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Mat. Sb., 1997, Volume 188, Number 9, Pages 127–156 (Mi msb260)  

This article is cited in 8 scientific papers (total in 8 papers)

On the classification of the maximal arithmetic subgroups of simply connected groups

A. A. Ryzhikov, V. I. Chernousov

Institute of Mathematics, National Academy of Sciences of the Republic of Belarus

Abstract: Let $G\subset \operatorname {GL}_n$ be a simply connected simple algebraic group defined over a field $K$ of algebraic numbers and let $T$ be the set of all non-Archimedean valuations $v$ of the field $K$. As is well known, each maximal arithmetic subgroup $\Gamma \subset G$ can be uniquely recovered by means of some collection of parachoric subgroups; to be more precise, there exist parachoric subgroups $M_v\subset G(K_v)$, $v\in T$, that have maximal types and satisfy the relation $\Gamma ={\mathrm N}_G(M)$, where $M=G(K)\cap \prod _{v\in T}M_v$. Thus, there naturally arises the following question: for what collections $\{M_v\}_{v\in T}$ of parachoric subgroups $M_v\subset G(K_v)$ of maximal types is the above subgroup $\Gamma \subset G$ a maximal arithmetic subgroup of $G$? Using Rohlfs's cohomology criterion for the maximality of an arithmetic subgroup, necessary and sufficient conditions for the maximality of the above arithmetic subgroup $\Gamma \subset G$ are obtained. The answer is given in terms of the existence of elements of the field $K$ with prescribed divisibility properties.

DOI: https://doi.org/10.4213/sm260

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English version:
Sbornik: Mathematics, 1997, 188:9, 1385–1413

Bibliographic databases:

UDC: 512.743
MSC: Primary 20G15; Secondary 11E57, 14L35, 14L40
Received: 30.12.1996

Citation: A. A. Ryzhikov, V. I. Chernousov, “On the classification of the maximal arithmetic subgroups of simply connected groups”, Mat. Sb., 188:9 (1997), 127–156; Sb. Math., 188:9 (1997), 1385–1413

Citation in format AMSBIB
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\pages 127--156
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    1. Belolipetsky, M, “On volumes of arithmetic quotients of SO(1, n)”, Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 3:4 (2004), 749  mathscinet  zmath  isi
    2. Belolipetsky, M, “Counting maximal arithmetic subgroups”, Duke Mathematical Journal, 140:1 (2007), 1  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    3. Agol, I, “Finiteness of arithmetic hyperbolic reflection groups”, Groups Geometry and Dynamics, 2:4 (2008), 481  crossref  mathscinet  zmath  isi
    4. Golsefidy, AS, “LATTICES OF MINIMUM COVOLUME IN CHEVALLEY GROUPS OVER LOCAL FIELDS OF POSITIVE CHARACTERISTIC”, Duke Mathematical Journal, 146:2 (2009), 227  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    5. Mikhail Belolipetsky, Alexander Lubotzky, “Manifolds counting and class field towers”, Advances in Mathematics, 229:6 (2012), 3123  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    6. Mohammadi A., Golsefidy A.S., “Discrete Subgroups Acting Transitively on Vertices of a Bruhat-Tits Building”, Duke Math J, 161:3 (2012), 483–544  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. Belolipetsky M., Emery V., “On Volumes of Arithmetic Quotients of Po (N, 1)(Degrees), N Odd”, Proc. London Math. Soc., 105:Part 3 (2012), 541–570  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    8. Vincent Emery, “On compact hyperbolic manifolds of Euler characteristic two”, Algebr. Geom. Topol, 14:2 (2014), 853  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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