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In fact, we can describe your group $G$ quite explicitly. The presentation has the property that each generator appears in exactly two relators. Therefore, the corresponding presentation complex $X$ is locally a surface everywhere except at the unique vertex. In particular, $X$ is homeomorphic to a surface $S$ with some number of points identified. To ...
There are some other types of groups for which this type of algorithm would work. The easiest examples are abelian groups. For example, with the free abelian group of rank $2$, $\langle x,y, \mid xy=yx \rangle$, if you systematically make the substitutions $y^ax^b \mapsto x^ay^b$ with $a,b = \pm 1$, together with free reductions, then you can transform any ...