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1. Applications of the algebraic geometry of the Putman–Wieland conjecture

   https://doi.org/10.1112/plms.12539  

   Landesman, Aaron; Litt, Daniel (May 2023, Proceedings of the London Mathematical Society)

   Abstract We give two applications of our prior work toward the Putman–Wieland conjecture. First, we deduce a strengthening of a result of Marković–Tošić on virtual mapping class group actions on the homology of covers. Second, let and let be a finite ‐cover of topological surfaces. We show the virtual action of the mapping class group of on an ‐isotypic component of has nonunitary image. 

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2. Virtual algebraic fibrations of surface-by-surface groups and orbits of the mapping class group

   Kropholler, Robert; Vidussi, Stefano; Walsh, Genevieve S (August 2024, Journal of algebra)

   We show that a conjecture of Putman–Wieland, which posits the nonexistence of finite orbits for higher Prym representations of the mapping class group, is equivalent to the existence of surface-by-surface and surface-by-free groups which do not virtually algebraically fiber. While the question about the existence of such groups remains open, we will show that there exist free-by-free and free-by-surface groups which do not algebraically fiber (hence fail to be virtually RFRS) 

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3. On the codimension-two cohomology of SLn(ℤ).

   https://doi.org/10.1016/j.aim.2024.109795  

   Brück, Benjamin; Miller, Jeremy; Patzt, Peter; Sroka, Robin J; Wilson, Jennifer CH (August 2024, Advances in Mathematics)

   Borel–Serre proved that SL_n(Z) is a virtual duality group of dimension (n choose 2) and the Steinberg module St_n(Q) is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to SL_n(Q). We determine the “relations among the relations” of this Steinberg module. That is, we construct an explicit partial resolution of length two of the SL_n(Z)-module St_n(Q). We use this partial resolution to show the codimension-2 rational cohomology group of SLn(Z) vanishes for n ≥ 3. This resolves a case of a conjecture of Church–Farb–Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of SLn(Z). 

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4. Stability in the high-dimensional cohomology of congruence subgroups

   https://doi.org/10.1112/s0010437x20007046  

   Miller, Jeremy; Nagpal, Rohit; Patzt, Peter (April 2020, Compositio Mathematica)

   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 $$\mathbf{SL}_{n}(\mathbb{Z})$$ . 

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5. Generating the homology of covers of surfaces

   https://doi.org/10.1112/blms.13026  

   Boggi, Marco; Putman, Andrew; Salter, Nick (March 2024, Bulletin of the London Mathematical Society)

   Abstract Putman and Wieland conjectured that if is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of under the action of lifts to of mapping classes on are infinite. We prove that this holds if is generated by the homology classes of lifts of simple closed curves on . We also prove that the subspace of spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2‐holed spheres, and we prove that is generated by the homology classes of lifts of loops on lying on subsurfaces homeomorphic to 3‐holed spheres. 

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