On the new bound for the number of solutions of polynomial equations in subgroups and the structure of graphs of Markoff triples

We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the “giant component” and on the size of individual connected components in the suitably defined functional graph of Markoff triples modulo $p$. This is a step towards the conjecture that there are no such nodes at all.