Finite two-relator groups and quotients of knot groups

Question

Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \rangle$ for some $n > 0$.

1. What are the finite two-relator groups?

This question might be unanswerable. An indication of why this problem might be difficult comes from a classical example due to Dehn. He shows that for any $n \in \mathbb{N}$ and $m \in \{ 2, 3, 4, 5\}$, the group $$ G_{m,n} = \langle a,b \mid a^2 = b^3, \: (ab)^m b^{3n} = 1 \rangle $$ is finite, i.e. a quotient of the trefoil knot group. Indeed, he shows that these groups fit in short exact sequences $$ 1 \longrightarrow C_k \longrightarrow G_{m,n} \longrightarrow H_m \longrightarrow 1 $$ where $k \in \{ 3n + 5, 4n + 10, 6n + 20, 12n + 50 \}$ and $H_m \in \{ S_3, A_4, S_4, A_5 \}$ depending on the value of $m$ (indexed as above). Thus the question is already quite non-trivial even when one of the relators is rather simple. A partial and valuable answer to Question 1 would hence be an answer to the following question:

2. What are the finite (relative) one-relator quotients of the trefoil knot group $\langle a, b \mid a^2 = b^3 \rangle$?

In particular, is $G_{m,n}$ finite for any $m>5$ and $n > 0$? (I suspect not). Any thoughts would be welcome!

Answer 1

Question 1:

As mentioned in comments, presentations with the same number of generators and relations are called balanced. The triviality problem for balanced presentations appears to be a question of Magnus (see Ian Leary's comment below, cf. Question 13.9 on Bestvina's problem list).

It is in a similar vein to the more famous Andrews--Curtis conjecture and, just like Andrews--Curtis, is wide open. Relaxing "triviality" to "finiteness" and restricting to two relations isn't going to help, so I feel confident in claiming that Question 1 (in the sense of algorithmic decidability) is an open problem.

Question 2:

Again, I doubt there is a reasonable full description.

Edit: apologies, my earlier answer was a little glib here, although I think the spirit -- there's unlikely to be a satisfying classification -- is correct.

Groups of this form are central extensions of generalised triangle groups in the sense of

Baumslag, Morgan, Shalen, Generalized triangle groups. Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 1, 25–31.

There's a disparate literature on this subject, but I'd be very surprised if it is completely understood which ones are finite. (This feels in a similar vein to the isomorphism problem for one-relator groups, say.)

For your specific question about Dehn's family, note that

$G_{m,n}/\langle b^3\rangle\cong \langle a,b\mid a^2,b^3,(ab)^m\rangle$

which is the $(2,3,m)$-triangle group. In particular, $G_{m,n}$ is infinite whenever $m>5$ and $n>0$.