List some cosets of infinite finitely presented group
Example:
F.<a,b,c,d,e> = FreeGroup();
H = [a^3,
a*b*a*b^(-1)*a^(-1)*b^(-1),
b*c*b*c^(-1)*b^(-1)*c^(-1),
c*d*c*d^(-1)*c^(-1)*d^(-1),
d*e*d*e^(-1)*d^(-1)*e^(-1),
a*c*a^(-1)*c^(-1),
a*d*a^(-1)*d^(-1),
a*e*a^(-1)*e^(-1),
b*d*b^(-1)*d^(-1),
b*e*b^(-1)*e^(-1),
c*e*c^(-1)*e^(-1),];
G = F/H
We take the freegroup on 5 generators and quotient out the braidrelations. This yields the braidgroup on 6 strands. Since we also quotient out a^3, we get that G is a factor group of the braidgroup on 6 strands. By a result from Coxeter (since 1/6
Geometrically, we are looking at braids up to the t_k move, which untwists k consecutive halftwists. The t_2 move forgets about over and undercrossings and thus B(n) mod t_2 = S(n). In 'Factorgroups of the braidgroup' coxeter studies those groups and proves that the factor group B(n)/a^k is finite iff 1/n + 1/3 <= 1/2) 1/k > 1/2.
In my case this means that B(6)/a^3 = G is infinite.infinite.
Is there a way to still get a finite list of representatives of the cosets?