superrigidity

Superrigidity is a concept in the field of geometric group theory and differential geometry that describes a remarkable form of structural rigidity within certain mathematical objects, particularly lattices in Lie groups. It was introduced by Grigori Margulis in the 1970s as a central component of his work on arithmeticity and the structure of discrete subgroups of Lie groups.

Superrigidity generalizes the notion of local or global rigidity, whereby a space or group is resistant to non-trivial deformations. In particular, a lattice—a discrete subgroup of a Lie group with finite covolume—is said to be superrigid if any representation of the lattice into a certain class of groups extends to a representation of the entire Lie group itself. This implies that the lattice's algebraic structure is deeply entwined with that of the ambient Lie group, leaving little room for variation.

The significance of superrigidity lies in its applications to classifying lattices and understanding arithmetic properties of groups. It plays a pivotal role in proving Margulis's arithmeticity theorem, which states that many lattices in higher-rank semisimple Lie groups are arithmetic, meaning they can be realized via algebraic groups defined over number fields.

Superrigidity typically occurs in higher-rank semisimple Lie groups, such as SL(n, R) for n ≥ 3, and is absent in rank-one groups like SL(2, R). Its implications extend to number theory, representation theory, and the study of manifolds, highlighting the profound link between symmetry, geometric structures, and algebraic properties in mathematics.