# Revision history [back]

### List some cosets of infinite finitely presented group

Example:

F.< a,b,c,d,e> = FreeGroup();

H = [a^3, abab^(-1)a^(-1)b^(-1), bcbc^(-1)b^(-1)c^(-1), cdcd^(-1)c^(-1)d^(-1), dede^(-1)d^(-1)e^(-1), aca^(-1)c^(-1), ada^(-1)d^(-1), aea^(-1)e^(-1), bdb^(-1)d^(-1), beb^(-1)e^(-1), cec^(-1)e^(-1),];

G = F/H

G is a factor group of the braidgroup on 6 strands. By a result from Coxeter (since 1/6 + 1/3 <= 2) G is infinite.

Is there a way to still get a finite list of representatives of the cosets?

### List some cosets of infinite finitely presented group

Example:

F.< a,b,c,d,e> = FreeGroup();

H = [a^3, abab^(-1)a^(-1)b^(-1), bcbc^(-1)b^(-1)c^(-1), cdcd^(-1)c^(-1)d^(-1), dede^(-1)d^(-1)e^(-1), aca^(-1)c^(-1), ada^(-1)d^(-1), aea^(-1)e^(-1), bdb^(-1)d^(-1), beb^(-1)e^(-1), cec^(-1)e^(-1),];

G = F/H

G is a factor group of the braidgroup on 6 strands. By a result from Coxeter (since 1/6 + 1/3 <= 2) 1/2) G is infinite.

Is there a way to still get a finite list of representatives of the cosets?

### List some cosets of infinite finitely presented group

Example:

F.< a,b,c,d,e> a,b,c,d,e > = FreeGroup();

H = [a^3, abab^(-1)a^(-1)b^(-1), bcbc^(-1)b^(-1)c^(-1), cdcd^(-1)c^(-1)d^(-1), dede^(-1)d^(-1)e^(-1), aca^(-1)c^(-1), ada^(-1)d^(-1), aea^(-1)e^(-1), bdb^(-1)d^(-1), beb^(-1)e^(-1), cec^(-1)e^(-1),];

G = F/H

G is a factor group of the braidgroup on 6 strands. By a result from Coxeter (since 1/6 + 1/3 <= 1/2) G is infinite.

Is there a way to still get a finite list of representatives of the cosets?

### List some cosets of infinite finitely presented group

Example:

F.< a,b,c,d,e >

F.<a,b,c,d,e> = FreeGroup();  H = [a^3,
abab^(-1)a^(-1)b^(-1),
a*b*a*b^(-1)*a^(-1)*b^(-1),
bcbc^(-1)b^(-1)c^(-1),
b*c*b*c^(-1)*b^(-1)*c^(-1),
cdcd^(-1)c^(-1)d^(-1),
c*d*c*d^(-1)*c^(-1)*d^(-1),
dede^(-1)d^(-1)e^(-1),
d*e*d*e^(-1)*d^(-1)*e^(-1),
a*c*a^(-1)*c^(-1), a*d*a^(-1)*d^(-1),
aea^(-1)e^(-1), a*e*a^(-1)*e^(-1),  bdb^(-1)d^(-1),
b*d*b^(-1)*d^(-1),
beb^(-1)e^(-1), b*e*b^(-1)*e^(-1),  cec^(-1)e^(-1),]; c*e*c^(-1)*e^(-1),];  G = F/HF/H


G is a factor group of the braidgroup on 6 strands. By a result from Coxeter (since 1/6 + 1/3 <= 1/2) G is infinite.

Is there a way to still get a finite list of representatives of the cosets?

### 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


G is a factor group of the braidgroup on 6 strands. By a result from Coxeter (since 1/6 + 1/3 <= 1/2) G is infinite.

Is there a way to still get a finite list of representatives of the cosets?

### 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


G is a factor group of the braidgroup on 6 strands. By a result from Coxeter (since 1/6 + 1/3 <= 1/2) G is infinite.

Is there a way to still get a finite list of representatives of the cosets?

### 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?

### 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 free group on 5 generators and quotient out the braidrelations. braid relations. This yields the braidgroup braid group on 6 strands. Since we also quotient out a^3, we get that G is a factor group of the braidgroup on 6 strands.

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' 'Factor groups of the braid group' coxeter studies those groups and proves that the factor group B(n)/a^k is finite iff 1/n + 1/k > 1/2.

In my case this means that B(6)/a^3 = G is infinite.

Is there a way to still get a finite list of representatives of the cosets?

### 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 free group on 5 generators and quotient out the braid relations. This yields the braid group on 6 strands. Since we also quotient out a^3, we get that G is a factor group of the braidgroup braid group on 6 strands.

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 'Factor groups of the braid group' coxeter studies those groups and proves that the factor group B(n)/a^k is finite iff 1/n + 1/k > 1/2.

In my case this means that B(6)/a^3 = G is infinite.

Is there a way to still get a finite list of representatives of the cosets?