Non-normal, standard subgroups of the Bianchi groups

Mason, A.; Odoni, R.

doi:10.1090/S0002-9939-96-03310-2

Non-normal, standard subgroups of the Bianchi groups
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by A. W. Mason and R. W. K. Odoni PDF

Proc. Amer. Math. Soc. 124 (1996), 721-726 Request permission

Abstract:

Let $S$ be a subgroup of $SL_n(K)$, where $K$ is a Dedekind ring, and let $\mathbf {q}$ be the $K$-ideal generated by $x_{ij},x_{ii}-x_{jj}$ $(i\ne j)$, where $(x_{ij})\in S$. The subgroup $S$ is called standard iff $S$ contains the normal subgroup of $SL_n(K)$ generated by the $\mathbf {q}$-elementary matrices. It is known that, when $n\ge 3$, $S$ is standard iff $S$ is normal in $SL_n(K)$. It is also known that every standard subgroup of $SL_2(K)$ is normal in $SL_2(K)$ when $K$ is an arithmetic Dedekind domain with infinitely many units. The ring of integers of an imaginary quadratic number field, $\mathcal {O}$, is one example (of three) of such an arithmetic domain with finitely many units. In this paper it is proved that every Bianchi group $SL_2(\mathcal {O})$ has uncountably many non-normal, standard subgroups. This result is already known for related groups like $SL_2(\mathbb {Z})$.

References

H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for $\textrm {SL}_{n}\,(n\geq 3)$ and $\textrm {Sp}_{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137. MR 244257
A. O. Gelfond and Yu. V. Linnik, Elementary methods in analytic number theory, George Allen and Unwin, London, 1966.
Fritz J. Grunewald and Joachim Schwermer, Free nonabelian quotients of $\textrm {SL}_{2}$ over orders of imaginary quadratic numberfields, J. Algebra 69 (1981), no. 2, 298–304. MR 617080, DOI 10.1016/0021-8693(81)90206-4
Bernhard Liehl, On the group $\textrm {SL}_{2}$ over orders of arithmetic type, J. Reine Angew. Math. 323 (1981), 153–171. MR 611449, DOI 10.1515/crll.1981.323.153
A. W. Mason, On subgroups of $\textrm {GL}(n,\,A)$ which are generated by commutators. II, J. Reine Angew. Math. 322 (1981), 118–135. MR 603028, DOI 10.1515/crll.1981.322.118
A. W. Mason, Standard subgroups of $\textrm {GL}_2(A)$, Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 3, 341–349. MR 908441, DOI 10.1017/S0013091500026730
A. W. Mason, Free quotients of congruence subgroups of $\textrm {SL}_2$ over a Dedekind ring of arithmetic type contained in a function field, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 3, 421–429. MR 878891, DOI 10.1017/S0305004100066809
A. W. Mason, Nonstandard, normal subgroups and nonnormal, standard subgroups of the modular group, Canad. Math. Bull. 32 (1989), no. 1, 109–113. MR 996131, DOI 10.4153/CMB-1989-016-5
A. W. Mason, Congruence hulls in $\textrm {SL}_n$, J. Pure Appl. Algebra 89 (1993), no. 3, 255–272. MR 1242721, DOI 10.1016/0022-4049(93)90056-Y
—, Normal subgroups of level zero of the Bianchi groups, Bull. London Math. Soc. 26 (1994), 263–267.
—, Free quotients of Bianchi groups and unipotent matrices (submitted).
A. W. Mason, R. W. K. Odoni, and W. W. Stothers, Almost all Bianchi groups have free, noncyclic quotients, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 1, 1–6. MR 1131473, DOI 10.1017/S0305004100075101
A. W. Mason and R. W. K. Odoni, Free quotients of subgroups of the Bianchi groups whose kernels contain many elementary matrices, Math. Proc. Cambridge Philos. Soc. 116 (1994), 253–273.
Jean-Pierre Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970), 489–527 (French). MR 272790, DOI 10.2307/1970630
L. N. Vaseršteĭn, The group $SL_{2}$ over Dedekind rings of arithmetic type, Mat. Sb. (N.S.) 89(131) (1972), 313–322, 351 (Russian). MR 0435293
R. Zimmert, Zur $\textrm {SL}_{2}$ der ganzen Zahlen eines imaginär-quadratischen Zahlkörpers, Invent. Math. 19 (1973), 73–81 (German). MR 318336, DOI 10.1007/BF01418852