List some cosets of infinite finitely presented group

asked 2024-04-17 16:30:45 +0200

karl111 gravatar image

updated 2024-04-17 23:13:13 +0200

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?

edit retag flag offensive close merge delete

Comments

The Braid Group with 6 strands is:

B.<a,b,c,d,e> = BraidGroup(6)

and in it we have for instance:

sage: a*c == c*a
True

My generators $a,b,c,d,e$ above are thus not the generators from the posted question. This group $B$ is something quite different. The given group $G$ is a quotient of the Coxeter group (https://en.wikipedia.org/wiki/Coxeter...) where the diagram is connecting any two different bullets. (Else we have at least two generators that commute...)

So please make clear what is $H$, how is $G$ a factor of a braid group $B_6$, which result of Coxeter is involved, so that the relation $\frac 16+\frac 13\le \frac 12$ should ring a bell, and how to get a finite list of representative for which finite group... Describing the mathematical problem can help.

dan_fulea gravatar imagedan_fulea ( 2024-04-29 15:34:15 +0200 )edit
add a comment