scholarly journals The second Johnson homomorphism and the second rational cohomology of the Johnson kernel

2007 ◽  
Vol 143 (3) ◽  
pp. 627-648 ◽  
Author(s):  
TAKUYA SAKASAI
Keyword(s):  
Representation Theory ◽  
Mapping Class Group ◽  
Symplectic Group ◽  
Class Group ◽  
Mapping Class ◽  
Closed Curves ◽  
Dehn Twists ◽  
Rational Cohomology ◽  
Cup Products

AbstractThe Johnson kernel is the subgroup of the mapping class group of a surface generated by Dehn twists along bounding simple closed curves, and has the second Johnson homomorphism as a free abelian quotient. In terms of the representation theory of the symplectic group, we give a complete description of cup products of two classes in the first rational cohomology of the Johnson kernel obtained by the rational dual of the second Johnson homomorphism.

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.

Dehn Twists

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Free Group ◽  
Mapping Class Group ◽  
Infinite Order ◽  
Class Group ◽  
Intersection Number ◽  
Mapping Class ◽  
Algebraic Operation ◽  
Closed Curves ◽  
Dehn Twists ◽  
Rank 2

This chapter deals with Dehn twists, the simplest infinite-order elements of Mod(S). It first defines Dehn twists and proves that they are nontrivial elements of the mapping class group. In particular, it considers the action of Dehn twists on simple closed curves. As one application of this study, the chapter proves that if two simple closed curves in Sɡ have geometric intersection number greater than 1, then the associated Dehn twists generate a free group of rank 2 in Mod(S). It also proves some fundamental facts about Dehn twists and describes the center of the mapping class group, along with algebraic relations that can occur between two Dehn twists. Finally, it explores three geometric operations on a surface that each induces an algebraic operation on the corresponding mapping class group: the inclusion homomorphism, the capping homomorphism, and the cutting homomorphism.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

scholarly journals Surface groups, infinite generating sets, and stable commutator length

2019 ◽  
Vol 150 (5) ◽  
pp. 2379-2386
Author(s):  
Dan Margalit ◽  
Andrew Putman
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Surface Group ◽  
Mapping Class ◽  
Surface Groups ◽  
Closed Curves ◽  
Generating Set ◽  
Generating Sets ◽  
Stable Commutator Length ◽  
Commutator Length

AbstractWe give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

scholarly journals Dehn filling Dehn twists

2020 ◽  
pp. 1-24 ◽  
Author(s):  
François Dahmani ◽  
Mark Hagen ◽  
Alessandro Sisto
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Hyperbolic Group ◽  
Low Complexity ◽  
Mapping Class ◽  
Dehn Twists ◽  
Hyperbolic Graph ◽  
Unbounded Orbits ◽  
Cocompact Subgroup ◽  
Sigma G

Abstract Let $\Sigma _{g,p}$ be the genus–g oriented surface with p punctures, with either g > 0 or p > 3. We show that $MCG(\Sigma _{g,p})/DT$ is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group $MCG(\Sigma _{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $\Sigma _{g,p}$ for suitable K. Moreover, we show that in low complexity $MCG(\Sigma _{g,p})/DT$ is in fact hyperbolic. In particular, for 3g − 3 + p ⩽ 2, we show that the mapping class group $MCG(\Sigma _{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some $L^q$ space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of $MCG(\Sigma _{g,p})$ is separable. The aforementioned results follow from general theorems about composite rotating families, in the sense of [13], that come from a collection of subgroups of vertex stabilizers for the action of a group G on a hyperbolic graph X. We give conditions ensuring that the graph X/N is again hyperbolic and various properties of the action of G on X persist for the action of G/N on X/N.

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 On stable commutator lengths of Dehn twists along separating curves

2017 ◽  
Vol 26 (10) ◽  
pp. 1750056 ◽  
Author(s):  
Naoyuki Monden ◽  
Kazuya Yoshihara
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Upper Bounds ◽  
Mapping Class ◽  
Oriented Surface ◽  
Dehn Twists ◽  
Closed Oriented Surface

We give new upper bounds on the stable commutator lengths of Dehn twists along separating curves in the mapping class group of a closed oriented surface. The estimates of these upper bounds are [Formula: see text], where [Formula: see text] is the genus of the surface.

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