Let $G=\langle S \mid R_1 \cup R_2 \cup R_3 \rangle$ be a group presentation with $S=\{a,b,c\}$, $R_1=\{aa{^{\text{-}1}}, bb{^{\text{-}1}}, c^2\}$, $R_2$ the set of all circular shifts of the word $w=a{^{\text{-}1}}bab{^{\text{-}1}}c$ and the set $R_3$ all circular shifts of the reverse inverse of $w$, $w^R=c{^{\text{-}1}}ba{^{\text{-}1}}b{^{\text{-}1}}a=cba{^{\text{-}1}}b{^{\text{-}1}}a$.
So $R_2 = {\{a{^{\text{-}1}}bab{^{\text{-}1}}c, \\\\ ca{^{\text{-}1}}bab{^{\text{-}1}}, \\\\ b{^{\text{-}1}}ca{^{\text{-}1}}ba, \\\\ ab{^{\text{-}1}}ca{^{\text{-}1}}b, \\\\ bab{^{\text{-}1}}ca{^{\text{-}1}} \}}$
and $R_3 = {\{cba{^{\text{-}1}}b{^{\text{-}1}}a, \\\\ acba{^{\text{-}1}}b{^{\text{-}1}}, \\\\ b{^{\text{-}1}}acba{^{\text{-}1}}, \\\\ a{^{\text{-}1}}b{^{\text{-}1}}acb, \\\\ ba{^{\text{-}1}}b{^{\text{-}1}}ac \}}$
There must be some much more concise way of defining $R_2$ and $R_3$. Can anyone help me out here?
As I said in my comment, we have $$G = \langle a,b,c \mid c^2,w \rangle,$$ which simplifies to $$G \cong \langle a,b \mid (ba^{-1}b^{-1}a)^2\rangle \cong \langle a,b \mid [a,b]^2\rangle,$$ which is a lot more concise than your version!