Two rigidity conjectures
fromTransformation groups
Yoshinobu Kamishima
Department of Mathematics,
Tokyo Metropolitan University

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Introduction

We recall two types of Rigidity Conjecture/Theorem posedin the days of topology.

I. The Borel conjecture (A. Borel 1960)
It expects that any two compact aspherical topologicalmanifolds with isomorphic fundamental groups must be
homeomorphic.

II. The Conformal rigidity(Obata & Lelong-Ferrand 1970)
If a closed noncompact group acts conformally on acompact Riemannian manifold of dimension  , thenit is conformal to  .

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Supporting Evidence to the topological case I

Any two closed aspherical topological manifolds withisomorphic fundamental groups of virtually abeliangroups are homeomorphic in dimension  (Farrell-Hsiang 1978).
Any two closed aspherical topological manifolds withisomorphic fundamental groups of virtually nilpotent
groups are homeomorphic in dimension  (Farrell-Hsiang 1983).
Any two closed aspherical topological manifolds withisomorphic fundamental groups  are homeomorphic indimension   (Farrell-Jones 1998). Here  isisomorphic to a discrete subgroup of  ( large).

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Supporting Evidence to the topological case

Note that this last statement covers the previous results. The
method to the proofs is based on the topological surgery
theory and the calculation of -groups. The previous results
to the topological case were inspired by the following smoothclassical results.

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Supporting Evidence to the smooth case I

Bieberbach Theorem (1911) - If two compact
Riemannian flat manifolds are homotopic, then they areaffinely diffeomorphic.
Mal’cev Theorem (1949) - If two compact nilmanifolds
are homotopic, then they are isomorphic.
Mostow Theorem (1954) - If two compact solvmanifolds
are homotopic, then they are diffeomorphic.
If two compact infranilmanifolds are homotopic, thenthey are affinely diffeomorphic (Auslander 1970’s,Kamishima-Lee-Raymond 1983).
If two compact infrasolvmanifolds are homotopic, thenthey are diffeomorphic (Baues 2004, Farrell-Jones1997).

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Supporting Evidence I to the smooth case I

Another class satisfying rigidity:
Mostow  -hyperbolic rigidity (      or  ) 1973.
Gromov-Margulis rigidity (Ballman-Schroeder’s book1985): If a compact Riemannian manifold of nonpositive
sectional curvature with flat dimension   is homotopic
to a compact locally symmetric Riemannian manifold,then two such Riemannian manifolds are isometric.

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Supporting Evidence to O& L-F II( -analogue of O&L-F) If a closed noncompact group
acts as  transformations on a compact
 -manifold of 
 , then it is  -isomorphic to  (Kamishima, J. Lee, R. Schoen 1996).
(Quaternionic  -analogue of O&L-F) If acts asquaternionic  transformations on a compact
quaternionic  -manifold of 
 , it ispseudo-conformally isomorphic to    . (1996, 2007.)(Quaternion Kähler analogue of O&L-F) If acts as
projective transformations on a compact quaternionic
Kähler manifold of   , it is projectively isomorphic to  (Alekseevsky -Marchiafava 1990’s).

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Aim

We shall develop these conjecture/theorem into theframework of Geometric Topology from the viewpoint ofTransformation groups. That is,— The Vague conjecture —(D’ambra and Gromov 1990)
If there exists a global geometric flow (relatively big Lieis rigid,group) on a compact geometric manifold , theni.e isomorphic to the standard model with flat -structure.

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Aim

In order to make vague conjecture clear, we formulate thefollowing two problems specifying the above
conjecture/theorem I,II.
(SI) The Smooth Borel conjecture: Any two compact
aspherical smooth manifolds with isomorphic
fundamental groups must be diffeomorphic.
(LII) The Obata & Lelong-Ferrand theorem to Lorentz
manifolds - If a closed group acts conformally on acompact Lorentz manifold of dimension  , then it isconformal to the conformally flat Lorentz model         .
At once we see that these two problems SI, LII are false.

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Aim
Nevertheless our purpose of this talk is to study the following

problems to establish affirmative results.
Which class of aspherical smooth manifolds satisfies
the smooth Borel conjecture?
Which class of compact Lorentz manifolds satisfies theanalogue of Obata & Lelong-Ferrand’ theorem?
Which kind of closed noncompact connected conformaltransformations assures that a compact Lorentz
manifold is conformal to   ?
First we observe the counterexamples of (SI) smooth Borel
rigidity and (LII) Lorentz Obata & Lelong-Ferrand theorem.

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(SI)The Smooth Borel conjecture.

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Counterexamples SI

It is well known thatThe connected sum  with an exotic sphere is notdiffeomorphic to   (Browder).
In fact, we presume that Borel thought at first it is true for smoothaspherical manifolds but Browder, Wall have studied smooth
structures of the connected sum of a manifold with homotopy
spheres at late 60’s. Then he knew a counterexample. Note that theconnected sum of exotic spheres does not cover all thediffeomorphism classes of smooth -torus from the -theory by thework of Wall. It is unknown how to construct the remaining explicitly.
Virtual smooth Borel conjecture !

A connected sum of closed hyperbolic manifold with an
exotic sphere (Farrell and Jones 1980’s).

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Rigidity results to Problem SI (with Oliver Baues)
Affirmative results: Take the aspherical homogeneous

manifold . A systematic study of generalaspherical homogeneous spaces was persisted by
Gorbatsevich in a series of papers. We shall answer to hisquestion.

  be compact aspherical homogeneous manifolds. If  
is an isomorphism, then  is induced by aTheorem A. Letdiffeomorphism .

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Rigidity results to Problem SI (with Oliver Baues)

We start with a fiber space with solv-geometry:    
where  acts properly on  and  is a simply connectedsolvable Lie group    and  principal bundle:      Baues observed the  -compatibility condition for thereductive group  of the algebraic hull .
We prove that  a simply connected nilpotent Lie groupsuch that the  -compatible triad     is equivalent tostandard -fiber space     whose fiber stabilisinggroup is a discrete standard subgroup of  .

ÍÍWe have the following rigidity forÍ .

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Rigidity results to Problem SI (with Oliver Baues)
Theorem B. Let        ,            
be standard homomorphisms. If  is an isomorphism of extensions, then          lifts to anevery equivariant diffeomorphism equivariant diffeomorphism          

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Rigidity results to Problem SI (with Oliver Baues)
Theorem B. Let        ,            
be standard homomorphisms. If  is an isomorphism of extensions, then          lifts to an
every equivariant diffeomorphism equivariant diffeomorphism          

Theorem A is obtained from Theorem B by showing
Aspherical homogeneous manifold has the structure of -compatible fiber space with solv-geometry.

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Rigidity results to Problem SI (with Oliver Baues)
Theorem B. Let        ,            
be standard homomorphisms. If  is an isomorphism of extensions, then          lifts to an
every equivariant diffeomorphism equivariant diffeomorphism          

Theorem A is obtained from Theorem B by showing
Aspherical homogeneous manifold has the structure of -compatible fiber space with solv-geometry.
This kind of rigidity was originally proved by Conner andRaymond for the fiber space with abelian geometry
such that(1970). In our case,  notnecessarily lattice but standard.

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Application-Double coset space

Let    be a connected simply connected Lie groupwith Levi decomposition such that  is the radical and  isa semisimple subgroup of noncompact type. Then there isthe canonical exact sequence:Here    is a (centerless) connected semisimple Liegroup without compact factor. Since is simply connected,
a maximal compact subgroup  maps isomorphically ontothat of  and   is a simply connected noncompact
Riemannian symmetric space.

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Application-Double coset spaces

Let  be a discrete cocompact subgroup of . If  istorsionfree, the double coset space  is a closed
aspherical manifold.
As      is a discrete cocompact subgroup, theMostow rigidity theorem says that given an isomorphism    , there exists an equivariant diffeomorphism            . Applying Theoem B,
Theorem C. Suppose that  is isomorphic to   . Then equivariantly diffeomorphic to        .

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LIIThe Obata & Lelong-Ferrand analogue toLorentz manifolds.

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Counterexamples LII

Recall the definition of conformally flat Lorentz model.
Let     be the canonical projection.

Take the quadric (Lorentz cone) in           :      Define the Lorentz model to be            . The correspondence is given;              .

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Counterexamples LII

As signature of  is   , denote by    thesubgroup of matrices preserving signature    .
The group    leaves  invariant.
        is the conformal group acting
transitively on      .
Note that the real pseudo-hyperbolic space  Ê  has
the compactification in   : Ê    Ê   (Pseudo-hyperbolic isometry of  Ê  extends toconformal Lorentz transformation of   .)Two rigidity conjecturesfromTransformation groups – p.22/47

-dimensional counterexample LII

Proposition A. compact Lorentz standard space form  onwhich a closed noncompact group acts as isometries.
In particular, the O& L-F conjecture to the Lorentz case is not true onlyby the existence of conformal .

Lorentz standard space form = complete Lorentz manifold ofnegative constant curvature. This is obtained as follows:                     which isidentified with       .
The group        acts as Lorentz
isometries on   (identified with  ) by           

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-dimensional counterexample LII

Choose a torsionfree discrete cocompact subgroupfrom     so that   is a compact Lorentz
manifold of curvature  . (In particular, it is a
conformally flat Lorentz manifold.)
Take a closed subgroup      .
As   is the group of Lorentz isometries, acts
as conformally on   .
As a consequence,      is a counterexample
because   is never Lorentz model     .

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Causal fields
Definition I. Let  be a vector field on a Lorentz manifold  .

  is spacelike     whenever    is lightlike       whenever     is timelike    whenever   Each  is called a causal vector field with respect to  .
The group generated by causal vector field is said to be
one-parameter group of causal transformations.
We can take the action on   as lightlike group ! orspacelike group  from the decomposition     ! .

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-dimensional counterexample LII

Recall that   is a spherical  manifold and     is an example of Fefferman Lorentz manifold. Itis proved thatProposition B.    is a compact conformally flat Lorentz

manifold which can admit a closed lightlike conformal Lorentz group   .
Here     !

     

.

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-dimensional counterexample LII

We observe that
Conformal Riemann case  non-elliptic behaviourof noncompact is determined.
Conformal Lorentz case  not true. (The Lorentz
structure does not control the non-ellipticity ofnoncompact groups.)

We take into account Fefferman Lorentz manifold todefine a class of fine Lorentz structure satisfyingnon-elliptic behaviour of noncompact groups.
We interpret the Fefferman Lorentz manifold in terms of-structure.

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Lorentz-

structure   " ##  " $¼ ¼
"  

$    #  .

Definition II. A 
-manifold
is said to be Lorentz- manifold
if it admits a -structure. That is, there is a reduction of the structuregroup ofto .

Note that if all #  , then the -structure defines a 
conformal structure of Lorentz metrics: .

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Lorentz-

structure

Recall the -dimensional conformally flat Lorentz    . Let   be thegeometry 
unitary Lorentz group with center   . Put     

where       .
The natural embedding   
embedding of the Lie groups:    induces the 

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Uniformization    restricts to a subgeometry  which is conformally flat Lorentz-geometry.

Remark that   of Proposition B is the group of conformal
Lorentz transformations but not conformal Lorentz-transformations,    .

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Conclusion LII

We obtain the Obata & Lelong-Ferrand’s theorem to theLorentz- manifolds.
Theorem X (Lorentz analogue). Suppose that a compact Lorentz-manifold
admits a closed subgroup   consisting of lightlike

conformal Lorentz- transformations. Then the universal covering is conformally isomorphic to the universal covering of  . Moreover,infinite cyclic subgroup  .


is the quotient of  by anMore precisely, there exists a representation   such that    .

Ê

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Remark LII

Such representations     #  %

¾ ½ Ê   are determined by%¾  

  # Ê  

The set of all such distinct homomorphisms is in one-to-one
correspondence with  #                          &The element          corresponds to   .

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Fin

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Part 2 - sketch of proofs
Proposition D. Let
be a Lorentz- manifold of dimension admits a -structure.
Ifis conformally flat, thenis conformally flat Lorentz- , i.e it is
uniformizable with respect to    .

Note that no particular Lorentz metric on

is specified.

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Results LII

We prove our vague conjecture to the Lorentz-manifolds affirmatively.
First of all, the existence of two dimensional lightlike vector
fields implies conformally flatness. Two dimensional lightlike
vector fields induces one timelike vector field.
Proposition E. Suppose that a compact Lorentz -manifold admits a closed subgroup   of conformal transformations.
If   contains a one-parameter subgroup of timelike conformaltransformations, thenis conformally flat.

 

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Results LII (Sketch)

Let  be the timelike vector field. A Riemannian metric 
is defined on a domain  the set of points at which
Weyl curvature tensor nonzero in :     '     '     '    

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Results LII (Sketch)

Let  be the timelike vector field. A Riemannian metric 
is defined on a domain  the set of points at which
Weyl curvature tensor nonzero in :     '     '     '    This defines  bundle   

 !  so there is a principal frame

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Results LII (Sketch)

Let  be the timelike vector field. A Riemannian metric is defined on a domain  the set of points at which
Weyl curvature tensor nonzero in :     '     '     '    This defines  bundle   

 !  so there is a principal frame

Let %   . The orbit map:   % defines a proper
embedding of    whose image will be compact by
the -structure theory of finite type. It is a contradiction,  .

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Results LII

Suppose that a compact conformally flat Lorentz-manifoldadmits a closed subgroup   consisting oflightlike conformal Lorentz- transformations.
By Proposition E, we have the developing pair: "#$      Here     .       .

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Results LII

Suppose that a compact conformally flat Lorentz-manifold
admits a closed subgroup   consisting oflightlike conformal Lorentz- transformations.
By Proposition E, we have the developing pair: "#$      Here     
shown that.    

  







 . Moreover, it is

The causality of lightlike fields implies that the lift of  to the universal covering  is   .

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Results LII

Suppose that a compact conformally flat Lorentz-manifold
admits a closed subgroup   consisting oflightlike conformal Lorentz- transformations.
By Proposition E, we have the developing pair: "#$      Here     
shown that.    

  





 . Moreover, it is

The causality of lightlike fields implies that the lift of  to the universal covering  is   .

   Ê which is the center of   .

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Results LII

We obtain the following commutative diagram of fiberspaces:Ê      Ê    Ê     Here       Ê.       Ê.
  

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CR-analogue

As a consequence, the group  acts properly
discontinuously on  such that the quotient   is anorbifold. In particular, a closed  orbifold   admits anoncompact  -transformations  .

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CR-analogue

As a consequence, the group  acts properly
discontinuously on  such that the quotient   is anorbifold. In particular, a closed  orbifold   admits anoncompact  -transformations  .
Now we use the analogue of Obata & Lelong-Ferrand’s
theorem to compact strictly pseudo-convex  -manifolds.
 is  -isomorphic.
The developing map "#$  "#$In particular,the diagram.

is conformally equivalent to    by

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Conclusion LII

By the diagram, there is the exact sequence: As   is a compact orbifold,  is a finite subgroup.
Moreover, we can prove that  is a cyclic group.

   is isomorphic to an infinite cyclic group  .
As a consequence,is the quotient of   by an    .
infinite cyclic subgroup  ;This proves Theorem X - Obata & Lelong-Ferrand’s theorem
to the Lorentz-manifolds.

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Rigidity results to Problem SI (with Oliver Baues)

We start with a fiber space with solv-geometry:    
where  acts properly on  and normalises a simplyconnected solvable Lie group .  A principal bundle:      The fiber stabilising subgroup % of  contains a solvable
subgroup of finite index and  % is compact. Associated togroup extension% there is a singular fibration:ÆÆ%      

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Rigidity results to Problem SI (with Oliver Baues)

If % is the connected component of %, then it induces agroup extension:

% % 

 %   

where is a virtually solvable group. Then Baues observedthe  -compatibility condition for the reductive group  ofthe algebraic hull . We prove that  a simply
connected nilpotent Lie group such that the  -compatible    is equivalent to standard -fiber space    whose fiber stabilising group is a discrete standard
subgroup of  .

ÍÍÍ

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