TITLE: Simplicity of infinite groups
QUESTION [7 upvotes]: Sorry about asking so many questions, but I am a bit further on in my classification, and I am up to the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10}, ([a,b]^4b)^7 \rangle$. It has no small quotients (up to 500000), and I suspect that it is simple. Is there a way to check for simplicity in infinite groups?

UPDATE (edit by YC)
The original question about the group $G$ has been answered below. Now I am up to the two groups $H := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10}, ([a,b]^4b)^8 \rangle$ and
$I := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10}, ([a,b]^4b)^9 \rangle$. I need to know if they are trivial, finite (the order would be good), or infinite.

REPLY [18 votes]: In fact your group is trivial. Here are two different computations with Magma, the first using coset enumeration over the subgroup $\langle ab \rangle$, and the second using the Knuth-Bendix completion algorithm.
 > G<a,b>:=Group<a,b|a^2,b^3,(a*b)^7,(a,b)^10, ((a,b)^4*b)^7 >;
 > H:=sub<G|a*b>;
 > time Index(G,H:CosetLimit:=100000000,Hard:=true); 
 1
 Time: 27.730

 > time  R := RWSGroup(G:MaxRelations:=100000, TidyInt:=1000);
 Time: 90.350
 > Order(R);
 1