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

TL;DR

This paper improves bounds on the number of solutions and the structure of graphs related to Markoff triples modulo prime p, advancing towards the conjecture that such solutions may not exist outside the main component.

Contribution

It refines previous bounds on solutions and component sizes in Markoff triple graphs modulo p, providing progress towards the conjecture of their nonexistence outside the giant component.

Findings

Sharpened bounds on nodes outside the giant component

Improved estimates on sizes of connected components

Progress towards the conjecture of no solutions outside the main component

Abstract

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.