Grigory Margulis

Grigory Margulis is a Russian-American mathematician whose work connected ergodic theory with the structure of Lie groups and arithmetic groups. He received the 1978 Fields Medal.

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Ergodic theory: The study of measure-preserving systems and their long-term statistical behavior. Margulis used dynamical methods to prove rigidity results about lattices.
Margulis superrigidity theorem: A theorem extending suitable linear representations of higher-rank lattices to representations of their ambient algebraic groups. This theorem is a signature result linking lattice representations to the surrounding Lie group.
David Kazhdan: A mathematician known for work in representation theory, property (T), and algebraic groups. Kazhdan’s property (T) is closely tied to rigidity questions Margulis pursued.
Borel density theorem: A theorem stating that certain lattices in semisimple groups are Zariski dense. It is a foundational result used in the setting of Margulis’s arithmeticity theorem.
Lie group: A group that is also a smooth manifold, with compatible multiplication and inversion. Continuous symmetries in Lie groups underpin many of the groups Margulis studied.
Margulis arithmeticity theorem: A theorem stating that irreducible lattices in higher-rank semisimple groups are arithmetic, subject to standard hypotheses. It turns rigidity of lattice embeddings into a classification by arithmetic origin.
G.A. Margulis’s doctoral advisor: Yakov Sinai is a Russian mathematician who made major contributions to ergodic theory and dynamical systems. Sinai advised Margulis at Moscow State University, where Margulis developed his early work.
Superrigidity: A phenomenon in which representations of a discrete group extend rigidly to a larger ambient group. Margulis’s theorem became the model for superrigidity across geometry and group theory.
Lattice (discrete subgroup): A discrete subgroup of a locally compact group whose quotient has finite invariant volume. Lattices are the central objects in Margulis’s rigidity and arithmeticity theorems.
Kazhdan property (T): A rigidity property of topological groups that forces unitary representations with almost-invariant vectors to have invariant vectors. Property (T) is among the rigidity phenomena connected to higher-rank groups and lattices.