Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem. (English) Zbl 1290.20039

This is a landmark paper. The authors prove a conjecture that has been around for decades. The precise statement of the rank theorem is as follows. Let \(K\) be a global function field and let \(\mathcal G\) be a linear algebraic group defined over \(K\). Let \(S\) be a finite set of places of \(K\) and let \(\Gamma=\mathcal G(\mathfrak o_S)\) be the group of its \(\mathfrak o_S\)-points. This group is well defined only up to commensurability, but everything said hereafter depends only on the commensurability class of \(\Gamma\). A group \(G\) is called of type \(FP_m\) if it admits a classifying space with finite \(m\)-skeleton. The finiteness length \(\Phi(G)\) of \(G\) is the supremum \(\leq\infty\) of those \(m\) for which \(G\) is of type \(FP_m\). Let \(K_p\) be the completion of \(K\) at \(p\). The local rank \(d_p\) of \(\mathcal G\) at \(p\) is the rank of \(\mathcal G\) over \(K_p\). If \(\mathcal G\) is isotropic and almost simple, the group \(\mathcal G(K_p)\) acts on its associated Bruhat-Tits building \(X_p\) and the dimension of \(X_p\) is the local rank \(d_p\).
Here is the statement of the rank theorem. Let \(\mathcal G\) be a connected non-commutative absolutely almost simple \(K\)-isotropic \(K\)-group. Then the finiteness length of the \(S\)-arithmetic group \(\Gamma=\mathcal G(\mathfrak o_S)\) is \(d-1\), where \(d\) is the sum of the local ranks \(d_p\) for \(p\in S\).
This also determines the finiteness length of \(S\)-arithmetic subgroups of isotropic reductive groups. This result is in sharp contrast with the situation for \(S\)-arithmetic groups over number fields. Here all \(S\)-arithmetic subgroups of reductive groups have infinite finiteness length. This was proved by Raghunathan in 1968 for arithmetic groups, and in 1976 by Borel and Serre for the general case of \(S\)-arithmetic groups. Interest in the positive characteristic case started with the result of Nagao in 1959, who showed that \(\mathrm{SL}(2,\mathbb F_q[t])\) is not finitely generated. Behr showed in 1969 that \(\Gamma\) as in the rank theorem is finitely generated (\(=FP_1\)) if and only if \(d>1\) and in 1998 that \(\Gamma\) is finitely presented (\(=FP_2\)) if and only if \(d>2\). After Serre had noticed in the early seventies that the Bruhat-Tits building can be used to study the homology of \(S\)-arithmetic groups and Behr and Stuhler had shown by examples in the early eighties that the finiteness properties for isotropic groups do not depend on the global rank but on the sum of the local ranks, the “rank theorem” emerged as a conjecture, but was not declared a conjecture in print, at the time. Several special cases had been proved in the meantime.
The proof of the rank theorem given by the authors uses, like most of the proofs of the earlier results, the action of \(\Gamma\) on the product \(X\) of the Bruhat-Tits buildings \(X_p\) and reduction theory, here recast in terms of the metric structure of Euclidean buildings. Another ingredient of the proof is Ken Brown’s criterion which allows to compute finiteness properties of a group from a (nice) action of the group on an (appropriate) filtered CW-complex. Technically speaking, the authors use a combinatorial Morse function on \(X\). Their Morse function is \(\Gamma\)-invariant and has cocompact sublevel sets. The main point in the application of Ken Brown’s criterion is to show that descending links are highly connected. It is not too difficult to show that this is true generically. To prove that this holds for all descending links takes up a large part of the paper, and is achieved by perturbing the initial Morse function in several steps.


20G30 Linear algebraic groups over global fields and their integers
51E24 Buildings and the geometry of diagrams
20F65 Geometric group theory
20E42 Groups with a \(BN\)-pair; buildings
20J05 Homological methods in group theory
11E57 Classical groups
Full Text: DOI arXiv


[1] H. Abels, ”Finiteness properties of certain arithmetic groups in the function field case,” Israel J. Math., vol. 76, iss. 1-2, pp. 113-128, 1991. · Zbl 0819.20051 · doi:10.1007/BF02782847
[2] P. Abramenko, Endlichkeitseigenschaften der Gruppen \({\mathrm{{SL}}}_n({\mathbb{F}}_q[t])\), 1987.
[3] P. Abramenko, Twin Buildings and Applications to \(S\)-Arithmetic Groups, New York: Springer-Verlag, 1996, vol. 1641. · Zbl 0908.20003 · doi:10.1007/BFb0094079
[4] P. Abramenko and K. S. Brown, Buildings: Theory and Applications, New York: Springer-Verlag, 2008, vol. 248. · Zbl 1214.20033 · doi:10.1007/978-0-387-78835-7
[5] J. M. Alonso, ”Finiteness conditions on groups and quasi-isometries,” J. Pure Appl. Algebra, vol. 95, iss. 2, pp. 121-129, 1994. · Zbl 0823.20034 · doi:10.1016/0022-4049(94)90069-8
[6] H. Bass and A. Lubotzky, Tree Lattices, Boston, MA: Birkhäuser, 2001, vol. 176. · Zbl 1053.20026
[7] H. Behr, ”Zur starken Approximation in algebraischen Gruppen über globalen Körpern,” J. Reine Angew. Math., vol. 229, pp. 107-116, 1968. · Zbl 0184.24404 · doi:10.1515/crll.1968.229.107
[8] H. Behr, ”Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern,” Invent. Math., vol. 7, pp. 1-32, 1969. · Zbl 0169.34802 · doi:10.1007/BF01418772
[9] H. Behr, ”Arithmetic groups over function fields. I. A complete characterization of finitely generated and finitely presented arithmetic subgroups of reductive algebraic groups,” J. Reine Angew. Math., vol. 495, pp. 79-118, 1998. · Zbl 0923.20038 · doi:10.1515/crll.1998.023
[10] H. Behr, ”Higher finiteness properties of \(S\)-arithmetic groups in the function field case I,” in Groups: Topological, Combinatorial and Arithmetic Aspects, Cambridge: Cambridge Univ. Press, 2004, vol. 311, pp. 27-42. · Zbl 1099.20022 · doi:10.1017/CBO9780511550706.004
[11] K. Behrend and A. Dhillon, ”Connected components of moduli stacks of torsors via Tamagawa numbers,” Canad. J. Math., vol. 61, iss. 1, pp. 3-28, 2009. · Zbl 1219.14030 · doi:10.4153/CJM-2009-001-5
[12] M. Bestvina, A. Eskin, and K. Wortman, Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups. · Zbl 1372.11055
[13] A. Borel and . J-P. Serre, ”Cohomologie d’immeubles et de groupes \(S\)-arithmétiques,” Topology, vol. 15, iss. 3, pp. 211-232, 1976. · Zbl 0338.20055 · doi:10.1016/0040-9383(76)90037-9
[14] A. Borel and J. Tits, ”Groupes réductifs,” Inst. Hautes Études Sci. Publ. Math., iss. 27, pp. 55-150, 1965. · Zbl 0145.17402 · doi:10.1007/BF02684375
[15] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, New York: Springer-Verlag, 1999, vol. 319. · Zbl 0988.53001
[16] K. S. Brown, ”Finiteness properties of groups. Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985),” J. Pure Appl. Algebra, vol. 44, pp. 45-75, 1987. · Zbl 0613.20033 · doi:10.1016/0022-4049(87)90015-6
[17] K. S. Brown, Buildings, New York: Springer-Verlag, 1989. · Zbl 0715.20017
[18] K. -U. Bux, R. Gramlich, and S. Witzel, Finiteness properties of Chevalley Groups over a Polynomial Ring over a Finite Field, 2009.
[19] K. Bux and K. Wortman, ”Finiteness properties of arithmetic groups over function fields,” Invent. Math., vol. 167, iss. 2, pp. 355-378, 2007. · Zbl 1126.20030 · doi:10.1007/s00222-006-0017-y
[20] K. Bux and K. Wortman, ”Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups,” Invent. Math., vol. 185, iss. 2, pp. 395-419, 2011. · Zbl 1237.20041 · doi:10.1007/s00222-011-0311-1
[21] C. Chevalley, ”Sur certains groupes simples,” Tôhoku Math. J., vol. 7, pp. 14-66, 1955. · Zbl 0066.01503 · doi:10.2748/tmj/1178245104
[22] A. Devillers, R. Gramlich, and B. Mühlherr, ”The sphericity of the complex of non-degenerate subspaces,” J. Lond. Math. Soc., vol. 79, iss. 3, pp. 684-700, 2009. · Zbl 1170.51006 · doi:10.1112/jlms/jdn088
[23] J. Dymara and D. Osajda, ”Boundaries of right-angled hyperbolic buildings,” Fund. Math., vol. 197, pp. 123-165, 2007. · Zbl 1177.20042 · doi:10.4064/fm197-0-6
[24] D. Gaitsgory, Contractibility of the space of rational maps. · Zbl 1263.14013 · doi:10.1007/s00222-012-0392-5
[25] G. Gandini, Bounding the homological finiteness length. · Zbl 1272.20052 · doi:10.1112/blms/bds047
[26] R. Godement, ”Domaines Fondamentaux des Groupes Arithmétiques,” in Séminaire Bourbaki, 1962/63. Fasc. 3, No. 257, Paris: Secrétariat mathématique, 1964, vol. 15. · Zbl 0136.30101
[27] G. Harder, ”Minkowskische Reduktionstheorie über Funktionenkörpern,” Invent. Math., vol. 7, pp. 33-54, 1969. · Zbl 0242.20046 · doi:10.1007/BF01418773
[28] G. Harder, ”Chevalley groups over function fields and automorphic forms,” Ann. of Math., vol. 100, pp. 249-306, 1974. · Zbl 0309.14041 · doi:10.2307/1971073
[29] G. Harder, ”Die Kohomologie \(S\)-arithmetischer Gruppen über Funktionenkörpern,” Invent. Math., vol. 42, pp. 135-175, 1977. · Zbl 0391.20036 · doi:10.1007/BF01389786
[30] A. von Heydebreck, ”Homotopy properties of certain complexes associated to spherical buildings,” Israel J. Math., vol. 133, pp. 369-379, 2003. · Zbl 1046.55011 · doi:10.1007/BF02773075
[31] V. G. Kac, Infinite-dimensional Lie algebras, Third ed., Cambridge: Cambridge Univ. Press, 1990. · Zbl 0716.17022 · doi:10.1017/CBO9780511626234
[32] M. Kneser, ”Starke Approximation in algebraischen Gruppen. I,” J. Reine Angew. Math., vol. 218, pp. 190-203, 1965. · Zbl 0143.04701 · doi:10.1515/crll.1965.218.190
[33] R. Köhl and S. Witzel, The sphericity of the Phan geometries of type \({\textB}_n\) and \({\textC}_n\) and the Phan-type theorem of type \({\textF}_4\). · Zbl 1280.51006 · doi:10.1090/S0002-9947-2012-05694-7
[34] G. A. Margulis, Discrete subgroups of semisimple Lie groups, New York: Springer-Verlag, 1991. · Zbl 0732.22008
[35] B. Mühlherr and H. Van Maldeghem, ”Codistances in buildings,” Innov. Incidence Geom., vol. 10, pp. 81-91, 2009. · Zbl 1264.51005
[36] H. Nagao, ”On \({ GL}(2,\,K[x])\),” J. Inst. Polytech. Osaka City Univ. Ser. A, vol. 10, pp. 117-121, 1959. · Zbl 0092.02504
[37] A. Pressley and G. Segal, Loop Groups, New York: The Clarendon Press Oxford University Press, 1986. · Zbl 0618.22011
[38] M. S. Raghunathan, ”A note on quotients of real algebraic groups by arithmetic subgroups,” Invent. Math., vol. 4, pp. 318-335, 1967/1968. · Zbl 0218.22015 · doi:10.1007/BF01425317
[39] B. Rémy, ”Groupes de Kac-Moody déployés et presque déployés,” Astérisque, vol. 277, 2002. · Zbl 1001.22018
[40] G. Rousseau, Immeubles des groupes réductives sur les corps locaux, 1977. · Zbl 0412.22006
[41] B. Schulz, Sphärische Unterkomplexe sphärischer Gebäude, 2005. · Zbl 1134.51300
[42] B. Schulz, Spherical subcomplexes of spherical buildings. · Zbl 1271.51006 · doi:10.2140/gt.2013.17.531
[43] . J-P. Serre, ”Cohomologie des groupes discrets,” in Prospects in Mathematics, Princeton, N.J.: Princeton Univ. Press, 1971, vol. 70, pp. 77-169. · Zbl 0235.22020
[44] . J-P. Serre, ”Arithmetic groups,” in Homological Group Theory, Cambridge: Cambridge Univ. Press, 1979, vol. 36, pp. 105-136. · Zbl 0432.20042
[45] . J-P. Serre, Trees, New York: Springer-Verlag, 1980. · Zbl 0548.20018
[46] T. A. Springer, ”Reduction theory over global fields,” Proc. Indian Acad. Sci. Math. Sci., vol. 104, iss. 1, pp. 207-216, 1994. · Zbl 0838.20057 · doi:10.1007/BF02830884
[47] T. A. Springer, Linear Algebraic Groups, Second ed., Boston, MA: Birkhäuser, 1998, vol. 9. · Zbl 0927.20024 · doi:10.1007/978-0-8176-4840-4
[48] U. Stuhler, ”Homological properties of certain arithmetic groups in the function field case,” Invent. Math., vol. 57, iss. 3, pp. 263-281, 1980. · Zbl 0432.14026 · doi:10.1007/BF01418929
[49] J. Tits, ”Uniqueness and presentation of Kac-Moody groups over fields,” J. Algebra, vol. 105, iss. 2, pp. 542-573, 1987. · Zbl 0626.22013 · doi:10.1016/0021-8693(87)90214-6
[50] A. Weil, Adeles and Algebraic Groups, Boston: Birkhäuser, 1982. · Zbl 0493.14028
[51] R. M. Weiss, The structure of affine buildings, Princeton, NJ: Princeton Univ. Press, 2009. · Zbl 1166.51001 · doi:10.1515/9781400829057
[52] S. Witzel, Finiteness Properties of Chevalley Groups over the ring of (Laurent) polynomials over a Finite Field, 2011. · Zbl 1213.20047
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