Let $G$ be finitely presented group with $n$ generators and $r$ relations where $n>r$. I want to show that $G$ has an element with infinite order.
My attempt:
Assume that $F_n$ is free group with $n$ generators. We know that $F_n/F^{\prime}_n\cong\mathbb Z ^n$. So $F_n$ has an element of infinite order, but I don't know how use the relations?
Hint: Consider the abelianisation of your group. This is a finitely generated abelian group, and you want to prove that its decomposition contains a $\mathbb{Z}$-term. Consider your abelian group as a $\mathbb{Z}$-module, form an appropriate matrix and diagonalise. What do you notice?
Sorry, I do not got the time to expand on this more just now. But my point is basically: compute the abelinisation, and do what you see here. What do you notice?