Hurwitz and (2,3)-generated matrix groups

A classical problem, which goes back to the XIXth century (Klein and others), asks for the description of normal subgroups and factors of the modular group $\mathrm{PSL}(2,\mathbb Z)$. The modular group is isomorphic to the free product of two cyclic groups of orders 2 and 3. Thus, the original problem can be reduced to determination of the groups that can be generated by an involution and an element of order 3. A related problem (considered by Hurwitz) arises if one requires that the order of the product of the above generators is 7.
In this general setting both problems are hopeless. Usually some important classes of groups (like finite simple or classical matrix groups) are considered. Even in this setting the problems remain open. For matrix groups of large rank the picture is more or less well understood, but the case of small ranks is more difficult and more interesting.
In the talk I shall present an overview of the problem and recent advancements due to the works of A. E. Zalesski, M. C. Tamburini and the speaker.