5

I am looking for an example of a word in a finitely-presented group $G = \langle S | R \rangle$ which represents the identity, but for which the proof that this is so is quite long.

That is, we need to cleverly insert a large number of relations, meanwhile increasing the size of the word, before we can discover that the word represents $1$. Not too long, though. I just want a nice explicit example for purposes of illustrating the phenomenon.

I suppose I could set about trying to manufacture such an example, but I figure that standard examples are out there in the books, and why reinvent the wheel!

Xam
  • 6,338
Mike F
  • 23,118

1 Answers1

5

I think the comment by angravian gives an excellent example.

The Higman group is typically given as a $4$ generator and $4$ relation presentation. It gave some examples of finite presented groups, without nontrivial finite quotients. At the time this was actually the first example.

The trivial group has an analogous presentation on three generators and three relations: $$ \langle x_1, x_2,x_3 \mid x_2x_1x_2^{-1}=x_1^2, x_3x_2x_3^{-1}=x_2^2, x_1x_3x_1^{-1}=x_3^2 \rangle.$$ Proving it is trivial though is quite difficult and all proofs I have seen go through clever manipulations. It is worth giving it a try, I know I learned presentations are difficult things trying to prove that group is trivial.

You can see this answer by Jim Belk which is the cleanest proof I have seen, and uses clever manipulations and ends up with things like $x_1^{2^{16}-2}$ showing up. There is also this answer.