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Explore the Tarski problem for hyperbolic groups with torsion in this 55-minute mathematical lecture. Delve into the groundbreaking work of Sela and Kharlampovich-Myasnikov who proved that non-abelian free groups are elementarily equivalent, solving Tarski's famous 1945 problem. Examine Sela's complete classification of finitely generated groups that are elementarily equivalent to torsion-free hyperbolic groups, while discovering why such classification remains elusive for hyperbolic groups with torsion. Learn about partial results addressing this challenging mathematical frontier and understand the new complexities introduced by non-trivial elements of finite order. Investigate the pathological behavior of morphisms from surface groups to hyperbolic groups with torsion and gain insight into the fundamental obstacles that make this problem particularly difficult in the presence of torsion elements.
