scholarly journals Generating the spin mapping class group by Dehn twists

2021 ◽  
Vol 4 ◽  
pp. 1619-1658
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Dehn Twists

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.

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 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.

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.

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