“Local rigidity”的意思、由来-开放百科全书

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

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

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.

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

and

(this means that its Lie algebra is not that of one of these two groups).

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

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 (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

) 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.^[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

Weil's original proof is by relating deformations of a subgroup

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

1. ^{{cite book | last1 = Selberg | first1 = Atle | title=Contributions to functional theory | year=1960 |publisher=Tata Institut, Bombay |pages=100–110 |chapter=On discontinuous groups in higher-dimensional symmetric spaces}}
2. ^{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=On discrete subgroups of Lie groups | jstor=1970140 | mr=0137792 | year=1960 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=72 | pages=369–384 | doi=10.2307/1970140}}
3. ^{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=On discrete subgroups of Lie groups. II | jstor=1970212 | mr=0137793 | year=1962 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=75 | pages=578–602 | doi=10.2307/1970212}}
4. ^{{cite journal| last = Garland | first = Howard | last2 = Raghunathan| first2 = M.~S. | date = 1970 | title = Fundamental domains for lattices in R-rank 1 Lie groups | journal = Annals of Mathematics | volume = 92 | pages = 279–326| doi=10.2307/1970838}}
5. ^{{Citation | last1=Goldman | first1=William | last2=Millson | first2=John | title=Local rigidity of discrete groups acting on complex hyperbolic space | year=1987 | journal=Inventiones Mathematicae | volume=88 | pages=495–520 | doi=10.1007/bf01391829| bibcode=1987InMat..88..495G }}
6. ^{{Citation | last1=Gelander| first1=Tsachik| last2=Levit | first2=Arie | title=Local rigidity of uniform lattices | year=2017| journal= Commentarii Mathematici Helvetici | arxiv= 1605.01693 }}
7. ^{{cite journal| last = Bergeron | first = Nicolas | last2 = Gelander | first2 = Tsachik | date = 2004 | title = A note on local rigidity | journal = Geometriae Dedicata | publisher = Kluwer | volume = 107| pages = 111–131| doi=10.1023/b:geom.0000049122.75284.06| arxiv= 1702.00342}}