Local rigidity theorems in the theory of discrete subgroups of Lie groups are results which show that small deformations of certain such subgroups are always trivial. It is different from Mostow rigidity and weaker than superrigidity.
History
The first such theorem was proven by Atle Selberg for co-compact discrete subgroups of the unimodular groups. Shortly afterwards a similar statement was proven by Eugenio Calabi in the setting of fundamental groups of compact hyperbolic manifolds. Finally, the theorem was extended to all co-compact subgroups of semisimple Lie groups by André Weil. The extension to non-cocompact lattices was made later by Howard Garland and Madabusi Santanam Raghunathan. The result is now sometimes referred to as Calabi—Weil rigidity.
Lattices in simple groups not of type A1 or A1 × A1
The simplest statement is when is a lattice in a simple Lie group and the latter is not locally isomorphic to or and . Whenever such a statement holds for a pair we will say that local rigidity holds.
Lattices in
Local rigidity holds for cocompact lattices in. A lattice in which is not cocompact has nontrivial deformations coming from Thurston's hyperbolic Dehn surgery theory. However, if one adds the restriction that a representation must send parabolic elements in to parabolic elements then local rigidity holds.
Lattices in
In this case local rigidity never holds. For cocompact lattices a small deformation remains a cocompact lattice but it may not be conjugated to the original one. Non-cocompact lattices are virtually free and hence have non-lattice deformations.
Semisimple Lie groups
Local rigidity holds for lattices in semisimple Lie groups providing the latter have no factor of type A1 or the former is irreducible.
Other results
There are also local rigidity results where the ambient group is changed, even in case where superrigidity fails. For example, if is a lattice in the unitary group and then the inclusion is locally rigid. A uniform lattice in any compactly generatedtopological group is topologically locally rigid, in the sense that any sufficiently small deformation of the inclusion is injective and is a uniform lattice in. An irreducible uniform lattice in the isometry group of any proper geodesically complete -space not isometric to the hyperbolic plane and without Euclidean factors is locally rigid.
Proofs of the theorem
Weil's original proof is by relating deformations of a subgroup in to the first cohomology group of with coefficients in the Lie algebra of, and then showing that this cohomology vanishes for cocompact lattices when has no simple factor of absolute type A1. A more geometric proof which also work in the non-compact cases uses Charles Ehresmann theory of structures.
