From: Marston Conder 
Date: Wed, 6 May 1998 22:20:51 +1200 (NZST)
Subject: Is SL(3,Z) a quotient of PSL(2,Z) ? 


Here's a question I've put to a couple of pubbers, but now seems worth 
asking around further: 

  Is the group SL(3,Z) a factor group of the modular group PSL(2,Z)?  
  and if not, then why not?  

Note the modular group is isomorphic to the free product C2*C3, so the 
question asks if SL(3,Z) can be generated by two elements u and v of 
orders 2 and 3 (e.g. the images of the transformations t -> -1/t and 
t -> (t-1)/t under some homomorphism from PSL(2,Z) to SL(3,Z)).  

It is well known that SL(3,Z) can be generated by the six transvections 
T_{ij} = I + E_{ij} (for distinct i,j in {1,2,3}), with defining relations  
   (T_{12}T_{21}^{-1}T_{12})^4 = 1,  
   [T_{ij},T_{ik}] = [T_{ij},T_{kj}] = 1  whenever i,j,k are distinct, 
   [T_{ij},T_{jk}] = T{ik}  whenever i,j,k are distinct. 

Not so well known is the alternative presentation 
 < x,y,z | x^3 = y^3 = z^2 = (xz)^3 = (yz)^3 
            = (x^{-1}zxy)^2 = (y^{-1}zyx)^2 = (xy)^6 = 1 > 
in terms of the 3 by 3 matrices   
   x = mat( 0, 1, 0 : 0, 0, 1 : 1, 0, 0 )  
   y = mat( 1, 0, 1 : 0,-1,-1 : 0, 1, 0 )
   z = mat( 0, 1, 0 : 1, 0, 0 :-1,-1,-1 ) .  

From this it may easily be verified that SL(3,Z) can be generated by two 
elements of order 3, for example xz and yz, and further, that there are 
several subgroups of small finite index in SL(3,Z) which can be generated 
by an involution and an element of order 3, 
e.g.  the subgroup generated by x^{-1}zxy and x has index 57 (and maps onto 
SL(3,p) by reduction mod p for every prime p other than 7), 
while the subgroup generated by z and (xy)^2 has index 117 (and maps onto 
SL(3,p) by reduction mod p for every prime p other than 3).  
  
Does this information help?  

All the best 
Marston Conder