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On Farrell-Tate cohomology of \(\mathrm{SL}_2\) over \(S\)-integers. (English) Zbl 1447.20017
Summary: In this paper, we provide number-theoretic formulas for Farrell-Tate cohomology for \(\mathrm{SL}_2\) over rings of \(S\)-integers in number fields satisfying a weak regularity assumption. These formulas describe group cohomology above the virtual cohomological dimension, and can be used to study some questions in homology of linear groups.
We expose three applications, to
(I) detection questions for the Quillen conjecture,
(II) the existence of transfers for the Friedlander-Milnor conjecture,
(III) cohomology of \(\mathrm{SL}_2\) over number fields.

MSC:
20G10 Cohomology theory for linear algebraic groups
11F75 Cohomology of arithmetic groups
20J05 Homological methods in group theory
20J06 Cohomology of groups
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[1] Adem, Alejandro; Milgram, R. James, Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften, vol. 309, (2004), Springer-Verlag Berlin, MR2035696 (2004k:20109) · Zbl 1061.20044
[2] Anton, Marian F., On a conjecture of Quillen at the prime 3, J. Pure Appl. Algebra, 144, 1, 1-20, (1999), MR1723188 (2000m:19003) · Zbl 0941.19004
[3] Anton, Marian; Roberts, Joshua, Unstable analogues of the lichtenbaum-Quillen conjecture, (Advances in Mathematics, (2013), Ed. Acad. Române Bucharest), 49-56, MR3203415
[4] Beauville, Arnaud, Finite subgroups of \(\operatorname{PGL}_2(K)\), (Vector Bundles and Complex Geometry, Contemp. Math., vol. 522, (2010), Amer. Math. Soc. Providence, RI), 23-29, MR2681719 (2011h:20096) · Zbl 1218.20030
[5] Bender, Edward A., Classes of matrices over an integral domain, Illinois J. Math., 11, 697-702, (1967), MR0218340 (36 #1427) · Zbl 0153.05303
[6] Brown, Kenneth S., Cohomology of groups, Graduate Texts in Mathematics, vol. 87, (1994), Springer-Verlag New York, Corrected reprint of the 1982 original, MR1324339 (96a:20072)
[7] Busch, Cornelia Minette, Conjugacy classes of p-torsion in symplectic groups over S-integers, New York J. Math., 12, (2006), 169-182 (electronic), MR2242531 (2007i:20068) · Zbl 1118.20045
[8] Conrad, Keith, Ideal classes and matrix conjugation over \(\mathbb{Z}\), unpublished work
[9] Dwyer, William G.; Friedlander, Eric M., Topological models for arithmetic, Topology, 33, 1, 1-24, (1994), MR1259512 (95h:19004) · Zbl 0792.19003
[10] Dwyer, W. G., Exotic cohomology for \(\operatorname{GL}_n(\mathbf{Z} [1 / 2])\), Proc. Amer. Math. Soc., 126, 7, 2159-2167, (1998), MR1443381 (2000a:57092) · Zbl 0894.55005
[11] Henn, Hans-Werner, The cohomology of \(\operatorname{SL}(3, \mathbf{Z} [1 / 2])\), K-Theory, 16, 4, 299-359, (1999), MR1683179 (2000g:20087) · Zbl 0930.20044
[12] Henn, Hans-Werner; Lannes, Jean; Schwartz, Lionel, Localizations of unstable A-modules and equivariant mod p cohomology, Math. Ann., 301, 1, 23-68, (1995), MR1312569 (95k:55036) · Zbl 0869.55016
[13] Knudson, Kevin P., Homology of linear groups, Progress in Mathematics, vol. 193, (2001), Birkhäuser Verlag Basel, MR1807154 (2001j:20070) · Zbl 0997.20045
[14] Krämer, Norbert, Die konjugationsklassenanzahlen der endlichen untergruppen in der norm-eins-gruppe von maximalordnungen in quaternionenalgebren, (1980), Mathematisches Institut, Universität Bonn, (in German)
[15] Latimer, Claiborne G.; MacDuffee, C. C., A correspondence between classes of ideals and classes of matrices, Ann. of Math. (2), 34, 2, 313-316, (1933), MR1503108 · Zbl 0006.29002
[16] Maclachlan, C., Torsion in arithmetic Fuchsian groups, J. Lond. Math. Soc. (2), 73, 1, 14-30, (2006), MR2197368 (2006j:20075) · Zbl 1125.20038
[17] Mislin, Guido, Tate cohomology for arbitrary groups via satellites, Topology Appl., 56, 3, 293-300, (1994), MR1269317 (95c:20072) · Zbl 0810.20040
[18] Mitchell, Stephen A., On the plus construction for \(B \operatorname{GL} \mathbf{Z} [\frac{1}{2}]\) at the prime 2, Math. Z., 209, 2, 205-222, (1992), MR1147814 (93b:55021) · Zbl 0773.55006
[19] Neukirch, Jürgen, Algebraische zahlentheorie, (1992), Springer-Verlag Berlin, (in German) · Zbl 0747.11001
[20] Prestel, Alexander, Die elliptischen fixpunkte der hilbertschen modulgruppen, Math. Ann., 177, 181-209, (1968), (in German), MR0228439 (37 #4019) · Zbl 0159.11302
[21] Quillen, Daniel, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2), Ann. of Math. (2), 94, 573-602, (1971), MR0298694 (45 #7743) · Zbl 0247.57013
[22] Rahm, Alexander D., The homological torsion of \(\operatorname{PSL}_2\) of the imaginary quadratic integers, Trans. Amer. Math. Soc., 365, 3, 1603-1635, (2013), MR3003276 · Zbl 1307.11065
[23] Rahm, Alexander D., Accessing the cohomology of discrete groups above their virtual cohomological dimension, J. Algebra, 404, 152-175, (2014), MR3177890 · Zbl 1296.11050
[24] Rahm, Alexander D.; Wendt, Matthias, A refinement of a conjecture of Quillen, C. R. Math. Acad. Sci. Paris, 353, 9, 779-784, (2015), MR3377672 · Zbl 1325.20040
[25] Schneider, Volker, Die elliptischen fixpunkte zu modulgruppen in quaternionenschiefkörpern, Math. Ann., 217, 1, 29-45, (1975), (in German), MR0384701 (52 #5574) · Zbl 0295.10022
[26] Serre, Jean-Pierre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., 15, 4, 259-331, (1972), (in French), MR0387283 (52 #8126) · Zbl 0235.14012
[27] Wendt, Matthias, Homology of \(\operatorname{SL}_2\) over function fields I: parabolic subcomplexes, J. Reine Angew. Math., 739, 159-205, (2018) · Zbl 1400.20044
[28] Wendt, Matthias, Homology of \(\operatorname{GL}_3\) of function rings of elliptic curves, (2015)
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