# Some non-maximal arithmetic groups

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Let k be a non-finite Dedekind domain, and σ be the ring of its integers. We shall assume that the ring R = σ- 2 is finite. Let us denote by Mn k resp. Mnσ the ring of all n by n matrices with entries in k resp. in σ, and Gln k its group of units.We denote by sln k the subgroup of Gln k whose elements g  have determinant, det g, equal to one. Let  H ε Mn  σ be a symmetric matrix, i.e., H = tH where tH denotes the transpose matrix of H. We let G = SO H = { g ε Sln k l tgHg = H }, and we let Gσ = G∩Mn σ. We want to exhibit certain H for which Gσ is not maxinal in G, in the sense that there exist a subgroup Δ contains Gσ properly and Δ : Gσ is finite.

Tipo de documento: Artículo - Article

Palabras clave: Teoría de los números, grupos discontinuos, grupos aritméticos

Fuente: http://www.bdigital.unal.edu.co

## Introducción

Revista Colombiana de Matematicas Volumen II,1968, pags.
21- 28 SOME NON-MAXIMAL ARITHMETIC GROUPS by Nelo D.
ALLAN Let k be a non-finite Dedekind domain, and ~ be the ring of its integers.
We shall assume that the ring R =)r-(2) Mn()r)) in is finite.
Let us denote by the ring of all k (resp.
in):f), and se n (k) We denote by ments t by n ox.
n (k) its group of units. n H denotes the transpose matrix of SO(H) = {g maximal in G, of r, H S~n (k) - tgHg E H We want to exhibit certain group A O£ (k) the subgroup of n whose ele- e~ual to one. Let H where be a symmetric matrix, i.e., (Jf) (resp. n matrices with entries have determinant, det g, g HEM n M (k) tH = H.
We let G and we let Off for which Gff = = GnMn (JY) • is not in the sense that there exists a sub- G such that t::,. contains Off properly and [e:.:GhJis finite. 1. E!~11~1E~Ei~~ Let L shall denote by all the L . be an order in Mn(k); we the fractional ideal generated by lJ (i,j)-entries of all the elements of , L we shall write L We shall say that dule L L is a direct summand if as an;.r-mo- is a direct sum of the units of nn L .
e . l.J l.J where e . l.J are M (k). n 21 It in ]V! n is well known that (k) are conjugate mmands and tl L nJ L I , i, j L ,J = nn 1 to the ones -hich =;,t:, for n, in our case the maximal orders I i,j n, are direct and some fractional su- =,a-l L. ln ·U of ideal k, i .e.
, L L If is pansion one of such orders, g-1 , of a group. g then by 100kinB at the ex- Sen (k ) E Consequently if j 11 SEn(k) is we see that L sf n (k), !J.= Gc then n G is L a group. For our purposes 1~M~~1.
If surable both to R we consider =~-~ follow The index r~ff: the order of the group 6 (it)] (v .
,) lJ g and -1 ,; g v , ,::::0 lJ -1 E g V, M (.e;). only finitely ly finitely 611 follows 22 all that finite. tg =...