How do you prove that a group specified by a presentation is infinite?

$\langle x,y \; | \; x^2=y^3=1 \rangle \cong \operatorname{PSL}_2(\mathbb Z)$ and this isomorphism identifies G with $\operatorname{PSL}_2/T^7=1$ (where $T:z\mapsto z+1$). Result is the symmetry group of the tiling of the hyperbolic plane. From this description one can see that G is infinite (e.g. because there are infinitely many triangles in the tiling and G acts on them transitively).

Grigory has already answered your particular question. However, I wanted to point out that your question "How do you prove that a group specified by a presentation is infinite?" has no good answer in general. Indeed, in general the question of whether a group presentation defines the trivial group is undecidable.

The group:

$$G = \left\langle x, y \mid x^l = y^m = (xy)^n = 1 \right\rangle$$ is triangular group so, if $\frac{1}{l} + \frac{1}{m} + \frac{1}{n} \lt 1$, then this group is infinite...