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Stability in the high-dimensional cohomology of congruence subgroups

Abstract

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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Jeremy Miller was supported in part by NSF grant DMS-1709726.

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[Ash94]Ash, A., Unstable cohomology of SL (n, O), J. Algebra 167 (1994), 330342; MR 1283290.10.1006/jabr.1994.1188
[AGM02]Ash, A., Gunnells, P. E. and McConnell, M., Cohomology of congruence subgroups of SL4(ℤ), J. Number Theory 94 (2002), 181212; MR 1904968.10.1006/jnth.2001.2730
[AGM08]Ash, A., Gunnells, P. E. and McConnell, M., Cohomology of congruence subgroups of SL(4, ℤ). II, J. Number Theory 128 (2008), 22632274; MR 2394820.10.1016/j.jnt.2007.09.002
[AGM10]Ash, A., Gunnells, P. E. and McConnell, M., Cohomology of congruence subgroups of SL4(ℤ). III, Math. Comp. 79 (2010), 18111831; MR 2630015.10.1090/S0025-5718-10-02331-8
[APS18]Ash, A., Putman, A. and Sam, S. V., Homological vanishing for the Steinberg representation, Compos. Math. 154 (2018), 11111130; MR 3797603.10.1112/S0010437X18007029
[AR79]Ash, A. and Rudolph, L., The modular symbol and continued fractions in higher dimensions, Invent. Math. 55 (1979), 241250; MR 553998.10.1007/BF01406842
[Bor74]Borel, A., Stable real cohomology of arithmetic groups, Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 235272; MR 0387496.10.24033/asens.1269
[BS73]Borel, A. and Serre, J.-P., Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436491; MR 0387495.10.1007/BF02566134
[Bro89]Brown, K. S., Buildings (Springer, New York, 1989); MR 969123.10.1007/978-1-4612-1019-1
[BC07]Brunetti, M. and Ciampella, A., A Priddy-type Koszulness criterion for non-locally finite algebras, Colloq. Math. 109 (2007), 179192; MR 2318516.10.4064/cm109-2-2
[Byk03]Bykovskiĭ, V. A., Generating elements of the annihilating ideal for modular symbols, Funktsional. Anal. i Prilozhen. 37 (2003), 2738; MR 2083229.10.1023/B:FAIA.0000015577.42722.21
[Cal15]Calegari, F., The stable homology of congruence subgroups, Geom. Topol. 19 (2015), 31493191; MR 3447101.10.2140/gt.2015.19.3149
[CS16]Chardin, M. and Symonds, P., Degree bounds on homology and a conjecture of Derksen, Compos. Math. 152 (2016), 20412049; MR 3569999.10.1112/S0010437X16007430
[CE17]Church, T. and Ellenberg, J., Homology of FI-modules, Geom. Topol. 21 (2017), 23732418; MR 3654111.10.2140/gt.2017.21.2373
[CEF15]Church, T., Ellenberg, J. S. and Farb, B., FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), 18331910; MR 3357185.10.1215/00127094-3120274
[CEFN14]Church, T., Ellenberg, J. S., Farb, B. and Nagpal, R., FI-modules over Noetherian rings, Geom. Topol. 18 (2014), 29512984; MR 3285226.10.2140/gt.2014.18.2951
[CFP14]Church, T., Farb, B. and Putman, A., A stability conjecture for the unstable cohomology of SLnℤ, mapping class groups, and Aut(F n), in Algebraic topology: applications and new directions, Contemporary Mathematics, vol. 620 (American Mathematical Society, Providence, RI, 2014), 5570; MR 3290086.
[CFP19]Church, T., Farb, B. and Putman, A., Integrality in the Steinberg module and the top-dimensional cohomology of SLn𝓞K, Amer. J. Math. 141 (2019), 13751419; MR 4011804.10.1353/ajm.2019.0036
[Cha84]Charney, R., On the problem of homology stability for congruence subgroups, Comm. Algebra 12 (1984), 20812123; MR 747219.10.1080/00927878408823099
[CMNR18]Church, T., Miller, J., Nagpal, R. and Reinhold, J., Linear and quadratic ranges in representation stability, Adv. Math. 333 (2018), 140.10.1016/j.aim.2018.05.025
[CP17]Church, T. and Putman, A., The codimension-one cohomology of SLn, Geom. Topol. 21 (2017), 9991032; MR 3626596.10.2140/gt.2017.21.999
[Dja15]Djament, A., De l’homologie stable des groupes de congruence, Preprint (2015), https://hal.archives-ouvertes.fr/hal-01565891v2.
[DSEV+19]Dutour Sikirić, M., Elbaz-Vincent, P., Kupers, A. and Martinet, J., Voronoi complexes in higher dimensions, cohomology of $GL_{N}(Z)$for $N\geqslant 8$and the triviality of $K_{8}(Z)$, Preprint (2019), arXiv:1910.11598.
[DSGG+19]Dutour Sikirić, M., Gangl, H., Gunnells, P. E., Hanke, J., Schürmann, A. and Yasaki, D., On the topological computation of K 4 of the Gaussian and Eisenstein integers, J. Homotopy Relat. Struct. 14 (2019), 281291; MR 3913976.10.1007/s40062-018-0212-8
[GKRW18]Galatius, S., Kupers, A. and Randal-Williams, O., Cellular $E_{k}$-algebras, Preprint (2018),arXiv:1805.07184.
[GL19]Gan, W. L. and Li, L., Linear stable range for homology of congruence subgroups via FI-modules, Selecta Math. (N.S.) 25 (2019), Art. 55 11; MR 3997138.10.1007/s00029-019-0500-0
[Hau78]Hausmann, J.-C., Manifolds with a given homology and fundamental group, Comment. Math. Helv. 53 (1978), 113134; MR 483534.10.1007/BF02566068
[LS76a]Lee, R. and Szczarba, R. H., The group K 3(Z) is cyclic of order forty-eight, Ann. of Math. (2) 104 (1976a), 3160; MR 0442934.10.2307/1971055
[LS76b]Lee, R. and Szczarba, R. H., On the homology and cohomology of congruence subgroups, Invent. Math. 33 (1976b), 1553; MR 0422498.10.1007/BF01425503
[LS78]Lee, R. and Szczarba, R. H., On the torsion in K 4(Z) and K 5(Z), Duke Math. J. 45 (1978), 101129; MR 0491893.10.1215/S0012-7094-78-04508-8
[MPP19]Miller, J., Patzt, P. and Putman, A., On the top dimensional cohomology groups of congruence subgroups of $\text{SL}_{n}(\mathbb{Z})$, Preprint (2019), arXiv:1909.02661.
[MPW19]Miller, J., Patzt, P. and Wilson, J. C. H., Central stability for the homology of congruence subgroups and the second homology of Torelli groups, Adv. Math. 354 (2019), 106740MR 3992366.
[MPWY18]Miller, J., Patzt, P., Wilson, J. C. H. and Yasaki, D., Non-integrality of some Steinberg modules, J. Topol., to appear. Preprint (2018), arXiv:1810.07683.
[Nag19]Nagpal, R., VI-modules in nondescribing characteristic, part I, Algebra Number Theory 13 (2019), 21512189; MR 4039499.10.2140/ant.2019.13.2151
[Par97]Paraschivescu, A., On a generalization of the double coset formula, Duke Math. J. 89 (1997), 18; MR 1458968.10.1215/S0012-7094-97-08901-8
[Pat17]Patzt, P., Central stability homology, Math. Z., to appear. Preprint (2017),arXiv:1704.04128v2.
[Pri70]Priddy, S. B., Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 3960; MR 0265437.10.1090/S0002-9947-1970-0265437-8
[PS19]Putman, A. and Studenmund, D., The dualizing module and top-dimensional cohomology group of $\text{GL}_{n}({\mathcal{O}})$, Preprint (2019), arXiv:1909.01217.
[Put15]Putman, A., Stability in the homology of congruence subgroups, Invent. Math. 202 (2015), 9871027; MR 3425385.10.1007/s00222-015-0581-0
[PS17]Putman, A. and Sam, S. V., Representation stability and finite linear groups, Duke Math. J. 166 (2017), 25212598; MR 3703435.10.1215/00127094-2017-0008
[Qui73]Quillen, D., Finite generation of the groups K i of rings of algebraic integers, in Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash. 1972), Lecture Notes in Mathematics, vol. 341 (Springer, Berlin, 1973), 179198; MR 0349812.
[SS15]Sam, S. V. and Snowden, A., Stability patterns in representation theory, Forum Math. Sigma 3 (2015), e11; MR 3376738.10.1017/fms.2015.10
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Stability in the high-dimensional cohomology of congruence subgroups

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