Let $\mathop{Mod}_{g,n}$ be the mapping class group of a genus-$g$ surface with $n$ punctures and let $\mathcal{C}_{g,n}$ be its curve complex. For a finite-index subgroup $G<\mathop{Mod}_{g,n}$, Boggi proved that $\widetilde{H}_k(\mathcal{C}_{g,n}/G;\mathbb{Q}) = 0$ for $g+n \gg k$. This plays an important role in the work of Putman-Wieland on the virtual first Betti number of $\mathop{Mod}_{g,n}$. The proof of Boggi's theorem uses mixed Hodge theory and is embedded in his flawed paper purporting to show that the mapping class group has the congruence subgroup property. We give a detailed exposition of Boggi's proof aimed at geometric topologists.
For a finite abelian group $G$ acting linearly on the plane let $\mathcal{X}$ be the associated quotient stack. We analyze a fine moduli space which parameterizes zero dimensional substacks of $\mathcal{X}$ whose corresponding subschemes of the plane have a given representation type, denoted by $\mathcal{X}^{[\rho]}$....
We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group...
Let $G$ be a compact, connected, simply connected semisimple Lie group with Weyl group $W$ and simple reflections $S$. For a simplicial complex $\mathcal K$ on $S$, form the homotopy colimit $X_{\mathcal K}(G)=\operatorname*{hocolim}_{I\in\mathcal K}G/G_I$ of standard partial flag manifolds. We compute its integral hom...
The shape of a number field $K$ of degree $n\geq2$ is the similarity class of the lattice obtained by projecting $\mathscr{O}_K$ orthogonally away from the rational line in Minkowski space. It defines a point in the space of shapes of rank-$(n-1)$ lattices, $\mathscr{S}_{n-1}:=\mathrm{GL}_{n-1}(\mathbb{Z})\backslash \m...
Let $p$ be an odd prime and let $\mathbf{G}/\mathbb Z$ be a split connected reductive group with $\operatorname{dim} Z(\mathbf{G})\leq1$. Assume that a split maximal torus of $\mathbf{G}$ admits a cocharacter whose pairing with every simple root is odd, and impose an explicit root-theoretic condition on the reduction o...
For a finite group $G$, define the field $\mathbb{Q}(G) := \mathbb{Q}(\{\chi(g) : \chi \in \operatorname{Irr}(G) , g \in G\})$. In this article we compute generating sets for the fields $\mathbb{Q}(G)$ when $G$ is a finite general or special linear group, a finite general or special unitary group, or one of the simple...
Eden Ketchum· 0 citations
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