scholarly journals The Torelli group and congruence subgroups of the mapping class group

2013 ◽  
pp. 169-196 ◽  
Author(s):  
Andrew Putman
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Congruence Subgroups

scholarly journals A lower bound of the distortion of the Torelli group in the mapping class group with boundary components

2017 ◽  
Vol 26 (08) ◽  
pp. 1750049
Author(s):  
Erika Kuno ◽  
Genki Omori
Keyword(s):  
Lower Bound ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Oriented Surface

We prove that the Torelli group of an oriented surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus–Farb–Putman’s techniques. Further we show that the distortion of the Torelli group in the level [Formula: see text] mapping class group is the same as that in the mapping class group.

scholarly journals The Johnson homomorphism and its kernel

2018 ◽  
Vol 2018 (735) ◽  
pp. 109-141 ◽  
Author(s):  
Andrew Putman
Keyword(s):  
Finite Index ◽  
General Result ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Finite Index Subgroups

AbstractWe give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to “subsurface Torelli groups”. Using this, we extend Johnson’s calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.

scholarly journals Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph

2018 ◽  
Vol 2020 (24) ◽  
pp. 9974-9987
Author(s):  
Hyungryul Baik ◽  
Hyunshik Shin
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Mapping Class ◽  
Torelli Group ◽  
Pure Braid Group ◽  
Curve Graph ◽  
Translation Length

Abstract In this paper, we show that the minimal asymptotic translation length of the Torelli group ${\mathcal{I}}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group ${\textrm{Mod}}(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translation length of the pure braid group ${\textrm{PB}}_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group ${\textrm{B}}_n$, which behaves like $1/n^2$.

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].

THE RESHETIKHIN-TURAEV REPRESENTATION OF THE MAPPING CLASS GROUP

1994 ◽  
Vol 03 (04) ◽  
pp. 547-574 ◽  
Author(s):  
GRETCHEN WRIGHT
Keyword(s):  
Mapping Class Group ◽  
Quadratic Forms ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Representation Space ◽  
Torelli Group ◽  
Spin Structures ◽  
Basis Vectors

The Reshetikhin-Turaev representation of the mapping class group of an orientable surface is computed explicitly in the case r = 4. It is then shown that the restriction of this representation to the Torelli group is equal to the sum of the Birman-Craggs homomorphisms. The proof makes use of an explicit correspondence between the basis vectors of the representation space, and the Z/2Z-quadratic forms on the first homology of the surface. This result corresponds to the fact, shown by Kirby and Melvin, that the three-manifold invariant when r = 4 is related to spin structures on the associated four-manifold.

scholarly journals Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

2008 ◽  
Vol 144 (3) ◽  
pp. 651-671 ◽  
Author(s):  
S. MORITA ◽  
R. C. PENNER
Keyword(s):  
Moduli Space ◽  
Mapping Class Group ◽  
Class Group ◽  
Main Tool ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Teichmuller Space ◽  
Moduli Space Of Curves ◽  
Torelli Group ◽  
Invariant Ideal

AbstractInfinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmüller space. The main tool for these results is the known mapping class group invariant ideal cell decomposition of the Teichmüller space.This new 1-cocycle is mapping class group equivariant, so various contractions of its powers yield various combinatorial (co)cycles of the moduli space of curves, which are also new. Our combinatorial construction can be related to former works of Kawazumi and the first-named author with the consequence that the algebra generated by the cohomology classes represented by the new cocycles is precisely the tautological algebra of the moduli space.There is finally a discussion of prospects for similarly finding cocycle lifts of the higher Johnson homomorphisms.

scholarly journals The cobordism group of homology cylinders

2010 ◽  
Vol 147 (3) ◽  
pp. 914-942 ◽  
Author(s):  
Jae Choon Cha ◽  
Stefan Friedl ◽  
Taehee Kim
Keyword(s):  
Mapping Class Group ◽  
Betti Number ◽  
Boundary Component ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Cobordism Group ◽  
Infinite Rank ◽  
Homology Cobordism

AbstractGaroufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.

scholarly journals Factoring in the hyperelliptic Torelli group

2015 ◽  
Vol 159 (2) ◽  
pp. 207-217 ◽  
Author(s):  
TARA E. BRENDLE ◽  
DAN MARGALIT
Keyword(s):  
Wide Class ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Algorithmic Approach ◽  
Hyperelliptic Involution ◽  
Dehn Twists

AbstractThe hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. Putman and the authors proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic types of elements. As one consequence, we answer an old question of Dennis Johnson.

THE RESHETIKHIN-TURAEV REPRESENTATION OF THE MAPPING CLASS GROUP AT THE SIXTH ROOT OF UNITY

1996 ◽  
Vol 05 (05) ◽  
pp. 721-739 ◽  
Author(s):  
GRETCHEN WRIGHT
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Roots Of Unity ◽  
Topological Invariants ◽  
Torelli Group ◽  
Intersection Pairing ◽  
Root Of Unity ◽  
Genus 2 ◽  
Group Construction

The quantum group construction of Reshetikhin and Turaev provides representations of the mapping class group, indexed by an integer parameter r. This paper presents computations of these representations when r=6, and analyzes their relationship to other topological invariants. It is shown that in genus 2, the representation splits into two summands. The first summand factors through the mapping class group action on the first homology of the surface with Z/3Z coefficients, while the second summand can be analyzed via its restriction to the subgroup of the mapping class group which is normally generated by the sixth power of a Dehn twist on a nonseparating curve. This analysis reveals a connection to the homology intersection pairing on the surface, and also yields information about the kernel and image of the representation. It is also shown that the representation yields a family of 2-dimensional nonabelian representations of the Torelli group. This paper continues the program established by the author in [Wr] to relate the Reshetikhin-Turaev representations at specific roots of unity to classical invariants.

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