Free Group - Tarski's Problems

Tarski's Problems

Around 1945, Alfred Tarski asked whether the free groups on two or more generators have the same first order theory, and whether this theory is decidable. Sela (2006) answered the first question by showing that any two nonabelian free groups have the same first order theory, and Kharlampovich & Myasnikov (2006) answered both questions, showing that this theory is decidable.

A similar unsolved (in 2011) question in free probability theory asks whether the von Neumann group algebras of any two non-abelian finitely generated free groups are isomorphic.