Grigory Margulis - Scientific Contributions

Scientific Contributions

Early work of Margulis dealt with Kazhdan's property (T) and the questions of rigidity and arithmeticity of lattices in semisimple algebraic groups of higher rank over a local field. It had been known since the 1950s (Borel, Harish-Chandra) that a certain simple-minded way of constructing subgroups of semisimple Lie groups produces examples of lattices, called arithmetic lattices. It is analogous to considering the subgroup SL(n,Z) of the real special linear group SL(n,R) that consists of matrices with integer entries. Margulis proved that under suitable assumptions on G (no compact factors and split rank greater or equal than two), any (irreducible) lattice Γ in it is arithmetic, i.e. can be obtained in this way. Thus Γ is commensurable with the subgroup G(Z) of G, i.e. they agree on subgroups of finite index in both. Unlike general lattices, which are defined by their properties, arithmetic lattices are defined by a construction. Therefore, these results of Margulis pave a way for classification of lattices. Arithmeticity turned out to be closely related to another remarkable property of lattices discovered by Margulis. Superrigidity for a lattice Γ in G roughly means that any homomorphism of Γ into the group of real invertible n × n matrices extends to the whole G. The name derives from the following variant:

If G and G', semisimple algebraic groups over a local field without compact factors and whose split rank is at least two and Γ and Γ ' are irreducible lattices in them, then any homomorphism f: Γ → Γ ' between the lattices agrees on a finite index subgroup of Γ with a homomorphism between the algebraic groups themselves.

(The case when f is an isomorphism is known as the strong rigidity.) While certain rigidity phenomena had already been known, the approach of Margulis was at the same time novel, powerful, and very elegant.

Margulis solved the long-standing Banach–Ruziewicz problem that asks whether the Lebesgue measure is the only normalized rotationally invariant finitely additive measure on the n-dimensional sphere. The affirmative solution for n ≥ 4, which was also independently and almost simultaneously obtained by Dennis Sullivan, follows from a construction of a certain dense subgroup of the orthogonal group that has property (T).

Margulis gave the first construction of expander graphs, which was later generalized in the theory of Ramanujan graphs.

In 1986, Margulis completed the proof of the Oppenheim conjecture on quadratic forms and diophantine approximation. This was a question that had been open for half a century, on which considerable progress had been made by the Hardy-Littlewood circle method; but to reduce the number of variables to the point of getting the best-possible results, the more structural methods from group theory proved decisive. He has formulated a further program of research in the same direction, that includes the Littlewood conjecture.