Alexander Yu. Olshanskii (; born 19 January 1946) is a Russian-American mathematician renowned for his contributions to combinatorial and geometric
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
. He is particularly noted for constructing infinite groups with unusual properties and for developing geometric methods in group theory. Olshanskii has also made significant contributions to the study of
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s and algebraic systems.
He served as the Centennial Professor of Mathematics at
Vanderbilt University
Vanderbilt University (informally Vandy or VU) is a private university, private research university in Nashville, Tennessee, United States. Founded in 1873, it was named in honor of shipping and railroad magnate Cornelius Vanderbilt, who provide ...
from 1999 until becoming professor emeritus in 2024.
Early life and education
Olshanskii was born in Saratov, Russia, into a family of a military aviation engineer and is one of three brothers. He completed high school in
Moscow State University
Moscow State University (MSU), officially M. V. Lomonosov Moscow State University,. is a public university, public research university in Moscow, Russia. The university includes 15 research institutes, 43 faculties, more than 300 departments, a ...
in 1968 with a B.S. in Mathematics. He earned his Ph.D. in Mathematics in 1971 under the supervision of Alfred Lvovich Shmelkin, with a dissertation titled "Some Questions of the Theory of Group Varieties." He later received his
Doctor of Sciences
A Doctor of Sciences, abbreviated д-р наук or д. н.; ; ; ; is a higher doctoral degree in the Russian Empire, Soviet Union and many Commonwealth of Independent States countries. One of the prerequisites of receiving a Doctor of Sciences ...
degree (Habilitation) in 1979.
Academic career
From 1970, Olshanskii was a faculty member in the Department of Mechanics and Mathematics at Moscow State University, becoming an associate professor in 1978 and a full professor in 1985. In 1983, he was an invited speaker at the
International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the IMU Abacus Medal (known before ...
in Warsaw, where he presented on geometric methods in combinatorial group theory.
In 1999, he joined Vanderbilt University as the Centennial Professor of Mathematics. He was named a Fellow of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
in 2014 for his contributions to group theory. He became professor emeritus at Vanderbilt University in 2024.
Research Contributions
Olshanskii's research has significantly influenced combinatorial and geometric group theory. In 1969, while still a graduate student, he resolved a problem posed by Bernard Neumann in 1935 concerning the existence of an infinite system of group identities not equivalent to any finite system. This achievement garnered recognition from Neumann, who was then at Vanderbilt University.
In the late 1970s and early 1980s, Olshanskii introduced graded van Kampen diagrams, a refinement of the classical van Kampen diagrams. Using this technique, he constructed the so-called Tarski monster groups: infinite groups of bounded exponent in which every proper subgroup is cyclic. These constructions provided counterexamples to several longstanding conjectures in group theory, including the von Neumann conjecture.
In 1980, he constructed a counterexample to the von Neumann–Day problem, demonstrating the existence of non-amenable groups without non-cyclic free subgroups.
Olshanskii also provided a new geometric proof of the Novikov--Adyan theorems addressing the Burnside problem. His proof for large odd exponents was notably concise—32 pages compared to the original's 300—and is still considered the most succinct.
His work has influenced the development of hyperbolic groups, a concept central to geometric group theory. Olshanskii extended his methods to study small cancellation theory and van Kampen diagrams over hyperbolic groups, and investigated factor groups of such groups.
In recent years, Olshanskii has focused on asymptotic invariants of groups, such as Dehn functions, distortion, and relative subgroup growth. These invariants are connected to the complexity of algorithmic problems in group theory. In collaboration with M. Birget, E. Rips, and M. Sapir, he developed a geometric criterion for determining when the word problem in a finitely presented group can be solved in nondeterministic polynomial time.
Olshanskii has authored over 100 scientific papers, including the influential monograph ''Geometry of Defining Relations in Groups''.
Honors and awards
* Invited Speaker at the
International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the IMU Abacus Medal (known before ...
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...