##
**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.

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.

Reviewer: Mikhail Belolipetsky (Durham)

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\textit{A. S. Golsefidy}, Duke Math. J. 146, No. 2, 227--251 (2009; Zbl 1161.22006)

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