scholarly journals On finite index subgroups of the mapping class group of a nonorientable surface

2014 ◽  
Vol 49 (2) ◽  
pp. 337-350
Author(s):  
Błażej Szepietowski ◽  
Keyword(s):  
Finite Index ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Nonorientable Surface ◽  
Finite Index Subgroups

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 Contractible, hyperbolic but non-CAT(0) complexes

2020 ◽  
Vol 30 (5) ◽  
pp. 1439-1463
Author(s):  
Richard C. H. Webb
Keyword(s):  
Isoperimetric Inequality ◽  
Finite Index ◽  
Mapping Class Group ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Index Subgroup ◽  
Analogous Statement ◽  
Almost All ◽  
Finite Index Subgroup

AbstractWe prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of at least n triangles. On the other hand we show that the curve complexes satisfy a linear combinatorial isoperimetric inequality, which answers a question of Andrew Putman.

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.

scholarly journals A minimal generating set of the level 2 mapping class group of a non-orientable surface

2014 ◽  
Vol 157 (2) ◽  
pp. 345-355
Author(s):  
SUSUMU HIROSE ◽  
MASATOSHI SATO
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Nonorientable Surface ◽  
Generating Set ◽  
Level 2 ◽  
Minimal Generating Set

AbstractWe construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus g, and determine its abelianization for g ≥ 4.

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