scholarly journals Exhausting curve complexes by finite rigid sets on nonorientable surfaces

2022 ◽  
pp. 1-29
Author(s):  
Elmas Irmak
Keyword(s):  
Curve Complex ◽  
Nonorientable Surface ◽  
Nonorientable Surfaces ◽  
Genus Formula ◽  
Curve Complexes

Let [Formula: see text] be a compact, connected, nonorientable surface of genus [Formula: see text] with [Formula: see text] boundary components. Let [Formula: see text] be the curve complex of [Formula: see text]. We prove that if [Formula: see text] or [Formula: see text], then there is an exhaustion of [Formula: see text] by a sequence of finite rigid sets. This improves the author’s result on exhaustion of [Formula: see text] by a sequence of finite superrigid sets.

scholarly journals Roots of Dehn twists on nonorientable surfaces

2019 ◽  
Vol 28 (12) ◽  
pp. 1950077
Author(s):  
Anna Parlak ◽  
Michał Stukow
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Maximal Degree ◽  
Simple Arithmetic ◽  
Nonorientable Surface ◽  
Dehn Twists ◽  
Nonorientable Surfaces ◽  
Genus Formula ◽  
Orientable Surfaces

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist [Formula: see text] about a nonseparating circle [Formula: see text] in the mapping class group [Formula: see text] of a nonorientable surface [Formula: see text] of genus [Formula: see text]. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer [Formula: see text], then for each sufficiently large [Formula: see text], [Formula: see text] has a root of degree [Formula: see text] in [Formula: see text]. Moreover, for any possible degree [Formula: see text], we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in [Formula: see text].

scholarly journals Automorphisms of curve complexes on nonorientable surfaces

2014 ◽  
Vol 8 (1) ◽  
pp. 39-68 ◽  
Author(s):  
Ferihe Atalan ◽  
Mustafa Korkmaz
Keyword(s):  
Nonorientable Surfaces ◽  
Curve Complexes

scholarly journals Finite rigid sets in curve complexes of nonorientable surfaces

2019 ◽  
Vol 206 (1) ◽  
pp. 83-103 ◽  
Author(s):  
Sabahattin Ilbira ◽  
Mustafa Korkmaz
Keyword(s):  
Nonorientable Surfaces ◽  
Curve Complexes

scholarly journals FINITE RIGID SETS IN CURVE COMPLEXES

2013 ◽  
Vol 05 (02) ◽  
pp. 183-203 ◽  
Author(s):  
JAVIER ARAMAYONA ◽  
CHRISTOPHER J. LEININGER
Keyword(s):  
Class Group ◽  
Topological Type ◽  
Orientable Surface ◽  
Mapping Class ◽  
Curve Complex ◽  
Finite Topological Type ◽  
Punctured Torus ◽  
Curve Complexes ◽  
Simplicial Map ◽  
Pointwise Stabilizer

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex 𝔛 of the curve complex [Formula: see text] such that every locally injective simplicial map [Formula: see text] is the restriction of an element of [Formula: see text], unique up to the (finite) pointwise stabilizer of 𝔛 in [Formula: see text]. Furthermore, if S is not a twice-punctured torus, then we can replace [Formula: see text] in this statement with the extended mapping class group Mod ±(S).

scholarly journals About Structure of Graph Obstructions for Klein Surface with 9 Vertices

2020 ◽  
pp. 65-86
Author(s):  
V.I. Petrenjuk ◽  
D.A. Petrenjuk
Keyword(s):  
Structural Properties ◽  
Graph Structure ◽  
Klein Surface ◽  
Oriented Surface ◽  
Nonorientable Surface ◽  
Nonorientable Surfaces ◽  
Surface Graph ◽  
Genus 2 ◽  
Minimal Embedding ◽  
Structure Graph

The structure of the 9 vertex obstructive graphs for the nonorientable surface of the genus 2 is established by the method of j-transformations of the graphs. The problem of establishing the structural properties of 9 vertex obstruction graphs for the surface of the undirected genus 2 by the method of j-transformation of graphs is considered. The article has an introduction and 5 sections. The introduction contains the main definitions, which are illustrated, to some extent, in Section 1, which provides several statements about their properties. Sections 2 – 4 investigate the structural properties of 9 vertex obstruction graphs for an undirected surface by presenting as a j-image of several graphs homeomorphic to one of the Kuratovsky graphs and at least one planar or projective-planar graph. Section 5 contains a new version of the proof of the statement about the peculiarities of the minimal embeddings of finite graphs in nonorientable surfaces, namely, that, in contrast to oriented surfaces, cell boundaries do not contain repeated edges. Also in section 5 the other properties peculiar to embeddings of graphs to non-oriented surfaces and the main result are given. The main result is Theorem 1. Each obstruction graph H for a non-oriented surface N2 of genus 2 satisfies the following. 1. An arbitrary edge u,u = (a,b) is placed on the Mebius strip by some minimal embedding of the graph H in N3 and there exists a locally projective-planar subgraph K of the graph H \ u which satisfies the condition: (tK({a,b},N3)=1)˄(tK\u({a,b},N2)=2), where tK({a,b},N) is the number of reachability of the set {a,b} on the nonorientable surface N; 2. There exists the smallest inclusion of many different subgraphs Ki of a 2-connected graph H homeomorphic to the graph K+e, where K is a locally planar subgraph of the graph H (at least K+e is homemorphic to K5 or K3,3), which covers the set of edges of the graph H. Keywords: graph, Klein surface, graph structure, graph obstruction, non-oriented surface, Möbius strip.

scholarly journals OUTER AUTOMORPHISMS OF MAPPING CLASS GROUPS OF NONORIENTABLE SURFACES

2010 ◽  
Vol 20 (03) ◽  
pp. 437-456 ◽  
Author(s):  
FERIHE ATALAN
Keyword(s):  
Automorphism Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Outer Automorphism ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Nonorientable Surface ◽  
Nonorientable Surfaces ◽  
Outer Automorphisms

Let Ng be the connected closed nonorientable surface of genus g ≥ 5 and Mod (Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod (Ng) is cyclic.

scholarly journals Distribution of points on Abelian covers over finite fields

2018 ◽  
Vol 14 (05) ◽  
pp. 1375-1401 ◽  
Author(s):  
Patrick Meisner
Keyword(s):  
Finite Fields ◽  
Galois Extension ◽  
Random Variables ◽  
Families Of Curves ◽  
Abelian Covers ◽  
Distribution Of Points ◽  
Genus Formula

We determine in this paper the distribution of the number of points on the covers of [Formula: see text] such that [Formula: see text] is a Galois extension and [Formula: see text] is abelian when [Formula: see text] is fixed and the genus, [Formula: see text], tends to infinity. This generalizes the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over [Formula: see text]. In all cases, the distribution is given by a sum of [Formula: see text] random variables.

Cuspidality in higher genus

2016 ◽  
Vol 12 (08) ◽  
pp. 2043-2060
Author(s):  
Dania Zantout
Keyword(s):  
Linear Operator ◽  
Half Space ◽  
Modular Forms ◽  
Cusp Forms ◽  
General Notion ◽  
Global Operator ◽  
Higher Genus ◽  
Genus Formula ◽  
Holomorphic Modular Forms

We define a global linear operator that projects holomorphic modular forms defined on the Siegel upper half space of genus [Formula: see text] to all the rational boundaries of lower degrees. This global operator reduces to Siegel's [Formula: see text] operator when considering only the maximal standard cusps of degree [Formula: see text]. One advantage of this generalization is that it allows us to give a general notion of cusp forms in genus [Formula: see text] and to bridge this new notion with the classical one found in the literature.

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