TITLE: How do you prove that a group specified by a presentation is infinite?
QUESTION [33 upvotes]: The group:
$$
G = \left\langle x, y \;  \left| \;  x^2 = y^3 = (xy)^7 = 1\right. \right\rangle
$$
is infinite, or so I've been told. How would I go about proving this? (To prove finiteness of a finitely presented group, I could do a coset enumeration, but I don't see how this helps if I want to prove that it's infinite.)

REPLY [4 votes]: One general way to do this, (which is not guaranteed to work, as Noah points out), is to exhibit infinitely many different homomorphic images of the group you start with . In this case, any group generated by an element of order $2$ and an element of order $3$ whose product has order $7$ is a homomorphic image of the group $G$. Such a group is a Hurwitz group, and these are well studied, for example in the work of M. Conder and of G. Higman, among others. Infinitely many finite simple groups are known to be Hurwitz groups, I believe.