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 (but holds more frequently) than superrigidity.
The first such theorem was proven by Atle Selberg for co-compact discrete subgroups of the unimodular groups
[1] 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.
[2][3] The extension to non-cocompact lattices was made later by Howard Garland and Madabusi Santanam Raghunathan.
[4] The result is now sometimes referred to as Calabi—Weil (or just Weil) rigidity.
be a group generated by a finite number of elements
is a lattice in a simple Lie group
(this means that its Lie algebra is not that of one of these two groups).
we will say that local rigidity holds.
Local rigidity holds for cocompact lattices 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.
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 (see Teichmüller space for more detail).
Non-cocompact lattices are virtually free and hence have non-lattice deformations.
Local rigidity holds for lattices in semisimple Lie groups providing the latter have no factor of type A1 (i.e. locally isomorphic to
There are also local rigidity results where the ambient group is changed, even in case where superrigidity fails.
is a lattice in the unitary group
in any compactly generated topological group
is topologically locally rigid, in the sense that any sufficiently small deformation
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.
[6] Weil's original proof is by relating deformations of a subgroup
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 (and William Thurston's) theory of