Local rigidity

History

The first such theorem was proven by Atle Selberg for co-compact discrete subgroups of the unimodular groups S L n ( R ) . Shortly afterwards a similar statement was proven by Eugenio Calabi in the setting of fundamental groups of compact hyperbolic manifolds. Finally, the theorem was extende 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 (or just Weil) rigidity.

Deformations of subgroups

Let Γ be a group generated by a finite number of elements g 1 , … , g n and G a Lie group. Then the map H o m ( Γ , G ) → G n defined by ρ ↦ ( ρ ( g 1 ) , … , ρ ( g n ) ) is injective and this endows H o m ( Γ , G ) with a topology induced by that of G n . If Γ is a subgroup of G then a deformation of Γ is any element in H o m ( Γ , G ) . Two representations ϕ , ψ are said to be conjugated if there exists a g ∈ G such that ϕ ( γ ) = g γ g − 1 for all γ ∈ Γ . See also character variety.

Lattices in simple groups not of type A1 or A1 × A1

The simplest statement is when Γ is a lattice in a simple Lie group g and the latter is not locally isomorphic to S L 2 ( R ) or S L 2 ( C ) and Γ (this means that its Lie algebra is not that of one of these two groups).

Whenever such a statement holds for a pair G ⊃ Γ we will say that local rigidity holds.

Lattices in S L 2 ( C ) {displaystyle mathrm {SL} _{2}(mathbb {C} )}

Local rigidity holds for cocompact lattices in S L 2 ( C ) . A lattice Γ in S L 2 ( C ) 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 Γ to parabolic elements then local rigidity holds.

Lattices in S L 2 ( R ) {displaystyle mathrm {SL} _{2}(mathbb {R} )}

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.

Semisimple Lie groups

Local rigidity holds for lattices in semisimple Lie groups providing the latter have no factor of type A1 (i.e. locally isomorphic to S L 2 ( R ) ) 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 Γ is a lattice in the unitary group S U ( n , 1 ) and n ≥ 2 then the inclusion Γ ⊂ S U ( n , 1 ) ⊂ S U ( n + 1 , 1 ) is locally rigid.

Proofs of the theorem

Weil's original proof is by relating deformations of a subgroup Γ in G to the first cohomology group of Γ with coefficients in the Lie algebra of G , and then showing that this cohomology vanishes for cocompact lattices when G 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 ( G , X ) structures.