How to show a group is infinite group by constructing epimorphism? 
Consider a group G with representation
  $$\langle a,b\mid abab^{-1}a^{-1}b^{-1}\rangle$$
  Prove that this group is an infinite group.

There is similar question here (Finding the kernel of an epimorphism onto $S_3$? ) asking how to construct an epimorphism into $S_3$ and give a representation of the kernel. I believe this is the right way to approach this problem. However, the answers the Reidemeister-Schreier algorithm.
So my question is:
1) How to give a representation of the kernel just using the basic of group theory?
2) With 1) being proved, how to use this fact to prove the group is infinite?
PS: I know this is called the braid group. However, this question is asked by one of my friends (I feel ashamed that I cannot answer his question) who just begin his journey in group theory. So I do not want to use advanced tools to perform an overkill, and I believe this is possible.
 A: *

*For point 1)


Finding the presentation of a kernel is relatively easy, as there is a standard algorithm, called the Reidemeister-Schreier algorithm. This has been implemented in, for example, GAP. I gave a rough explaination of the algorithm here, where there are also links to worked examples.


*

*For point 2)


I'll explain two ways of showing this group is infinite. Only the second uses the Reidemeister-Schreier algorithm (which is an "advanced tool" and definitely "overkill"!). The first method is better here.
Method 1.
One way to show that a group $G$ is infinite is to find an epimorphism onto a known infinite group. You can do this here by computing the abelianization, which is $\mathbb{Z}$:
$$
\begin{align*}
\langle a,b\mid abab^{-1}a^{-1}b^{-1}\rangle^{ab}
&=\langle a,b\mid abab^{-1}a^{-1}b^{-1}, [a, b]\rangle\\
&=\langle a,b\mid ab^{-1}, [a, b]\rangle\\
&\cong\langle a\mid -\rangle\\
&\cong\mathbb{Z}.
\end{align*}
$$
Method 2.
Another way to show that a group is infinite is to find an infinite subgroup. This seems to be the method which the question is getting at, but is harder in this case. The link in the question proves that for $G=\langle a,b\mid abab^{-1}a^{-1}b^{-1}\rangle$ the group here, we have that $G\twoheadrightarrow S_3$ and the kernel of this map has presentation $$\langle x,y,z \mid y^{-1}x^{-1}yzxz^{-1}, y^{-1}zxyx^{-1}z^{-1}\rangle.$$
(This is the precise example in the linked question!) As Derek Holt points out in his answer to the linked question, this is a presentation for the Pure Braid Group on $3$ strands $P_3$, which is a known infinite group.
That said, the question states that the given group is $B_3$ itself. It is relatively clear that $B_3$ is infinite, as twisting two strands corresponds to a copy of $\mathbb{Z}$...
