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Finite presentability of arithmetic groups over global function fields

Published online by Cambridge University Press:  20 January 2009

Helmut Behr
Helmut Behr
Affiliation:
Universität Frankfurt am Main, West Germany
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Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but over global function fields this is not always true. All known exceptions are “small” groups, which means that either the rank of the algebraic group or the set S of the underlying S-arithmetic ring has to be small. There exists now a complete list of all such groups which are not finitely generated, whereas we onlyhave a conjecture which groups are finitely generated but not finitely presented.

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Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 1 , February 1987 , pp. 23 - 39
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1.Abels, H., Kompakt definierbare topologische Gruppen, Math. Ann. 197 (1972), 221–233.CrossRefGoogle Scholar
2.Abels, H., Generators and relations for groups of homeomorphisms, in: Transformation groups, Proceedings Newcastle 1976 (London Mathematical Society Lecture Notes 26, 3–20).Google Scholar
3.Abels, H., An example of a finitely presented solvable group, in: Homological Group Theory, Proceedings Durham 1977 (London Mathematical Society Lecture Notes 36, 205–211).Google Scholar
4.Abels, H., Finite presentability of S-arithmetic groups—compact presentability of solvable groups, to appear.Google Scholar
5.Behr, B., Zur starken Approximation in algebraischen Gruppen über globalen Körpern, J. Reine Angew. Math. 229 (1968), 107–116.Google Scholar
6.Behr, H., Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkorpern, Invent. Math. 7 (1969), 1–32.CrossRefGoogle Scholar
7.Behr, H., is not finitely presentable, in: Homological Group Theory, Proceedings Durham 1977 (London Mathematical Society Lecture Notes 36, 213–224).Google Scholar
8.Borel, A. and Tits, J., Groupes reductifs, Publ. Math. I.H.E.S. 11 (1965 ), 55–152.CrossRefGoogle Scholar
9.Bourbaki, N., Groupes et algebres de Lie, chap. IV; Elements de mathematique XXXIV, Paris 1968.Google Scholar
10.Bruhat, F. and Tits, J., Groupes reductifs sur en corps local I, Publ. Math. I.H.E.S. 41 (1972), 5–252.Google Scholar
11.Godement, R., Domaines fondamentaux des groupes arithmetiques, Seminaire Bourbaki, vol. 15, exp. 257, Paris1962–63.Google Scholar
12.Harder, G., Minkowskische Reduktionstheorie in Funktionenkörpern, Invent. Math. 7 (1969), 33–54.CrossRefGoogle Scholar
13.Hurrelbrink, J., Endlich präsentierte arithmetische Gruppen und K2 Über Laurent-Polynomringen, Math. Ann. 225 (1977), 123–129.CrossRefGoogle Scholar
14.Keller, K., Nicht endlich erzeugbare arithmetische Gruppen über Funktionenkorpern, Dissertation Frankfurt a.M. 1980.Google Scholar
14.aBEHR, H., Addendum to [14], preprint 1981.Google Scholar
15.Kneser, M., Erzeugende und Relationen verailgemeinerter Einheitengruppen, J. Reine Angew. Math. 214 /15 (1964), 345–349.CrossRefGoogle Scholar
16.Mchardy, G., Endliche und fast-endliche Präsentierbarkeit einiger arithmetischer Gruppen, Dissertation Frankfurt a.M. 1982.Google Scholar
16.aBehr, H., Addendum to ], preprint 1982.Google Scholar
17.Rehmann, U. and Soule, C., Finitely presented groups of matrices, Algebraic K-theory, Evanston 1976 (Springer Lecture Notes 551 (1976), 164–169).CrossRefGoogle Scholar
18.Serre, J. P., Trees (Springer-Verlag, Berlin-Heidelberg-New York, 1980).Google Scholar
19.Splitthoff, S., Finite presentability of Steinberg groups and related Chevalley groups, Dissertation Bielefeld (1985 ).CrossRefGoogle Scholar
20.Stuhler, U., Zur Frage der endlichen Prasentierbarkeit gewisser arithmetischer Gruppen im Funktiönenkorperfall, Math. Ann. 224 (1976), 217–232.CrossRefGoogle Scholar
21.Tits, J., Classification of algebraic semisimple groups, Proc. Sympos. Pure Math. IX (1966), 33–62.Google Scholar
22.Tits, J., Reductive groups over local fields, Proc. Sympos. Pure Math. XXXIII, part 1 (1979), 29–69.Google Scholar