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Alexander Yu. Olshanskii

A.Yu. Olshanskii is a Centennial Professor at the Department of Mathematics of Vanderbilt University. Before joining Vanderbilt University, he was a Professor of Mathematics in Moscow State University. His research expertise is mostly in combinatorial and geometric group theory although he has made significant contributions to other areas (finite groups and Lie algebras, in particular). There are very few specialists in group theory whose contributions to the modern understanding of group theory is comparable to Olshanskii's. He solved several key problems in group theory including

• B.H. Neumann's problem about existence of non-finitely based varieties of groups,
• Shmidt-Tarski's problem about existence of infinite non-cyclic groups with all proper subgroups cyclic of prime order,
• von Neumann's problem about existence of non-amenable groups without free non-cyclic subgroups,
• Gromov's problem about existence of infinite quotients of finite exponent for non-elementary hyperbolic groups,
• Gromov's problem about possible distortions of subgroups of finitely presented groups.

Olshanskii's geometric method of graded van Kampen diagrams allowed him and his students to solve many other old and well-known problems in group theory. This includes the solution of Burnside problem for even exponents by S. Ivanov, a former student of Olshanskii, and a construction of a finitely generated non-trivial divisible group by V. Guba, another former student of Olshanskii. The latest applications of his method were a construction of a finitely presented non-amenable group without free non-abelian subgroups (by A.Yu. Olshanskii and M. V. Sapir), and the construction of an infinite finitely generated group with exactly two conjugacy classes (by D. Osin, also a former student of Olshanskii).

Many of the monster groups constructed by Olshanskii and his students are, in modern terms, inductive limits of Gromov-hyperbolic groups. Hyperbolicity plays an important role in Olshanskii's method, and several well known facts about Gromov-hyperbolic groups can be traced back to papers of Olshanskii. After hyperbolic groups were formally introduced into group theory by Gromov, Olshanskii established several key facts about them including the (strong) genericity of hyperbolic groups (conjectured by Gromov), SQ-universality of non-elementary hyperbolic groups, and others.

A.Yu. Olshanskii has more than 20 PhD students . He wrote a very influential book Geometry of defining relations in groups , and several big survey papers. He was an invited speaker at the ICM in Warsaw, 1982, and many other international conferences. Olshanskii is a recipient of several prizes including the Malcev's prize of the Russian Academy of Sciences, Kargapolov prize, and the prize of the Moscow Mathematical Society.