Other than "a natural generalization of $SL(2,\mathbb{Z})$" (or any sort of better understood $SL(2,?)$" (such as $SL(2,\mathbb{R})$ which is acts isometrically on the upper half plane which lives in the crossroad of many fields), is there any direct reason why one would want to understand this group (and its cohomology ring)?
(In case you're wondering, I'm looking at this paper by Adem and Naffah: https://arxiv.org/pdf/math/9503230.pdf)
Thanks in advance, and please pardon my ignorance.