Local rigidity theorems in the theory of discrete subgroups of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s are results which show that small deformations of certain such subgroups are always trivial. It is different from
Mostow rigidity Mostow may refer to: People
* George Mostow (1923–2017), American mathematician
** Mostow rigidity theorem
* Jonathan Mostow
Jonathan Mostow (born November 28, 1961) is an American film director, screenwriter, and producer. He has directed f ...
and weaker (but holds more frequently) than
superrigidity In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group ''G'' can, under some circumstances, be as good as a representation of ''G' ...
.
History
The first such theorem was proven by
Atle Selberg
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded t ...
for co-compact discrete subgroups of the unimodular groups
. Shortly afterwards a similar statement was proven by
Eugenio Calabi
Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...
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
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
. The extension to non-cocompact lattices was made later by Howard Garland and
Madabusi Santanam Raghunathan
Madabusi Santanam Raghunathan FRS is an Indian mathematician. He is currently Head of the National Centre for Mathematics, Indian Institute of Technology, Mumbai. Formerly Professor of eminence at TIFR in Homi Bhabha Chair. Raghunathan receiv ...
.
The result is now sometimes referred to as Calabi—Weil (or just Weil) rigidity.
Statement
Deformations of subgroups
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
In the mathematics of moduli theory, given an algebraic, reductive, Lie group G and a finitely generated group \pi, the G-''character variety of'' \pi is a space of equivalence classes of group homomorphisms from \pi to G:
:\mathfrak(\pi,G)=\ ...
.
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
(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)
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)
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
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
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
or
) 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
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an ...
and
then the inclusion
is locally rigid.
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.
Proofs of the theorem
Weil's original proof is by relating deformations of a subgroup
in
to the first
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
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
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurston ...
's) theory of
structures.
References
{{reflist
Discrete groups
Hyperbolic geometry