Geometry & Topology

The stable homology of congruence subgroups

Frank Calegari

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We relate the completed cohomology groups of SLN(OF), where OF is the ring of integers of a number field, to K–theory and Galois cohomology. Various consequences include showing that Borel’s stable classes become infinitely p–divisible up the p–congruence tower if and only if a certain p–adic zeta value is nonzero. We use our results to compute H2(ΓN(p), Fp) (for sufficiently large N), where ΓN(p) is the full level-p congruence subgroup of SLN().

Article information

Geom. Topol., Volume 19, Number 6 (2015), 3149-3191.

Received: 7 November 2013
Revised: 27 December 2014
Accepted: 26 January 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F75: Cohomology of arithmetic groups 19F99: None of the above, but in this section
Secondary: 11F80: Galois representations

arithmetic groups stable homology completed homology $K$–theory


Calegari, Frank. The stable homology of congruence subgroups. Geom. Topol. 19 (2015), no. 6, 3149--3191. doi:10.2140/gt.2015.19.3149.

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  • A Ash, Galois representations attached to mod $p$ cohomology of ${\rm GL}(n,{\bf Z})$, Duke Math. J. 65 (1992) 235–255
  • H Bass, J Milnor, J-P Serre, Solution of the congruence subgroup problem for ${\rm SL}_{n}\,(n\geq 3)$ and ${\rm Sp}_{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. (1967) 59–137
  • A Besser, P Buckingham, R de Jeu, X-F Roblot, On the $p$–adic Beilinson conjecture for number fields, Pure Appl. Math. Q. 5 (2009) 375–434
  • A Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. 7 (1974) 235–272
  • N Boston, J S Ellenberg, Pro-$\!p$ groups and towers of rational homology spheres, Geom. Topol. 10 (2006) 331–334
  • W Browder, J Pakianathan, Cohomology of uniformly powerful $p$–groups, Trans. Amer. Math. Soc. 352 (2000) 2659–2688
  • F Calegari, Irrationality of certain $p$–adic periods for small $p$, Int. Math. Res. Not. 2005 (2005) 1235–1249
  • F Calegari, N M Dunfield, Automorphic forms and rational homology $3$–spheres, Geom. Topol. 10 (2006) 295–329
  • F Calegari, M Emerton, Hecke operators on stable cohomology
  • F Calegari, M Emerton, Mod–$p$ cohomology growth in $p$–adic analytic towers of 3-manifolds, Groups Geom. Dyn. 5 (2011) 355–366
  • F Calegari, M Emerton, Completed cohomology –- a survey, from: “Non-abelian fundamental groups and Iwasawa theory”, (J Coates, M Kim, F Pop, M Saidi, P Schneider, editors), London Math. Soc. Lecture Note Ser. 393, Cambridge Univ. Press (2012) 239–257
  • F Calegari, D Geraghty, Modularity lifting beyond the Taylor–Wiles method
  • F Calegari, A Venkatesh, A torsion Jacquet–Langlands correspondence
  • R M Charney, Homology stability for ${\rm GL}\sb{n}$ of a Dedekind domain, Invent. Math. 56 (1980) 1–17
  • R Charney, On the problem of homology stability for congruence subgroups, Comm. Algebra 12 (1984) 2081–2123
  • T Church, J S Ellenberg, B Farb, FI–modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015) 1833–1910
  • T Church, J S Ellenberg, B Farb, R Nagpal, FI–modules over Noetherian rings, Geom. Topol. 18 (2014) 2951–2984
  • T Church, B Farb, Representation theory and homological stability, Adv. Math. 245 (2013) 250–314
  • H Cohen, H W Lenstra, Jr, Heuristics on class groups of number fields, from: “Number theory, Noordwijkerhout 1983”, (H Jager, editor), Lecture Notes in Math. 1068, Springer, Berlin (1984) 33–62
  • H Darmon, F Diamond, R Taylor, Fermat's last theorem, from: “Elliptic curves, modular forms & Fermat's last theorem”, (J Coates, S T Yau, editors), Int. Press, Cambridge, MA (1997) 2–140
  • P Deligne, Extensions centrales non résiduellement finies de groupes arithmétiques, C. R. Acad. Sci. Paris Sér. A-B 287 (1978) A203–A208
  • L Evens, E M Friedlander, On $K\sb\ast({\bf Z}/p\sp{2}{\bf Z})$ and related homology groups, Trans. Amer. Math. Soc. 270 (1982) 1–46
  • O Gabber, $K\!$–theory of Henselian local rings and Henselian pairs, from: “Algebraic $K\!$–theory, commutative algebra, and algebraic geometry”, (R K Dennis, C Pedrini, M R Stein, editors), Contemp. Math. 126, Amer. Math. Soc. (1992) 59–70
  • R Greenberg, Iwasawa theory for $p$–adic representations, from: “Algebraic number theory”, (J Coates, R Greenberg, B Mazur, I Satake, editors), Adv. Stud. Pure Math. 17, Academic Press, Boston, MA (1989) 97–137
  • L Hesselholt, I Madsen, On the $K\!$–theory of local fields, Ann. of Math. 158 (2003) 1–113
  • M Lazard, Groupes analytiques $p$–adiques, Inst. Hautes Études Sci. Publ. Math. (1965) 389–603
  • R Lee, R H Szczarba, The group $K_{3}(Z)$ is cyclic of order forty-eight, Ann. of Math. 104 (1976) 31–60
  • J McCleary, A user's guide to spectral sequences, 2nd edition, Cambridge Studies in Advanced Mathematics 58, Cambridge Univ. Press (2001)
  • J S Milne, Arithmetic duality theorems, 2nd edition, BookSurge, Charleston, SC (2006)
  • J W Milnor, J C Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965) 211–264
  • J A Neisendorfer, Homotopy groups with coefficients, J. Fixed Point Theory Appl. 8 (2010) 247–338
  • I A Panin, The Hurewicz theorem and $K\!$–theory of complete discrete valuation rings, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) 763–775, 878 In Russian; translated in Math. USSR-Ivz. 29 (1987) 119–131
  • A Putman, Stability in the homology of congruence subgroups
  • D Quillen, On the cohomology and $K\!$–theory of the general linear groups over a finite field, Ann. of Math. 96 (1972) 552–586
  • D Quillen, Finite generation of the groups $K_{i}$ of rings of algebraic integers, from: “Algebraic $K\!$–theory, I: Higher $K\!$–theories”, Lecture Notes in Math. 341, Springer, Berlin (1973) 179–198
  • P Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979) 181–205
  • P Schneider, Über die Werte der Riemannschen Zetafunktion an den ganzzahligen Stellen, J. Reine Angew. Math. 313 (1980) 189–194
  • P Scholze, On torsion in the cohomology of locally symmetric varieties
  • C Soulé, $K\!$–théorie des anneaux d'entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979) 251–295
  • C Soulé, On higher $p$–adic regulators, from: “Algebraic $K\!$–theory”, (E M Friedlander, M R Stein, editors), Lecture Notes in Math. 854, Springer, Berlin (1981) 372–401
  • J Tate, Duality theorems in Galois cohomology over number fields, from: “Proc. Internat. Congr. Mat. (Stockholm, 1962)”, Inst. Mittag-Leffler, Djursholm (1963) 288–295
  • V Voevodsky, On motivic cohomology with $\mathbf{Z}/l$–coefficients, Ann. of Math. 174 (2011) 401–438
  • J B Wagoner, Continuous cohomology and $p$–adic $K\!$–theory, from: “Algebraic $K$\!–theory”, (M R Stein, editor), Lecture Notes in Math. 551, Springer, Berlin (1976) 241–248
  • J B Wagoner, A $p$–adic regulator problem in algebraic $K\!$–theory and group cohomology, Bull. Amer. Math. Soc. 10 (1984) 101–104
  • C Weibel, Algebraic $K\!$–theory of rings of integers in local and global fields, from: “Handbook of $K\!$–theory, Vol. 1, 2”, (E M Friedlander, D R Grayson, editors), Springer, Berlin (2005) 139–190
  • C A Weibel, The $K\!$–book: An introduction to algebraic $K\!$–theory, Graduate Studies in Mathematics 145, Amer. Math. Soc. (2013)
  • A Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990) 493–540
  • A Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. 141 (1995) 443–551
  • E C Zeeman, A note on a theorem of Armand Borel, Proc. Cambridge Philos. Soc. 54 (1958) 396–398