Zur Theorie der Siegelschen Modulgruppe

“…Mennicke [31] has shown that, as a subgroup of Γ 2 (1), Γ 2 (n) is normally generated by the single element Note that a, b ∈ Γ 2 (n). Then direct calculation shows that a −1 b −1 ab = g 2n .…”

Section: Further Resultsmentioning

confidence: 99%

Cohomology of the boundary of Siegel modular varieties of degree two, with applications

Hoffman

Weintraub

2003

Fund. Math.

View full text Add to dashboard Cite

“…The purpose of thi s no te is to point out that just th e opposite is true. In fact we will show that no normal subgroup nor any s ubgroup of f, = SL(l , Z ) of finite ind e x can be free for t 3 …”

mentioning

confidence: 85%

“…Here the relevant fact is that two commuting elements in a free product of cyclic groups must belong to a common cyclic subgroup. Final ly, Mennicke has shown in [3] that the symp lectic group Sp(2t , Z) also has the "congruence subgroup property" for t :;", 2. …”

mentioning

confidence: 99%

Subgroups of SL(t, Z)

Newman¹

1969

J. RES. NATL. BUR. STAN. SECT. B. MATH. SCI.

View full text Add to dashboard Cite

It is s hown that if t ~ 3, the n no subgroup of SLit, Z) of finite inde x is free (in fact is not even the fre e produ c t of cyc li c groups). H e r e SL(t, Z ) is the multipli ca tive group of t X t matri ces o ver the integers o f d ete rmin a nt 1.Key word s: Co ngru e nce s ubgro up property; fr ee g roup s; fr ee products; hi ghe r modular gTOUpS.It is kn ow n that th e class ica l modul ar group f = PSL (2, Z ) possesses man y s ub groups whi c h are free: In fact (see [4])1 e ve ry normal s ubgroup of f is free, apart from the three exce ptions f, P , f 3, (fp is th e fully invariant s ubgro up of f ge nerated b y th e pth powers of the ele me nts of f). Th e r eason for thi s is that f is th e free product of tw o cyclic groups, one of ord e r 2 and the other of order 3, and the Kuros h s ub group th eore m the n implies that a s ub gro up of f is free if and only if it co ntain s 'no elements of finite orde r. Thi s res ult at once implies that e ve ry norm al s ub group of SL (2, Z) whi c h does not co ntain -/ mu st be free.It is natural to ask if a simular situati on exis ts for the hi ghe r modular groups. The purpose of thi s no te is to point out that just th e opposite is true. In fact we will show that no normal subgroup nor any s ubgroup of f, = SL(l , Z ) of finite ind e x can be free for t 3 3.If n is a ny positive integer , th en f,(n ) is the prin cipal co ngrue nce subgroup of f t of level n; that is, th e totality of matri ces A E f t s uc h that A == I mod ii. Epq stands for the tXt matrix with a 1 in po sition (p, q) and ° else wh ere.The proof mak es use of a deep res .. .!lt prov ed rece ntly by Me nnicke [2], and by Bass, Lazard , and Serre [1] , toge th e r with so me e le me ntary re mark s on free groups.We assume throu ghout the following that t 3 3. We first prove LEMMA 1. Let F be a free group. Then any two commuting elements ofF are each powers of the same element of F.PROOF. Suppose that x , yare co mmutin g elements of F. Put G= {x, y}. Th e n G is an abelian subgroup of F of rank 1 or 2. Furthermore G is free, by th e Kuros h s ubgro up th eore m. Thus G can· not be of rank 2, since a free group of rank 2 is not abelian. He nce G mu st be of rank 1 and there· fore cyclic: G= {z}. Thus x and yare each powers of z, and the le mm a is proved. LEMMA 2. Let n be any positive integer. Put A= I + nE1 2' B = 1+ nE 13. Then A, B are commuting elements of ft(n ) which are not powers of the same element C, for any CEft(n).PROOF. Th e fa c t that A and B are co mmuting elements of ft(n) is clear, since E I2E I3 = EI3EI2 =0. Suppose that there is an ele me nt CEft(n) such that A=C", B=Cs, for some integers r, s. The n rs oft 0, and A s=BI"=Crs. Since E'f2=EY3=O, we have 1 Fig ures in brac kets indi ca te th e lit e rature refere nces at the e nd of thi s pa per.143

show abstract

Zur Theorie der Siegelschen Modulgruppe

Abstract: Wenn man also zu ailgemeineren Koeffizientenbereichen iibergehen will, so wird man zuerst versuehen, Brenners Ergebnis zu verallgemeinern.

Cited by 60 publications

References 6 publications

On singular modular forms for principal congruence subgroups

On singular modular forms for principal congruence subgroups

Cohomology of the boundary of Siegel modular varieties of degree two, with applications

Subgroups of SL(t, Z)

Contact Info

Product

Resources

About