ON POWERS OF HALF-TWISTS IN M(0, 2n)

2017 ◽  
Vol 60 (2) ◽  
pp. 333-338 ◽  
Author(s):  
GREGOR MASBAUM
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Normal Closure ◽  
Mapping Class ◽  
Infinite Index ◽  
Skein Theory ◽  
Half Twist

AbstractWe use elementary skein theory to prove a version of a result of Stylianakis (Stylianakis, The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere, arXiv:1511.02912) who showed that under mild restrictions on m and n, the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.

scholarly journals The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

2019 ◽  
Vol 11 (02) ◽  
pp. 273-292
Author(s):  
Charalampos Stylianakis
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Mapping Class Group ◽  
Abelian Subgroup ◽  
Infinite Order ◽  
Class Group ◽  
Normal Closure ◽  
Mapping Class ◽  
Infinite Index ◽  
Half Twist

In this paper we show that the normal closure of the [Formula: see text]th power of a half-twist has infinite index in the mapping class group of a punctured sphere if [Formula: see text] is at least five. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the [Formula: see text]th power of a Dehn twist has infinite order, and for some integers [Formula: see text] the quotient contains a free group. As a second corollary we recover a result of Coxeter: the normal closure of the [Formula: see text]th power of a half-twist in the braid group of at least four strands has infinite index. Our method is to reformulate the Jones representation of the mapping class group of a punctured sphere, using the action of Hecke algebras on [Formula: see text]-graphs, as introduced by Kazhdan–Lusztig.

scholarly journals SIMPLY INTERSECTING PAIR MAPS IN THE MAPPING CLASS GROUP

2012 ◽  
Vol 21 (11) ◽  
pp. 1250107
Author(s):  
LEAH R. CHILDERS
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Infinite Index ◽  
Index Subgroup ◽  
Torelli Group ◽  
Topological Characterization

The Torelli group, [Formula: see text], is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying [Formula: see text]: bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on [Formula: see text], while SIP-maps have received relatively little attention until recently, due to an infinite presentation of [Formula: see text] introduced by Putman that uses all three types of elements. We will give a topological characterization of the image of an SIP-map under the Johnson homomorphism and Birman–Craggs–Johnson homomorphism. We will also classify which SIP-maps are in the kernel of these homomorphisms. Then we will look at the subgroup generated by all SIP-maps, SIP (Sg), and show it is an infinite index subgroup of [Formula: see text].

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 The local-to-global property for Morse quasi-geodesics

2021 ◽  
Author(s):  
Jacob Russell ◽  
Davide Spriano ◽  
Hung Cong Tran
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Type Theorem ◽  
Hyperbolic Groups ◽  
Global Property ◽  
Mapping Class ◽  
Relatively Hyperbolic Groups ◽  
Convex Cocompact ◽  
Relatively Hyperbolic ◽  
Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

scholarly journals Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

2018 ◽  
Vol 27 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Paul P. Gustafson
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Fusion Category ◽  
Mapping Class ◽  
Group Representations ◽  
Colored Graphs ◽  
Finite Image ◽  
Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

AUTOMORPHISMS OF BRAID GROUPS ON S2, T2, P2 AND THE KLEIN BOTTLE K

2008 ◽  
Vol 17 (01) ◽  
pp. 47-53 ◽  
Author(s):  
PING ZHANG
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Klein Bottle ◽  
Closed Surface ◽  
Group Extension ◽  
Braid Groups ◽  
Mapping Class ◽  
Group B ◽  
Central Automorphisms

It is shown that for the braid group Bn(M) on a closed surface M of nonnegative Euler characteristic, Out (Bn(M)) is isomorphic to a group extension of the group of central automorphisms of Bn(M) by the extended mapping class group of M, with an explicit and complete description of Aut (Bn(S2)), Aut (Bn(P2)), Out (Bn(S2)) and Out (Bn(P2)).

scholarly journals An Infinite Genus Mapping Class Group and Stable Cohomology

2009 ◽  
Vol 287 (3) ◽  
pp. 787-804 ◽  
Author(s):  
Louis Funar ◽  
Christophe Kapoudjian
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Stable Cohomology

scholarly journals The homology of the stable nonorientable mapping class group

2008 ◽  
Vol 8 (3) ◽  
pp. 1811-1832
Author(s):  
Oscar Randal-Williams
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class
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