My adviser recently shared a problem with me that seeks to establish non-elementary* hyperbolic quotients for mapping class groups. They told me that this could be useful for establishing results on separability or omnipotence, and that these could be relevant for examining profinite rigidity of hyperbolic 3-manifolds. Unfortunately, I'm not fully read up on these topics.
In this recent paper, Behrstock et. al., Hagen, Martin and Sisto also seek to make headway on the question of hyperbolic quotients for mapping class groups. They have a discussion in their introduction of the relevance of this question, mentioning again separability and omnipotence, profinite rigidity, and placing things in the context of the virtual Haken conjecture. Again, I'm a little ignorant of these topics and their history of this point. So my question:
Q: Can anyone explain with some detail (or point me to some nice references as to) why it's relevant that mapping class groups have hyperbolic quotients? Or why it's helpful that any group has such a quotient?
*A Gromov-hyperbolic group is non-elementary if it is not virtually cyclic, i.e. is infinite and not virtually $\mathbb{Z}$.
