February 2012 Discrete subgroups acting transitively on vertices of a Bruhat–Tits building
Amir Mohammadi, Alireza Salehi Golsefidy
Duke Math. J. 161(3): 483-544 (February 2012). DOI: 10.1215/00127094-1507430


We describe all the discrete subgroups of Ad(G0)(F)Aut(F) that act transitively on the set of vertices of B=B(F,G0), the Bruhat–Tits building of a pair (F,G0) of a characteristic 0 nonarchimedean local field, and a simply connected, absolutely almost simple F-group if B is of dimension at least 4. In fact, we classify all such maximal subgroups. We show that there are exactly eleven families of such subgroups and explicitly construct them. Moreover, we show that four of these families act simply transitively on the vertices. In particular, we show that there is no such action if either the dimension of the building is larger than 7, if F is not isomorphic to Qp for some prime p, or if the building is associated to SLn,D0, where D0 is a noncommutative division algebra. Along the way we also give a new proof of the Siegel–Klingen theorem on the rationality of certain Dedekind zeta functions and L-functions.


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Amir Mohammadi. Alireza Salehi Golsefidy. "Discrete subgroups acting transitively on vertices of a Bruhat–Tits building." Duke Math. J. 161 (3) 483 - 544, February 2012. https://doi.org/10.1215/00127094-1507430


Published: February 2012
First available in Project Euclid: 1 February 2012

zbMATH: 1254.22015
MathSciNet: MR2881229
Digital Object Identifier: 10.1215/00127094-1507430

Primary: 11J83
Secondary: 11K60

Rights: Copyright © 2012 Duke University Press


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Vol.161 • No. 3 • February 2012
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