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.

History

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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.

Statement

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Deformations of subgroups

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Let   be a group generated by a finite number of elements   and   a Lie group. Then the map   defined by   is injective and this endows   with a topology induced by that of  . If   is a subgroup of   then a deformation of   is any element in  . Two representations   are said to be conjugated if there exists a   such that   for all  . See also character variety.

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

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

There exists a neighbourhood   in   of the inclusion   such that any   is conjugated to  .

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

Lattices in SL(2,C)

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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 SL(2,R)

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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

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Local rigidity holds for lattices in semisimple Lie groups providing the latter have no factor of type A1 (i.e. locally isomorphic to   or  ) or the former is irreducible.

Other results

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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.[5]

A uniform lattice   in any compactly generated topological 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.[6]

Proofs of the theorem

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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 (and William Thurston's) theory of   structures.[7]

References

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  1. ^ Selberg, Atle (1960). "On discontinuous groups in higher-dimensional symmetric spaces". Contributions to functional theory. Tata Institut, Bombay. pp. 100–110.
  2. ^ Weil, André (1960), "On discrete subgroups of Lie groups", Annals of Mathematics, Second Series, 72 (2): 369–384, doi:10.2307/1970140, ISSN 0003-486X, JSTOR 1970140, MR 0137792
  3. ^ Weil, André (1962), "On discrete subgroups of Lie groups. II", Annals of Mathematics, Second Series, 75 (3): 578–602, doi:10.2307/1970212, ISSN 0003-486X, JSTOR 1970212, MR 0137793
  4. ^ Garland, Howard; Raghunathan, M.~S. (1970). "Fundamental domains for lattices in R-rank 1 Lie groups". Annals of Mathematics. 92: 279–326. doi:10.2307/1970838. JSTOR 1970838.
  5. ^ Goldman, William; Millson, John (1987), "Local rigidity of discrete groups acting on complex hyperbolic space", Inventiones Mathematicae, 88 (3): 495–520, Bibcode:1987InMat..88..495G, doi:10.1007/bf01391829, S2CID 15347622
  6. ^ Gelander, Tsachik; Levit, Arie (2017), "Local rigidity of uniform lattices", Commentarii Mathematici Helvetici, arXiv:1605.01693
  7. ^ Bergeron, Nicolas; Gelander, Tsachik (2004). "A note on local rigidity". Geometriae Dedicata. 107. Kluwer: 111–131. arXiv:1702.00342. doi:10.1023/b:geom.0000049122.75284.06. S2CID 54064202.