How to prove this group is infinite order？

Question

Given a group $G$ with presentation $$G=\langle u,v\mid u^3=v^3=(uv)^3=1\rangle$$ How to prove $G$ is infinite? In general, do we have any systematic methods on finding the order of a group given by presentation?

Answer 1

It's undecidable from a finite presentation whether a finitely presented group is finite, by the Adyan-Rabin theorem (the trivial group is a positive witness and $\mathbb{Z}$ is a negative witness). In other words, one must use cleverness.

In general, we can try to prove that a group $G$ given by a presentation is infinite by finding a homomorphism into some other group which we know is infinite, and which has infinite image. In this case we are lucky: this group is the von Dyck group $D(3, 3, 3)$, which is known to be the group of orientation-preserving symmetries of the following tesselation of $\mathbb{R}^2$ by triangles:

$u$ and $v$ can be taken to be rotations about two of the vertices of any of the triangles, and $uv$ is a rotation about the third vertex (or so the Wikipedia article claims, but I'm having trouble actually visualizing this...). This gives a homomorphism from $G$ to the oriented Euclidean group $\mathbb{R}^2 \rtimes SO(2)$ (which is even injective but we don't need this) whose image contains translations, and so in particular is infinite.

In general the von Dyck group

$$D(\ell, m, n) = \langle x, y, z \mid x^\ell = y^m = z^n = xyz = 1 \rangle$$

is known to be infinite iff $\frac{1}{\ell} + \frac{1}{m} + \frac{1}{n} \ge 3$; this comes from its interpretation as the group of orientation-preserving symmetries of a tiling of either the sphere, the Euclidean plane, or the hyperbolic plane by triangles, depending on the value of this sum.

Answer 2

Let $\omega=\frac{-1+i\sqrt3}2$ be a third root of unity and consider the bijective maps $U,V\colon \Bbb C\to \Bbb C$ given by $U(z)=\omega z$ and $V(z)=\omega (z-1)+1$ — in other words rotation by 120 degrees around $0$ and $1$, respectively. Show that $U\circ V$ is a rotation by 240 degrees around some point. Conclude that there is a homomorphism $\phi$ from $G$ to the group of permutations of $\Bbb C$ with $\phi(u)=U$ and $\phi(v)=V$. Show that $V\circ U$ is a translation. Conclude.

A possible general strategy is to find a homomorphism to a concretely group. But beware: We did not show that $\phi$ is injective, but that was good enough for showing infinite order.