Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics
Tatiana Bandman, Shelly Garion, Boris Kunyavskii

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.
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Taxonomy
TopicsFinite Group Theory Research · Coding theory and cryptography · Geometric and Algebraic Topology
