TITLE: How to approach proofs similar to "Show a group, $G$, is infinite if $G = \langle r, s, t\mid rst = 1\rangle $"
QUESTION [7 upvotes]: How to approach proofs similar to "Show a group, $G$, is infinite if $G = \langle r, s, t\mid rst = 1\rangle $"

I have not worked much with relations and tend to get lost in notation. I am practicing solving problems like the one in the title but am having a hard time as I am not sure the tricks to try or areas to investigate first in trying to make a proof. What are some hints for starting a proof about some quality of a group defined by a relation? 
So far the only relations I know about are the dihedral groups of order $2n$, the quaternions, and cyclically generated groups so comparisons to how we show properties of those might be illuminating.

REPLY [3 votes]: I want to stick to generalities with this answer, but the underlying point is: you can see that your group is infinite, and indeed is "large", simply by looking at the presentation. No calculations are needed.

One way to try and prove that a group is infinite is to compute the abelianization of the group (that is, force the generators to pairwise commute) and see if the resulting group is infinite (this is a special case of @lulu's answer). The abelianisation of the group you have here is $\mathbb{Z}\times\mathbb{Z}$. For general methods to compute abelinisations, you might find this question useful.
Now, by considering the abelinisation it can be proven that a presentation with more generators than relators defines an infinite group. In particular, every group with at least two generators and a single defining relation is infinite (these are called "one relator groups", and there is a rich theory of these groups). Equipped with this result, you can see that your group is infinite without having to do any calculations.

A group is large if it has a finite index subgroup which maps onto a non-abelian free group. Clearly, large groups are infinite. In a pleasingly short paper, Benjamin Baumslag and Stephen J. Pride* proved that a presentation with two more generators than relators defines a large group. Hence, your group is large. Gromov then proved that a presentation with more generators than relators such that one relator is a proper power (so of the form $w^n$, $n>1$) defines a large group. Equipped with the Baumslag-Pride result, you can see that your group is large without having to do any calculations (this observation is weaker than @lhf's answer).
*"Groups with two more generators than relators." Journal of the London Mathematical Society 2.3 (1978): 425-426