Frank Harou writes:

I would like computing the rank of K/[K,K] where K is the kernel of a morphisme \psi defined as follow :

un:=Z(7)^0;
zero:=0*Z(7);

gk1:=[[1,1],[0,1]];
gk2:=[[1,0],[-E(3),1]];

gi1:=[[un, un], [zero, un]];
gi2:=[[un, zero], [2*un, un]];

GammaK:=Group([[1,1],[0,1]], [[1,0],[-E(3),1]]);
GammaI:=Group(gi1,gi2);

psi:=GroupHomomorphismByImages(GammaK,GammaI,
GammaK.generators,GammaI.generators);

Does a trick exist to avoid the instruction kernel(psi) ?

My feeling is that this is difficult without more information on the group
GammaK. It is difficult to carry out nontrivial calculations in an infinite
matrix group without further information, since all one can do is multiply
elements and check for equality.
Of course, since the image of your map psi has finite order (336), one could
work out Schreier generators for the kernel K, but there would be 337 of
those, and it might be hard to do much with them.
The most satisfactory form would be a presentation of GammaK on
the defining generators. I know a presentation of the Bianchi group SL_2(O_3),
where O_3 is the ring Z[E(3)], but that has more generators, and Gammak
may well be a proper subgroup of SL_2(O_3).

Derek Holt.