We prove that the Steinberg representation of a connected reductive group over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.
A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a sur
face group. We also give a simple representation-theoretic description of the structure of the abelianizations of these commutator subgroups and calculate their homology.
Let $M_n$ be the connect sum of $n$ copies of $S^2 times S^1$. A classical theorem of Laudenbach says that the mapping class group $text{Mod}(M_n)$ is an extension of $text{Out}(F_n)$ by a group $(mathbb{Z}/2)^n$ generated by sphere twists. We prove
that this extension splits, so $text{Mod}(M_n)$ is the semidirect product of $text{Out}(F_n)$ by $(mathbb{Z}/2)^n$, which $text{Out}(F_n)$ acts on via the dual of the natural surjection $text{Out}(F_n) rightarrow text{GL}_n(mathbb{Z}/2)$. Our splitting takes $text{Out}(F_n)$ to the subgroup of $text{Mod}(M_n)$ consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of $M_n$. Our techniques also simplify various aspects of Laudenbachs original proof, including the identification of the twist subgroup with $(mathbb{Z}/2)^n$.
We prove that various subgroups of the mapping class group $Mod(Sigma)$ of a surface $Sigma$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the point-pushing and surface braid subgroups,
and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of $H_1(Sigma;mathbb{Z})$.
For a finite ring $R$, not necessarily commutative, we prove that the category of $text{VIC}(R)$-modules over a left Noetherian ring $mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$. As an appli
cation, we prove a very general twisted homology stability for $text{GL}_n(R)$ with $R$ a finite noncommutative ring.
For a number ring $mathcal{O}$, Borel and Serre proved that $text{SL}_n(mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $text{GL}_n(mathcal{O})$ is a virtual duality group. In contrast to
$text{SL}_n(mathcal{O})$, we prove that the dualizing module of $text{GL}_n(mathcal{O})$ is sometimes the Steinberg module, but sometimes instead is a variant that takes into account a sort of orientation. Using this, we obtain vanishing and nonvanishing theorems for the cohomology of $text{GL}_n(mathcal{O})$ in its virtual cohomological dimension.
We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group
preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.
We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,mathbb{Z})$-module.
We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.
We calculate the abelianizations of the level $L$ subgroup of the genus $g$ mapping class group and the level $L$ congruence subgroup of the $2g times 2g$ symplectic group for $L$ odd and $g geq 3$.
