Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

TL;DR

This survey reviews recent advances in solving word equations in simple groups, highlighting interdisciplinary methods from algebra, geometry, number theory, and dynamics that led to significant breakthroughs.

Contribution

It synthesizes diverse approaches and results from the past decade, emphasizing the interplay of different mathematical disciplines in understanding equations in simple groups.

Findings

Solutions to several long-standing problems

Interdisciplinary methods advancing the field

Connections between algebra, geometry, and dynamics

Abstract

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods, group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.