# presentations of Sp(n, Z) and cocycles.

Question: where can I find explicit presentations of the group $Sp(n, \mathbb Z)$, for small $n$?

It is known that $Sp(n, \mathbb Z)$ admits a $2$-cocycle $h$ with values in $\mathbb Z/2\mathbb Z$ which I'd like to view in the following way. If we fix a presentation with relations forming a set $R$, then $h$ is a function $R \to \mathbb Z/2\mathbb Z$ (which fulfils some condition).

Question: where can I find an explicit description of this cocycle, in the form of what values does it take on elements of some presentation?

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There is a simple and explicit presentation of $SP(2n,\mathbb{Z})$ in Theorem 9.2.13 of Hahn-O’Meara's book "The classical groups and K-theory". In some sense, this presentation is related to the ideas of Klingen mentioned by Rivin; however, just following Klingen's method leads to an enormous presentation (something on the order of 5 families of generators and 67 families of relations), as is shown in the paper

MR0280606 (43 #6325) Birman, Joan S. On Siegel's modular group. Math. Ann. 191 1971 59–68.

Another simple presentation can be found in the paper

MR1152494 (93b:11054) Lu, Ning(1-RICE) A simple presentation of the Siegel modular groups. Linear Algebra Appl. 166 (1992), 185–194.

The basic idea is that there exist nice presentations for the mapping class group, which surjects onto the symplectic group.

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Nice references (though does not the N. Lu paper precede the REALLY simple Wajnryb presentation of the MCG?) – Igor Rivin Oct 24 '11 at 18:43
@Igor : Waynryb's presentation is contained in MR0719117 (85g:57007) Wajnryb, Bronislaw(IL-TECH) A simple presentation for the mapping class group of an orientable surface. Israel J. Math. 45 (1983), no. 2-3, 157–174. – Andy Putman Oct 25 '11 at 4:30
(though he also has a 1999 paper called "An elementary approach to the mapping class group of a surface" which derives a similar presentation without using Cerf theory, which appeared in work of Hatcher and Thurston that he quoted in his first paper). – Andy Putman Oct 25 '11 at 4:32
(also, there is an error in the statement of the main theorem of Wajnryb's 1983 paper which is corrected in Birman, Joan S.; Wajnryb, Bronislaw, Presentations of the mapping class group. Errata: "3-fold branched coverings and the mapping class group of a surface'' [in Geometry and topology (College Park, MD, 1983/84), 24--46, Lecture Notes in Math., 1167, Springer, Berlin, 1985] and "A simple presentation of the mapping class group of an orientable surface'' [Israel J. Math. 45 (1983), no. 2-3, 157--174] by Wajnryb. Israel J. Math. 88 (1994), no. 1-3, 425–427. ) – Andy Putman Oct 25 '11 at 4:34
@Andy: Actually, I was talking about the '96 Wajnryb paper (two-generator), which is slightly tweaked in Korkmaz '06. Presumably this can be processed as per N. Lu to get a reasonably simple two-generator symplectic group presentation. – Igor Rivin Oct 25 '11 at 7:52

An explicit presentation was given by Behr in MR0369562 (51 #5795) Behr, Helmut Eine endliche Präsentation der symplektischen Gruppe Sp4(Z). (German) Math. Z. 141 (1975), 47–56. 20H25

And a slightly shorter one by P. Bender in Bender, Peter Eine Präsentation der symplektischen Gruppe Sp(4,Z) mit 2 Erzeugenden und 8 definierenden Relationen. (German) J. Algebra 65 (1980), no. 2, 328–331.

General $SP(2n, Z)$ can be reduced to $SP(4, Z),$ the argument is in MR0133303 (24 #A3137) Klingen, Helmut Charakterisierung der Siegelschen Modulgruppe durch ein endliches System definierender Relationen.

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For $Sp_4(\mathbf{Z})$, there is a presentation in H. Behr "Eine endliche Präsentation der symplektischen Gruppe $Sp_4(\mathbf{Z})$", Math. Z. 141 (1975), 47--56.

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