A finite set of generators for the homeotopy group of a non-orientable surface

1969 ◽  
Vol 65 (2) ◽  
pp. 409-430 ◽  
Author(s):  
D. R. J. Chillingworth
Keyword(s):  
Normal Subgroup ◽  
Quotient Group ◽  
Closed Surface ◽  
Orientable Surface ◽  
Finite Set

Let X be a closed surface, i.e. a compact connected 2-manifold without boundary. If Gx denotes the group of all homeomorphisms of X to itself, and Nx is the normal subgroup consisting of homeomorphisms which are isotopic to the identity, then the quotient group Gx/Nx is called the homeotopy group of X and is denoted by ∧x.

EXHAUSTION OF THE CURVE GRAPH VIA RIGID EXPANSIONS

2018 ◽  
Vol 61 (1) ◽  
pp. 195-230 ◽  
Author(s):  
JESÚS HERNÁNDEZ HERNÁNDEZ
Keyword(s):  
Topological Type ◽  
Orientable Surface ◽  
Finite Topological Type ◽  
Curve Graph ◽  
Finite Set ◽  
Rigid Set

AbstractFor an orientable surfaceSof finite topological type with genusg≥ 3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph$\mathcal{C}$(S). The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set in Aramayona and Leininger,J. Topology Anal.5(2) (2013), 183–203 and Aramayona and Leininger,Pac. J. Math.282(2) (2016), 257–283, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.

C-Valuations and Normal C-Orderings

1989 ◽  
Vol 41 (1) ◽  
pp. 14-67 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Division Ring ◽  
Maximal Ideal ◽  
Quotient Group ◽  
Valuation Ring ◽  
Multiplicative Group ◽  
Invertible Elements ◽  
Natural Way

Let D stand for a division ring (or skewfield), let G stand for an ordered abelian group with positive infinity adjoined, and let ω: D → G. We call to a valuation of D with value group G, if ω is an onto mapping from D to G such that(i) ω(x) = ∞ if and only if x = 0,(ii) ω(x1 + x2) = min(ω (x1), ω (x2)), and(iii) ω (x1 x2) = ω (x1) + ω (x2).Associated to the valuation ω are its valuation ringR = ﹛x ∈ Dω(x) ≧ 0﹜,its maximal idealJ = ﹛x ∈ |ω(x) > 0﹜, and its residue division ring D = R/J.The invertible elements of the ring R are called valuation units. Clearly R and, hence, J are preserved under conjugation so that 1 + J is also preserved under conjugation. The latter is thus a normal subgroup of the multiplicative group Dm of D and hence, the quotient group D˙/1 + J makes sense (the residue group of ω). It enlarges in a natural way the residue division ring D (0 excluded, and addition “forgotten“).

scholarly journals ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

2020 ◽  
pp. 1-10
Author(s):  
MARK GRANT ◽  
AGATA SIENICKA
Keyword(s):  
Braid Group ◽  
Class Group ◽  
Closed Surface ◽  
Orientable Surface ◽  
Mapping Class ◽  
Closed Orientable Surface ◽  
Outer Action ◽  
Positive Genus ◽  
Closed Sphere ◽  
Group Modulo

Abstract The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

scholarly journals Limit sets of Teichmüller geodesics with minimal nonuniquely ergodic vertical foliation, II

2020 ◽  
Vol 2020 (758) ◽  
pp. 1-66
Author(s):  
Jeffrey Brock ◽  
Christopher Leininger ◽  
Babak Modami ◽  
Kasra Rafi
Keyword(s):  
Closed Surface ◽  
Maximal Dimension ◽  
Curve Complex ◽  
Limit Sets ◽  
Ergodic Measures ◽  
Finite Set ◽  
Geodesic Rays ◽  
Thurston Boundary ◽  
Ending Lamination ◽  
Gromov Boundary

AbstractGiven a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus g for which the space of measures has the maximal dimension {3g-3}, for example.We also study the limit sets in the Thurston boundary of Teichmüller geodesic rays defined by quadratic differentials whose vertical foliations are obtained from the constructions mentioned above. We prove that such examples exist for which the limit is a cycle in the 1-skeleton of the simplex of projective classes of measures visiting every vertex.

On the homeotopy group of a non-orientable surface

1972 ◽  
Vol 71 (3) ◽  
pp. 437-448 ◽  
Author(s):  
Joan S. Birman ◽  
D. R. J. Chillingworth
Keyword(s):  
Projective Planes ◽  
Double Cover ◽  
Orientable Surface ◽  
Factor Group ◽  
Simple Type ◽  
Defining Relations ◽  
Entire Structure ◽  
Finite Set ◽  
Orientable Surfaces ◽  
Made In

Let X be a closed, compact connected 2-manifold (a surface), which we will denote by O or N if we wish to stress that X is orientable or non-orientable. Let G(X) denote the group of all homeomorphisms X → X, D(X) the normal subgroup of homeomorphisms isotopic to the identity, and H(X) the factor group G(X)/D(X), i.e. the homeotopy group of X. The problem of determining generators for H(O) was considered by Lickorish in (7, 8), and the second of these papers specifies a finite set of generators of a particularly simple type. In (10) and (11) Lickorish considered the analogous problem for non-orientable surfaces, and, using Lickorish's partial results, Chilling-worth (4) determined a finite set of generators for H(N). While the generators obtained for H(O) and H(N) were strikingly similar, it was noteworthy that the techniques used in the two cases were different, and in particular that little use was made in the non-orientable case of the earlier results obtained on the orientable case. The purpose of this note is to show that the results of Lickorish and Chillingworth on non-orientable surfaces follow rather easily from the work in (7, 8) by an application of some ideas from the theory of covering spaces (2). Moreover, while Lickorish and Chillingworth sought only to find generators, we are able to show (Theorem 1) how in fact the entire structure of the group H(N) is determined by H(O), where O is an orientable double cover of N. Finally, we are able to determine defining relations for H(N) for the case where N is the connected sum of 3 projective planes (Theorem 3).

Groupes géométriques de rang de Morley fini

2008 ◽  
Vol 7 (4) ◽  
pp. 751-792 ◽  
Author(s):  
Olivier Frécon
Keyword(s):  
Normal Subgroup ◽  
Large Class ◽  
Quotient Group ◽  
Characteristic Subgroup ◽  
Morley Rank ◽  
Connected Group ◽  
Finite Morley Rank ◽  
Linearity Conjecture

AbstractWe consider a new subgroup In(G) in any group G of finite Morley rank. This definably characteristic subgroup is the smallest normal subgroup of G from which we can hope to build a geometry over the quotient group G/ In(G). We say that G is a geometric group if In(G) is trivial.This paper is a discussion of a conjecture which states that every geometric group G of finite Morley rank is definably linear over a ring K1 ⊕…⊕ Kn where K1,…,Kn are some interpretable fields. This linearity conjecture seems to generalize the Cherlin–Zil'ber conjecture in a very large class of groups of finite Morley rank.We show that, if this linearity conjecture is true, then there is a Rosenlicht theorem for groups of finite Morley rank, in the sense that the quotient group of any connected group of finite Morley rank by its hypercentre is definably linear.

scholarly journals On Structural Properties of ξ -Complex Fuzzy Sets and Their Applications

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Aneeza Imtiaz ◽  
Umer Shuaib ◽  
Hanan Alolaiyan ◽  
Abdul Razaq ◽  
Muhammad Gulistan
Keyword(s):  
Normal Subgroup ◽  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Quotient Group ◽  
Physical Situation ◽  
The Novel ◽  
Complex Fuzzy Set ◽  
Fuzzy Subgroup ◽  
Fuzzy Subgroups ◽  
Necessary And Sufficient

Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of ξ -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of α , δ -cut sets of ξ -complex fuzzy sets and justify the representation of an ξ -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of ξ -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an ξ -complex fuzzy set is ξ -complex fuzzy subgroup. Furthermore, we extend the idea of ξ -complex fuzzy normal subgroup to define the quotient group of a group G by this particular ξ -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of G by a specific normal subgroup G A ξ .

scholarly journals On the normal structure of a one-point stabilizer in a doubly transitive permutation group

1978 ◽  
Vol 25 (2) ◽  
pp. 145-166
Author(s):  
M. D. Atkinson ◽  
Cheryl E. Praeger
Keyword(s):  
Normal Subgroup ◽  
Permutation Group ◽  
Closed Subgroup ◽  
Normal Structure ◽  
Subject Classification ◽  
Transitive Permutation Group ◽  
Normal Extension ◽  
Extra Information ◽  
Finite Set ◽  
Point Stabilizer

Let G be a doubly transitive permutation group on a finite set Ω, and let Kα be a normal subgroup of the stabilizer Gα of a point α in Ω. If the action of Gα on the set of orbits of Kα in Ω − {α} is 2-primitive with kernel Kα it is shown that either G is a normal extension of PSL(3, q) or Kα ∩ Gγ is a strongly closed subgroup of Gαγ in Gα, where γ ∈ Ω − {α}. If in addition the action of Gα on the set of orbits of Kα is assumed to be 3-transitive, extra information is obtained using permutation theoretic and centralizer ring methods. In the case where Kα has three orbits in Ω − {α} strong restrictions are obtained on either the structure of G or the degrees of certain irreducible characters of G. Subject classification (Amer. Math. Soc. (MOS) 1970: 20 B 20, 20 B 25.

ON THE AUTOMORPHISM GROUP OF A FREE METABELIAN LIE ALGEBRA

2008 ◽  
Vol 18 (02) ◽  
pp. 209-226 ◽  
Author(s):  
VITALY ROMAN'KOV
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Lie Algebra ◽  
Free Subgroup ◽  
Finite Set ◽  
Free Metabelian Lie Algebra ◽  
Inner Automorphisms ◽  
Infinite Set ◽  
Wild Automorphisms

Let K be a field of any characteristic. We prove that a free metabelian Lie algebra M3 of rank 3 over K admits wild automorphisms. Moreover, the subgroup I Aut M3 of all automorphisms identical modulo the derived subalgebra [Formula: see text] cannot be generated by any finite set of IA-automorphisms together with the sets of all inner and all tame IA-automorphisms. In the case if K is finite the group Aut M3 cannot be generated by any finite set of automorphisms together with the sets of all tame, all inner automorphisms and all one-row automorphisms. We present an infinite set of wild IA-automorphisms of M3 which generates a free subgroup F∞ modulo normal subgroup generated by all tame, all inner and all one-row automorphisms of M3.

scholarly journals Almost congruence extension property for subgroups of free groups

2018 ◽  
Vol 21 (1) ◽  
pp. 125-146
Author(s):  
Lev Glebsky ◽  
Nevarez Nieto Saul
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Free Groups ◽  
Extension Property ◽  
Normal Closure ◽  
Sufficient Condition ◽  
Congruence Extension Property ◽  
Finitely Generated ◽  
Finite Set

AbstractLetHbe a subgroup ofFand{\langle\kern-1.422638pt\langle H\rangle\kern-1.422638pt\rangle_{F}}the normal closure ofHinF. We say thatHhas the Almost Congruence Extension Property (ACEP) inFif there is a finite set of nontrivial elements{\digamma\subset H}such that for any normal subgroupNofHone has{H\cap\langle\kern-1.422638pt\langle N\rangle\kern-1.422638pt\rangle_{F}=N}whenever{N\cap\digamma=\emptyset}. In this paper, we provide a sufficient condition for a subgroup of a free group to not possess ACEP. It also shows that any finitely generated subgroup of a free group satisfies some generalization of ACEP.

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