scholarly journals Towards representation stability for the second homology of the Torelli group

2012 ◽  
Vol 16 (3) ◽  
pp. 1725-1765 ◽  
Author(s):  
Søren K Boldsen ◽  
Mia Hauge Dollerup
Keyword(s):  
Torelli Group ◽  
Representation Stability

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

scholarly journals A Birman exact sequence for the Torelli subgroup of Aut(Fn)

2016 ◽  
Vol 26 (03) ◽  
pp. 585-617 ◽  
Author(s):  
Matthew Day ◽  
Andrew Putman
Keyword(s):  
Exact Sequence ◽  
Recent Work ◽  
Homology Group ◽  
Entire Group ◽  
Torelli Group

We develop an analogue of the Birman exact sequence for the Torelli subgroup of [Formula: see text]. This builds on earlier work of the authors, who studied an analogue of the Birman exact sequence for the entire group [Formula: see text]. These results play an important role in the authors’ recent work on the second homology group of the Torelli group.

scholarly journals Generating the Torelli group

2012 ◽  
Vol 58 (1) ◽  
pp. 165-188 ◽  
Author(s):  
Allen Hatcher ◽  
Dan Margalit
Keyword(s):  
Torelli Group

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 Stability in the high-dimensional cohomology of congruence subgroups

2020 ◽  
Vol 156 (4) ◽  
pp. 822-861
Author(s):  
Jeremy Miller ◽  
Rohit Nagpal ◽  
Peter Patzt
Keyword(s):  
Congruence Subgroup ◽  
Stability Result ◽  
High Dimensional ◽  
Vanishing Theorem ◽  
Congruence Subgroups ◽  
Codimension One ◽  
Representation Stability ◽  
Steinberg Module ◽  
Finiteness Properties ◽  
Special Case

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

scholarly journals Representation Stability of Power Sets and Square Free Polynomials

2015 ◽  
Vol 67 (5) ◽  
pp. 1024-1045
Author(s):  
Samia Ashraf ◽  
Haniya Azam ◽  
Barbu Berceanu
Keyword(s):  
Symmetric Group ◽  
Braid Group ◽  
Point Of View ◽  
Pure Braid Group ◽  
Representation Stability ◽  
Stability Point ◽  
Power Set ◽  
The Stability

AbstractThe symmetric group 𝓢n acts on the power set 𝓟(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

scholarly journals Homological Stability for Spaces of Commuting Elements in Lie Groups

2020 ◽  
Author(s):  
Daniel A Ramras ◽  
Mentor Stafa
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Stable Range ◽  
Rational Homology ◽  
Infinite Dimensional ◽  
Fixed Group ◽  
Character Varieties ◽  
Representation Stability ◽  
Equivariant Homology ◽  
Homological Stability

Abstract In this paper, we study homological stability for spaces $\textrm{Hom}({{\mathbb{Z}}}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, $\textrm{Comm}(G)$ and $B_{\textrm{com}} G$, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability—in particular, the theory of $\textrm{FI}_W$-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.

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