zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the groups $\mathrm{SL}\sb 2(\mathbb{Z}[x])$ and $\mathrm{SL}\sb 2(k[x,y])$. (English) Zbl 0805.20042
Let $R = \bbfZ[x]$ or $k[x,y]$, where $\bbfZ$ is the ring of rational integers and $k$ is a finite field, and let $U\sb 2(R)$ be the subgroup of $\text{SL}\sb 2(R)$ generated by the unipotent matrices. The authors prove that $\text{SL}\sb 2(R)/U\sb 2(R)$ has a free quotient of any finite rank. This result is very much a two-dimensional anomaly since Suslin has proved that, when $n \geq 3$, $\text{SL}\sb n(R) = E\sb n(R)$, where $E\sb n(R)$ is the subgroup generated by the elementary matrices. Using a technique involving symplectic matrices it is shown that, if $S$ is a quotient of $R$, then the corresponding natural map $\phi: \text{SL}\sb 2(R) \to \text{SL}\sb 2(S)$ is “almost surjective”. The proof of the main result follows from looking at $\text{Im }\phi$ for various $S$. When $R = \bbfZ[x]$, $S$ is the ring of integers of some imaginary quadratic number fields. When $R = k[x,y]$, $S$ is a ring of integers in a hyperelliptic function field over $k$. The paper also contains results on the exact stable range of various polynomial rings, together with structure theorems for the group $U\sb 2(R)/\widehat{E}\sb 2(R)$, when $\widehat{E}\sb 2(R)$ is the normal subgroup of $SL\sb 2(R)$ generated by $E\sb 2(R)$.

MSC:
20H25Other matrix groups over rings
20H10Fuchsian groups and their generalizations (group theory)
20E07Subgroup theorems; subgroup growth
20E05Free nonabelian groups
20F05Generators, relations, and presentations of groups
19B14Stability for linear groups ($K_1$)
11R58Arithmetic theory of algebraic function fields
20G35Linear algebraic groups over adèles and other rings and schemes
WorldCat.org
Full Text: DOI
References:
[1] H. Bass,K-theory and stable algebra, Publ. Math., Inst. Hautes Étud. Sci.22 (1964), 485--544.
[2] H. Bass,Algebraic K-theory, W. A. Benjamin, New York, Amsterdam, 1968.
[3] H. Bass, J. Milnor and J. P. Serre,Solution of the congruence subgroup problem for SL n (A) (n 3)and Sp2n (A) (n 2), Publ. Math. Inst. Hautes Étud. Sci.33 (1967), 59--137. · Zbl 0174.05203 · doi:10.1007/BF02684586
[4] P. M. Cohn,On the structure of the GL2 of a ring, Publ. I. H. E. S.30 (1966), 5--53.
[5] G. Cooke and P. Weinberger,On the construction of division chains in algebraic number rings, with application to SL2, Commun. in Algebra3 (1975), 481--524. · Zbl 0315.12001 · doi:10.1080/00927877508822057
[6] J. Dieudonné,La géometrie des groupes classiques, Springer-Verlag, Berlin, Heidelberg, New York, 1971.
[7] D. Estes and J. Ohm,Stable range in commutative rings, J. Algebra7 (1967), 343--362. · Zbl 0156.27303 · doi:10.1016/0021-8693(67)90075-0
[8] F. Grunewald, H. Helling and J. Mennicke, SL2 over complex quadratic numberfields, I, Algebra i Logica17 (1978), 512--580.
[9] F. Grunewald, J. Mennicke and L. Vaserstein,On symplectic groups over polynomial rings, Math. Z.206 (1990), 35--56. · Zbl 0725.20038 · doi:10.1007/BF02571323
[10] F. Grunewald and J. Schwermer,Free non-abelian quotients of SL2 over orders of imaginary quadratic numberfields, J. Algebra69 (1981), 298--304. · Zbl 0461.20026 · doi:10.1016/0021-8693(81)90206-4
[11] F. Grunewald and J. Schwermer,Arithmetic quotients of hyperbolic 3-space, cups forms, and link complements, Duke Math. J.48 (1981), 351--358. · Zbl 0485.57005 · doi:10.1215/S0012-7094-81-04820-1
[12] E.-U. Gekeler,Drinfeld Modular Curves, Lecture Notes in Math., Vol. 1231, Springer-Verlag, Berlin, Heidelberg, New York, 1987. · Zbl 0848.11029
[13] E.-U. Gekeler,Le genre des courbes modulaires de Drinfeld, C. R. Acad. Sci. Paris300 (1985), 647--650.
[14] B. Liehl,On the group SL2 over orders of arithmetic type, J. Reine Angew. Math.323 (1981), 153--171. · Zbl 0447.20035 · doi:10.1515/crll.1981.323.153
[15] R. Maschinsky,Die Operation der Gruppen SL2 und GL2 auf dem Baum der Gitterklassen einer elliptischen Kurve, Diplomarbeit, Wuppertal, 1989.
[16] C. Queen,Some arithmetic properties of subrings of function fields over finite fields, Archiv der Mathematik26 (1975), 51--56. · Zbl 0303.12008 · doi:10.1007/BF01229702
[17] J. P. Serre,Trees, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
[18] J. P. Serre,Local Fields, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
[19] J. P. Serre,Le problème des groupes de congruence pour SL2, Annals of Math.92 (1970), 489--527. · Zbl 0239.20063 · doi:10.2307/1970630
[20] R. Swan,Generators and relations for certain linear groups, Advances in Math.6 (1971), 1--77. · Zbl 0221.20060 · doi:10.1016/0001-8708(71)90027-2
[21] U. Stuhler,Über die Faktorkommutatorgruppe der Gruppen SL2(O)im Funktionenkörperfall, Archiv der Mathematik42 (1984), 314--324. · Zbl 0578.20035 · doi:10.1007/BF01246121
[22] A. Suslin,On the structure of the special linear group over polynomial rings, Math. USSR, Izv11 (1977), 221--238. · Zbl 0378.13002 · doi:10.1070/IM1977v011n02ABEH001709
[23] W. Terhalle,Zur Gruppe SL2($\mathbb{Z}$[x]), Diplomarbeit, Bielefeld, 1989.
[24] L. Tavgen,On bounded generation of Chevalley groups over S-integer algebraic numbers, Izv. Nauk, 1990. · Zbl 0697.20032
[25] L. Vaserstein,Stabilization for classical groups over rings, Mat. Sbornik81 (1970), 328--351 (Translated as: Math. USSR, Sb.10 (1970), 307--326).
[26] L. Vaserstein,On the SL2 over Dedekind rings of arithmetic type, Math. USSR Sb.18 (1972), 321--332. · Zbl 0359.20027 · doi:10.1070/SM1972v018n02ABEH001775
[27] L. Vaserstein,Structure of the classical arithmetic groups of rank greater than 1, Math USSR Sb.20 (1973), 465--492. · Zbl 0291.14016 · doi:10.1070/SM1973v020n03ABEH001885
[28] L. Vaserstein,On the normal subgroups of GL n over a ring, Lecture Notes in Math., Vol. 854, Springer-Verlag, Berlin, Heidelberg, New York, 1972, pp. 456--465.
[29] L. Vaserstein,Stabilization of unitary and orthogonal groups over a ring with involution, Mat. Sbornik81 (1970), 307--326. · Zbl 0253.20066 · doi:10.1070/SM1970v010n03ABEH001673
[30] L. Vaserstein and A. Suslin,Serre’s problem on projective modules over polynomial rings and algebraic K-theory, Izv. Akad. Nauk SSSR10 (1976), 937--1000. · Zbl 0338.13015
[31] A. Weil,Basic Number Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1967. · Zbl 0176.33601