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1. Borel and Volume Classes for Dense Representations of Discrete Groups

   https://doi.org/10.1093/imrn/rnab078  

   Farre, James (April 2021, International Mathematics Research Notices)

   Abstract We show that the bounded Borel class of any dense representation $$\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $$G$$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $$\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$$ is uniformly separated in semi-norm from any other representation $$\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$$ for which there is a subgroup $$H\le G$$ on which $$\rho $$ is still dense but $$\rho ^{\prime}$$ is discrete or indiscrete but stabilizes a point, line, or plane in $${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of $${\operatorname{PSL}}_2{\mathbb{R}}$$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties. 

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2. The cohomology of semi-infinite Deligne–Lusztig varieties

   https://doi.org/10.1515/crelle-2019-0039  

   Chan, Charlotte (January 2020, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes X h {X_{h}} . Boyarchenko’s two conjectures are on the maximality of X h {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant 1 / n {1/n} in the case h = 2 {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of X h {X_{h}} attains its Weil–Deligne bound, so that the cohomology of X h {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group H c i ⁢ ( X h ) {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence θ ↦ H c i ⁢ ( X h ) ⁢ [ θ ] {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p -adic groups in general. 

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3. Reciprocity maps with restricted ramification

   Sharifi, Romyar T. (August 2022, Transactions of the American Mathematical Society)

   We compare two maps that arise in study of the cohomology of global fields with ramification restricted to a finite set S of primes. One of these maps, which we call an S-reciprocity map, interpolates the values of cup products in S-ramified cohomology. In the case of p-ramified cohomology of the pth cyclotomic field for an odd prime p, we use this to exhibit an intriguing relationship between particular values of the cup product on cyclotomic p-units. We then consider higher analogues of the S-reciprocity map and relate their cokernels to the graded quotients in augmentation filtrations of Iwasawa modules. 

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4. The eleventh cohomology group of $$\overline {\mathcal {M}}_{g,n}$$

   https://doi.org/10.1017/fms.2023.59  

   Canning, Samir; Larson, Hannah; Payne, Sam (January 2023, Forum of Mathematics, Sigma)

   Abstract We prove that the rational cohomology group$$H^{11}(\overline {\mathcal {M}}_{g,n})$$vanishes unless$$g = 1$$and$$n \geq 11$$. We show furthermore that$$H^k(\overline {\mathcal {M}}_{g,n})$$is pure Hodge–Tate for all even$$k \leq 12$$and deduce that$$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$$is surprisingly well approximated by a polynomial inq. In addition, we use$$H^{11}(\overline {\mathcal {M}}_{1,11})$$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology. 

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5. The cohomology of Torelli groups is algebraic

   https://doi.org/10.1017/fms.2020.41  

   Kupers, Alexander; Randal-Williams, Oscar (January 2020, Forum of Mathematics, Sigma)

   Abstract The Torelli group of $$W_g = \#^g S^n \times S^n$$ is the group of diffeomorphisms of $$W_g$$ fixing a disc that act trivially on $$H_n(W_g;\mathbb{Z} )$$ . The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $$\text{Sp}_{2g}(\mathbb{Z} )$$ or $$\text{O}_{g,g}(\mathbb{Z} )$$ . In this article we prove that for $$2n \geq 6$$ and $$g \geq 2$$ , they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent. 

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