## Lattices of minimum covolume in Chevalley groups over local fields of positive characteristic.(English)Zbl 1161.22006

The main result of the paper states that if $$\mathbb{G}$$ is a simply connected Chevalley group of either classical type of rank bigger than 1 or type E$$_6$$ and if $$q$$ is a power of a prime number $$p>5$$ which is larger than $$9$$ when $$\mathbb{G}$$ is not of type A, then $$G=\mathbb{G}\left( \mathbb{F}_q((t^{-1}))\right)$$, up to an automorphism, has a unique lattice of minimum covolume, which is $$\mathbb{G}(\mathbb{F}_q[t])$$. In the introduction the author also mentions an extension of this theorem to simply connected, absolutely almost simple groups $$\mathbb{G}$$ over $$\mathbb{F}_q((t^{-1}))$$, which is a subject of a forthcoming paper.
The paper starts with a short historical introduction which gives an excellent brief account of the previous work on arithmetic subgroups of minimum covolume from C. L. Siegel to F. W. Gehring and G. J. Martin. It explains the motivation underlying the current work. The author then gives an outline of the proof of the main theorem which splits into several steps. One of the main features of the argument which has to be pointed out is that it almost entirely avoids any case-by-case considerations. This is achieved through a number of structural results some of which could be of the independent interest. Another important feature of the current article is that the author uses the opportunity to prove several facts needed for Chevalley groups directly instead of deducing them from more general theorems. Here we can mention the derivation of the formula for the covolume of $$\Gamma = \mathbb{G}(\mathbb{F}_q[t])$$ in $$G=\mathbb{G}\left(\mathbb{F}_q((t^{-1}))\right)$$ in Section 4, the elementary proof of the fact that $$\mathrm{H}^1(\mathrm{SL}_n(\mathbb{F}_q), \mathfrak{gl}_n(\mathbb{F}_q)) = 0$$ in Section 5, and the proof that $$N_G(\Gamma) = \Gamma$$ using reduction theory in Section 10. The argument in Section 4 can be used as an introduction to the work of G. Prasad [Publ. Math. Inst. Hautes Étud. Sci. 69, 91–117 (1989; Zbl 0695.22005)] on volumes of arithmetic quotients of semisimple groups.
The main result of the paper implies that the lattices of minimal covolume in Chevalley groups over local fields of positive characteristic are usually nonuniform, i.e. non-cocompact. This surprising phenomenon was first discovered by A. Lubotzky [J. Am. Math. Soc. 3, No. 4, 961–975 (1990; Zbl 0731.22009)] for lattices in $$G = \mathrm{SL}_2(\mathbb{F}_q[t])$$. In [Ann. Sci. Norm. Super. Pisa, Cl. Sci. (5) 3, No. 4, 749–770 (2004; Zbl 1170.11307) and ibid. 6, 263–268 (2007)], the reviewer showed that the same happens for odd orthogonal groups of high enough dimension over $$\mathbb{R}$$ and conjectured that generically (for the groups which contain nonuniform lattices) the minimal covolume is always attained on a nonuniform lattice. The results of this paper give further support to the conjecture.

### MSC:

 2.2e+41 Discrete subgroups of Lie groups 1.1e+58 Classical groups

### Keywords:

minimal covolume; Chevalley groups; positive characteristic

### Citations:

Zbl 0695.22005; Zbl 0731.22009; Zbl 1170.11307
Full Text:

### References:

 [1] E. Artin, Algebraic Numbers and Algebraic Functions , Gordon and Breach, New York, 1967. · Zbl 0194.35301 [2] E. Bombieri, “Counting points on curves over finite fields (d’après S. A. Stepanov)” in Séminare Bourbaki (1972/1973) , exp. no. 430, Lecture Notes in Math. 383 , Springer, Berlin, 1974, 234–241. · Zbl 0307.14011 [3] A. Borel, Linear Algebraic Groups , 2nd ed., Grad. Texts in Math. 126 , Springer, New York, 1991. · Zbl 0726.20030 [4] A. Borel and G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups , Inst. Hautes Études Sci. Publ. Math. 69 (1989), 119–171. · Zbl 0707.11032 [5] A. Borel and J. Tits, Éléments unipotents et sous-groupes paraboliques de groupes réductifes, I , Invent. Math. 12 (1971), 95–104. · Zbl 0238.20055 [6] R. W. Carter, Simple Groups of Lie Type , Pure Appl. Math. 28 , Wiley, London, 1972. · Zbl 0248.20015 [7] C. Chabauty, Limite d’ensembles et géométrie des nombres , Bull. Soc. Math. France 78 (1950), 143–151. · Zbl 0039.04101 [8] V. I. Chernousov and A. A. Ryzhkov, On the classification of maximal arithmetic subgroups of simply connected groups , Sb. Math. 188 (1997), 1385–1413. · Zbl 0899.20026 [9] E. Cline, B. Parshall, and L. Scott, Cohomology of finite groups of Lie type, I , Inst. Hautes Études Sci. Publ. Math. 45 (1975), 169–191. · Zbl 0412.20044 [10] -, Cohomology of finite groups of Lie type, II , J. Algebra 45 (1977), 182–198. · Zbl 0412.20045 [11] F. W. Gehring and G. J. Martin, Precisely invariant collars and the volume of hyperbolic $$3$$- folds , J. Differential Geom. 49 (1998), 411–435. · Zbl 0989.57010 [12] -, The volume of hyperbolic $$3$$-folds with $$p$$- torsion, $$p \geq 6$$, Quart. J. Math. Oxford Ser. (2) 50 (1999), 1–12. · Zbl 0926.30027 [13] -,“$$(p, g, r)$$-Kleinian groups and the Margulis constant” in Complex Analysis and Dynamical Systems, II (Nahariya, Israel, 2003), Contemp. Math. 382 , Amer. Math. Soc., Providence, 2005, 149–169. · Zbl 1088.30044 [14] -, Minimal co- volume hyperbolic lattices, I: The spherical points of a Kleinian group, to appear in Ann. of Math. (2) (2009). · Zbl 1171.30014 [15] P. Gille, Unipotent subgroups of reductive groups in characteristic $$p>0$$ , Duke Math. J. 114 (2002), 307–328. · Zbl 1013.20040 [16] G. Harder, Minkowskische Reduktionstheorie über Funktionenkörpern , Invent. Math. 7 (1969), 33–54. · Zbl 0242.20046 [17] -, Chevalley groups over function fields and automorphic forms , Ann. of Math. (2) 100 (1974), 249–306. JSTOR: · Zbl 0309.14041 [18] G. M. D. Hogeweij, Almost-classical Lie algebras I , Nederl. Akad. Wetensch. Indag. Math. 44 (1982), 441–452.; II, 453–460. · Zbl 0512.17003 [19] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $$\mathfrakp$$-adic Chevalley groups , Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. · Zbl 0228.20015 [20] D. A. KažDan and G. A. Margulis, A proof of Selberg’s hypothesis (in Russian), Mat. Sb. (N.S.) 75 ( 117 ) (1968), 163–168. [21] S. Lang, Algebraic groups over finite fields , Amer. J. Math. 78 (1956), 555–563. JSTOR: · Zbl 0073.37901 [22] A. Lubotzky, Lattices of minimal covolume in SL $$_2$$: A non-Archimedean analogue of Siegel’s theorem $$\mu\geq\pi/21$$, J. Amer. Math. Soc. 3 (1990), 961–975. JSTOR: · Zbl 0731.22009 [23] A. Lubotzky and T. Weigel, Lattices of minimal covolume in $$\mathrm SL_2$$ over local fields , Proc. London Math. Soc. (3) 78 (1999), 283–333. · Zbl 1026.22013 [24] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups , Ergeb. Math. Grenzgeb. (3) 17 , Springer, Berlin, 1991. · Zbl 0732.22008 [25] T. H. Marshall and G. J. Martin, Minimal co-volume hyperbolic lattices, II: Simple torsion in Kleinian groups , preprint, 2008. [26] R. Meyerhoff, The cusped hyperbolic $$3$$-orbifold of minimum volume , Bull. Amer. Math. Soc. (N.S.) 13 (1985), 154–156. · Zbl 0602.57009 [27] H. Niederreiter and C. Xing, Rational points on curves over finite fields: Theory and applications , London Math. Soc. Lecture Note Ser. 285 , Cambridge Univ. Press, Cambridge, 2001. · Zbl 0971.11033 [28] A. Prasad, Reduction theory for a rational function field , Proc. Indian Acad. Sci. Math. Sci. 113 (2003), 153–163. · Zbl 1044.11038 [29] G. Prasad, Volumes of $$S$$-arithmetic quotients of semi-simple groups , with an appendix by M. Jarden and P. Gopal, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 91–117. · Zbl 0695.22005 [30] J. Rohlfs, Die maximalen arithmetisch definierten Untergruppen zerfallender einfacher Gruppen , Math. Ann. 244 (1979), 219–231. · Zbl 0426.20030 [31] A. Salehi Golsefidy, Lattices of minimum covolume in Chevalley groups over a local field of positive characteristic , Ph.D. dissertation, Yale University, New Haven, 2006. [32] -, Lattices of minimum co-volume in positive characteristic are non-uniform , in preparation. [33] C. L. Siegel, Some remarks on discontinuous groups , Ann. of Math. (2) 46 (1945), 708–718. JSTOR: · Zbl 0061.04505 [34] T. A. Springer, Reduction theory over global fields , Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 207–216. · Zbl 0838.20057 [35] R. Steinberg, Generators, relations and coverings of algebraic groups, II , J. Algebra 71 (1981), 527–543. · Zbl 0468.20038 [36] J. Tits, “Unipotent elements and parabolic subgroups of reductive groups, II” in Algebraic Groups (Utrecht, Netherlands, 1986) , Lecture Notes in Math. 1271 , Springer, Berlin, 1987, 265–284. · Zbl 0658.20025 [37] A. Weil, Adèles and Algebraic Groups , with appendices by M. Demazure and T. Ono, Progr. Math. 23 , Birkhäuser, Boston, 1982. · Zbl 0493.14028 [38] B. Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups , Ann. of Math. (2) 120 (1984), 271–315. JSTOR: · Zbl 0568.14025
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