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I have a set of relators $[r_1, r_2, r_3, r_4]$ of a free group on two generators $a$ and $b$, where $\begin{cases} r_1 = a^{2p} \text{ (where $p$ is some positive integer) } \\ r_2 = b^4 \\ t_3 = (ab)^2,\\ r_4 = (ab^{-1})^2 \end{cases}$

how one can express words like $w = a^{-2}ba^2b^{-1}$ that lie in the group generated by the $r_i$ as words formed by the $r_i$. What I tried was to take a $r$ that is an $r_i$ up to a certain power or up to a cyclic rotation of its generators (in this example $ba$ or $ba^{-1}$) and make the product $wr$ such that there are less generators in the original word. I don't know if this method always works as I'm afraid that the resulting words will become bigger and bigger. Is there an specific method that always gives a result?

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    Question: you talk about "the group generated by the $r_i$". What is usually regarded as of primary significance is "the group *normally* generated by the $r_i$". For example, the definition of the group with generators $a,b$ and relators $r_1,r_2,r_3,r_4$ is the quotient of the free group $F(a,b)$ modulo the group *normally* generated by $r_1,r_2,r_3,r_4$. Was that your intent? – 2017-02-08
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    @LeeMosher: I meant the subgroup of the free group. – 2017-02-08

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This is in general a hard problem. You might want to read up on Dehn's algorithm (which is a bit like what you hope for, but requires special circumstances):

https://en.wikipedia.org/wiki/Small_cancellation_theory

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    I was just reading your paper *Notes on Computational Group Theory* online. I'm now working the other way round and building permutations and force them to obey the relator restrictions, using GAP on examples. – 2017-02-09