In mathematics, in the theory of discrete group

Discrete group

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...

s, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group

Algebraic group

In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

 G can, under some circumstances, be as good as a representation of G itself. That this phenomenon happens for certain broadly-defined classes of lattice

Lattice (discrete subgroup)

In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...

s inside semisimple groups was the discovery of Grigory Margulis

Grigory Margulis

Gregori Aleksandrovich Margulis is a Russian mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the...

, who proved some fundamental results in this direction.

There is more than one result that goes by the name of Margulis superrigidity. One statement is this: take G to be a simply-connected semisimple real algebraic group in GL_n, such that the Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 of its real points has real rank at least 2 and no compact factors. For a local field

Local field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

 F and ρ a linear representation of the lattice Γ of the Lie group, into GL_n (F), assume the image ρ(Γ) is not relatively compact (in the topology arising from F) and such that its closure in the Zariski topology

Zariski topology

In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...

 is connected. Then F is the real numbers or the complex numbers, and there is a rational representation

Rational representation

In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map of algebraic varieties....

of G giving rise to ρ by restriction.