plane Curve ◽  
Orientable Surface ◽  
Mapping Class ◽  
Torelli Group ◽  
Free Products ◽  
Dehn Twists ◽  
Curve Singularities ◽  
Vanishing Cycles

Let F be an orientable surface with or without boundary and let M(F) be the mapping class group of F, i.e. the group of isotopy classes of orientation preserving diffeomorphisms of F. To each essential simple closed curve c on F we can associate an element C of M(F) called the Dehn twist about c. We refer the reader to [1] for definitions. It is well known (see [1]) that, at least in the case where F has no more than one boundary component, M(F) is generated by Dehn twists. Further, there are important subgroups of M(F) which are also generated by Dehn twists or simple products of Dehn twists; for example the Torelli group, the kernel of the homology action map M(F)→ Aut(H1(F;Z)) = Sp(H1(F;Z)), where Sp(H1(F;Z)) denotes the symplectic group, is known to be generated by Dehn twists about bounding curves and by “bounding pairs”. See [8] for proofs and definitions. Also Dehn twists crop up as geometric monodromy maps associated to Picard–Lefschetz vanishing cycles for plane curve singularities (see [5]).

scholarly journals Normal closures of powers of Dehn twists in mapping class groups

1992 ◽  
Vol 34 (3) ◽  
pp. 314-317 ◽  
Author(s):  
Stephen P. Humphries
Keyword(s):  
Class Group ◽  
Simple Closed Curve ◽  
Isotopy Class ◽  
Dehn Twist ◽  
Closed Curve ◽  
Mapping Class ◽  
Class Groups ◽  
Closed Curves ◽  
Dehn Twists ◽  
Definition Of

LetF = F(g, n)be an oriented surface of genusg≥1withn<2boundary components and letM(F)be its mapping class group. ThenM(F)is generated by Dehn twists about a finite number of non-bounding simple closed curves inF([6, 5]). See [1] for the definition of a Dehn twist. Letebe a non-bounding simple closed curve inFand letEdenote the isotopy class of the Dehn twist aboute. LetNbe the normal closure ofE2inM(F). In this paper we answer a question of Birman [1, Qu 28 page 219]:Theorem 1.The subgroup N is of finite index in M(F).

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