Deformations of group actions

Abstract

Let $G$ be a noncompact real algebraic group and $\G<G$ a lattice. One purpose of this paper is to show that there is an smooth, volume preserving, mixing action of $G$ or $\G$ on a compact manifold which admits a smooth deformation. We also describe some other, rather special, deformations when $G=SO(1,n)$ and provide a simple proof that any action of a compact Lie group is locally rigid.

Full text

Page 1

arXiv:math/0407417v2 [math.DS] 16 Jul 2006
DEFORMATIONS OF GROUP ACTIONS
DAVID FISHER
Abstract. Let G be a noncompact real algebraic group and Γ < G a lattice.
One purpose of this paper is to show that there is a smooth, volume preserv-
ing, mixing action of G or Γ on a compact manifold which admits a smooth
deformation. In fact, we prove a stronger statement by exhibiting large fi-
nite dimensional spaces of deformations. We also describe some other, rather
special, deformations when G = SO(1,n).
1. Introduction
In recent years, many results have been proven concerning local rigidity of group
actions. Most known results concern actions of semi-simple Lie groups with all
simple factors of real rank at least two and lattices in such groups. For the best
known results in that category and some historical references, see [6]. More recently
some partial results have been proven for more general groups, including lattices
in Sp(1,n) and F−20
4
, see [7] and [11]. In this context it is interesting to ask when
Lie groups and their lattices have actions by diffeomorphisms which admit non-
trivial perturbations or deformations. Actions of SL(n,Z) constructed by Katok
and Lewis admit non-trivial deformations and their construction was modified by
Benveniste to produce actions of some higher rank simple Lie groups with non-
trivial volume preserving deformations [2, 13]. Benveniste’s deformations are also
non-trivial when all actions in question are restricted to any lattice in the acting
group. Throughout this paper, by a connected real algebraic group, I mean the
connected component of the real points of an algebraic group defined over R. One
of the main results of this paper is the following generalization of Benveniste’s
theorem:
Theorem 1.1. Let G be a non-compact, simple, connected real algebraic group
and Γ < G a lattice. Then there is an a smooth, volume preserving, mixing, action
of G which admits non-trivial, volume preserving deformations. Furthermore, the
deformations remain non-trivial when restricted to Γ.
Remarks:
(1) It is immediate from the construction that we can construct actions where
the space of deformations has arbitrarily large, finite, dimension. It would
be interesting to know when the deformation space is infinite dimensional.
For any group G which contains a lattice with a homomorphism onto Z, it
is easy to build examples with infinite dimensional families of perturbation
using simple induction constructions and Lemma 2.3 below.
(2) The construction of the actions and deformations is a refinement of Ben-
veniste’s which is based on an earlier construction of Katok and Lewis
Author partially supported by NSF grant DMS-0226121 and a PSC-CUNY grant.
1

Page 2

2DAVID FISHER
[2, 13]. The proof that the deformations are non-trivial is new and uses a
consequence of Ratner’s theorem due to N.Shah. Benveniste’s proof used
commutativity coming from a higher rank assumption on G.
(3) Benveniste’s explicit deformations are non-trivial in a sense that is weaker
than what is meant by Theorem 1.1, see below for discussion.
We now introduce some basic notions in order to describe in detail the meaning
of the statement of Theorem 1.1 and also to clarify Remark (1) and show that we
produce “many” deformations. Let ρ be an action of a group D on a manifold
M. Then the space of actions of D on M is naturally the space Hom(D,Diff(M)).
The group Diff(M) acts on this space by conjugation, and two group actions are
conjugate if and only if they are in the same Diff(M) orbit. Note that if D is
finitely generated, Hom(D,Diff(M)) is naturally a closed subset of the product of
a finite number of copies of Diff(M). In the finite dimensional setting, i.e. where
we replace Diff(M) by an algebraic group H, the space Hom(D,H) is an algebraic
variety, and there is a construction of a quotient scheme Hom(D,H)//H which
allows one to study the space of non-trivial deformations. In the infinite dimen-
sional setting it is unclear that there is any meaningful algebraic geometry, or that
Hom(D,Diff(M)) is a manifold at “most” (or even any) points. We now describe
in more detail the space of deformations produced in the proof of Theorem 1.1. We
do this for Γ actions, where the space Hom(Γ,Diff(M)) is at least a closed subset
of a comprehensible topological space, the k-fold product Diff(M)×···×Diff(M)
where k is the number of generators of Γ. (There are various ways in which one can
topologize Hom(G,Diff(M)) but this is more complicated, and we will not discuss
it here.) Our construction produces an action of Γ on a manifold M and a collection
of deformations which provide an embedding of a small, finite dimensional ball in
Hom(Γ,Diff(M)) which intersects the Diff(M) orbit of ρ only at ρ. Benveniste’s
construction, even using our proof in place of his, only produces an action ρ with
deformations which define a map of a ball into Hom(Γ,Diff(M)) which intersects
the Diff(M) orbit of ρ at a countable set which could, a priori, accumulate on ρ.
Since we do not know, in this context, that the representation variety is even locally
a manifold, it is unclear that the two statements are equivalent. As will be clear
below, given Γ and a positive integer n we can construct a manifold M and an
action ρ such that the ball discussed above has dimension n. By “blowing up and
gluing” along many distinct pairs of closed orbits, one can even arrange that n is
much larger than dim(M).
In addition to removing the rank restriction in Benveniste’s theorem, our use of
Ratner theory is sufficiently robust to allow us to use slight modifications of the
examples for Theorem 1.1 to prove the following stronger theorem:
Theorem 1.2. Let G be a connected, non-compact real algebraic group and Γ < G
a lattice. Then there is a faithful, smooth, volume preserving, ergodic, action of
G which admits non-trivial, volume preserving deformations.
deformations remain non-trivial when restricted to Γ.
Furthermore, the
Remarks:
(1) As will be clear in the proof, Theorem 1.2 is only really difficult if the
abelianization of G is compact. See Proposition 2.2.
(2) Theorem 1.2 is in a sense sharp, since any smooth action of a compact
group is locally rigid by a theorem of Palais [17].

Page 3

DEFORMATIONS OF GROUP ACTIONS3
(3) Though the author knows of no claims in the literature, it is quite likely that
one can prove local rigidity theorems for groups which are not semisimple.
In particular, the results of [6] seem likely to hold for G any algebraic group
all of whose non-trivial factors are semisimple groups with all simple factors
of real rank at least two and for lattices Γ in such G. Any such G or Γ can be
shown to have property (T), so the arguments in [7] go through verbatim.
The main difficulty in proving local rigidity for quasi-affine actions appears
to be in generalizing some of the results of [8]. The results in [22] should
be relevant to this generalization.
With the exception of some of the results in [3, 6, 7], all local rigidity theorems
known to the author are for actions on manifolds which are homogeneous. The
deformations constructed to prove Theorems 1.1 and 1.2 above are on manifolds
which are not homogeneous. In section 5 we show that some affine actions of lattices
in SO(1,n) on homogeneous manifolds have non-trivial deformations. This then
implies that the same is true for some affine actions of SO(1,n) as well.
One motivation for this paper is the so-called Zimmer program to classify actions
of higher rank lattices, see [24, 25] for more details. In the context of both this
program and other older rigidity results, it is not surprising that actions of rank 1
lattices exhibit greater flexibility than higher rank lattices.
Acknowledgements: I would like to thank E.J.Benveniste and K.Whyte for many
useful conversations concerning the construction and properties of “exotic actions”
discussed in this paper. I learned a great deal about these examples during the work
on [4] and [9]. I would also like to thank Dave Witte Morris for some useful emails
concerning algebraic groups and lattices. A proof that “bending deformations” gave
non-trivial deformations of affine actions of lattices and Lie groups was discovered
jointly with R.Spatzier during an extremely pleasant visit to the University of
Michigan in Spring of 2002. This proof uses the normal form theory of Hirsch-
Pugh-Shub for partially hyperbolic diffeomorphisms and is sketched in section 5.
That proof led me to the proof using Ratner theory given in section 5, which was the
inspiration for the use of Ratner theory in the proofs of Theorem 1.1 and Theorem
1.2. Theorem 1.1 was motivated by a question of Y.Shalom concerning the existence
of deformations of actions of lattices in SU(1,n) and SP(1,n).
2. Preliminaries
2.1. Invariant measures and centralizers of group actions. In this section
we recall a consequence of work of Ratner and Shah on invariant measures which is
instrumental in our proofs. For the proofs of Theorems 1.1 and 1.2, our use of this
result will be based on the fact that the perturbations we construct are conjugate
back to the original action on a set of full measure. Other applications in Section
5 will be to deformations that are trivial by construction on a “large” subgroup of
the acting group. We let Aff(H/Λ) denote the affine group of H/Λ. This group
consists of diffeomorphisms of H/Λ which are covered by maps of the form A◦Lh
where A is an automorphism of H such that A(Λ) = Λ and Lhis left translation
by H. The group Aff(H/Λ) is a Lie group and is, in fact a quotient of a a subgroup
of Aut(H)⋉H, see [8] for more discussion.
Corollary 2.1. Let H be a connected real algebraic group, Λ < H a lattice, µHthe
measure on H/Λ induced by a fixed Haar measure on H and F < H a group that
either

Page 4

4DAVID FISHER
(1) contains a subgroup F′< F generated by unipotent elements,
(2) contains a subgroup F′< F such that F′is a lattice in a subgroup F′′< H
generated by unipotent elements.
Further assume F′acts ergodically on H/Λ. If φ is an essentially surjective, es-
sentially injective measurable map from H/Λ to H/Λ which commutes with the F
action, then φ is translation by an element of the centralizer of F′in Aff(H/Λ).
Though this corollary is essentially noted in Witte’s paper [21] where a more general
result is proven. For completeness, we briefly recall the proof.
Proof. If φ commutes with F′, then the graph N of φ is preserved by F′. Letting
˜φ : H/Λ→H/Λ×H/Λ be given by˜φ = (Id,φ), it is straightforward to check that
˜φ∗µH is an F′invariant, ergodic measure on N that projects to µH on each H/Λ.
Given a subgroup L < H, we denote by ∆(L) the diagonal embedding of L in H×H.
By Shah’s extensions of Ratner’s theorem to groups generated by unipotents and
their lattices [19, 20], this implies that that N is a closed L orbit in H/Λ×H/Λ where
L is a closed subgroup of H×H containing ∆(F′). Since˜φ composed with projection
on the second factor is essentially injective and essentially surjective, we have that
L is exactly ∆(H). This implies that N is a closed ∆(H) orbit in H/Λ×H/Λ, which
implies that φ is translation by an element of H composed with an automorphism
of H. This implies that φ is in fact an element of ZAff(H/Λ)(F′).
?
2.2. Reductions for the proof of Theorem 1.2. We recall briefly some facts
concerning (real) algebraic groups. Given a connected real algebraic group G,
we can write G as a semi-direct product L⋉U where U,L < G and U is the
unipotent radical of G and L is a reductive real algebraic group which is called a
Levi complement.
Proposition 2.2. To prove Theorem 1.2, it suffices to consider the case when the
Levi complement of G has compact center.
Proof. We assume the center of the Levi complement is non-compact and produce
the family of perturbations required in Theorem 1.2. We have a projection π1 :
G→L, so any L action defines a G action. We can write L as SZ(L) where S is
semisimple, S∩Z(L) = F is finite and the product is almost direct. Therefore S is
normal in L and there is a projection π2: L→Z(L)/F. The group Z(L)/F is an
abelian Lie group and non-compact whenever Z(L) is non-compact, so we can find
a third projection π3: Z(L)/F→R. We write π = π3◦π2◦π1and note that any R
action defines a G action via composition with π.
There are many R actions which are smooth and mixing and which have non-
trivial perturbations, we give an example for completeness. Let Λ < SL(2,R) be
a cocompact lattice and let ρR be the action of R on SL(2,R)/Λ defined by the
horocycle flow. (I.e. defined by identifying R with strictly upper triangular matrices
in SL(2,R) and acting by left translation.) It is easy to perturb this action, even as
an action by left translations, since this action has zero entropy and many nearby
actions defined by “nearby” subgroups of SL(2,R) have positive entropy. It is
straightforward to construct families on which the entropy is a strictly increasing
as one moves awayfrom ρR. Since Lyapunov exponents are also conjugacy invariants
of diffeomorphisms, by choosing an R action on a larger manifold, one can construct
large dimensional families of deformations which are easily seen to be non-trivial

Page 5

DEFORMATIONS OF GROUP ACTIONS5
and to give rise to non-trivial deformations of the faithful actions described in the
next paragraph.
To construct a faithful action of G with pertubations, we embed G in SL(N +
1,R), choose a cocompact lattice ∆ < SL(N+1,R) and let ρFbe the left translation
action of G on SL(N +1,R)/∆. We then form M = SL(2,R)/Λ×SL(N +1,R)/∆
and let ρ be the diagonal G action defined by π◦ρRand ρF. It is straightforward
to check that this satisfies the hypotheses of Theorem 1.2. The fact that the action
is mixing follows from the Howe-Moore theorem, see e.g. [23]. It is also straight-
forward to check that ρ|Γ provides the necessary action of Γ when Γ < G is a
lattice.
?
We now turn to the case where the entire Levi complement is compact. In order
to prove the existence of deformations in this case, we will need to use induced
actions. Let F be a locally compact topological group and D < F a closed subgroup.
Assume D acts continuously on a compact space X. We can then form the space
(F×X)/D where D acts by (f,x)d = (fd−1,dx). This space is clearly equipped
with a left F action which we refer to as the induced F action. It is easy to check
that if F is a Lie group, X is a smooth manifold and D acts smoothly on X, then
the induced F action is also smooth.
Lemma 2.3. If H is a real Lie group and L < H is a closed cocompact subgroup,
then if L has an action on a compact manifold M which admits non-trivial de-
formations (resp. perturbations), the induced H action on (H×M)/L also admits
non-trivial deformations (resp. perturbations).
Proof. Let ρ0 be an L action on M and ρt a non-trivial one parameter deforma-
tion. We can form the induced actions ˜ ρt of H on (H×M)/L which are again
smooth actions and it is clear that ˜ ρtis a one parameter deformation of ˜ ρ0. For a
contradiction, assume that deformations ˜ ρtare trivial. Then there exist diffeomor-
phisms φtof (H×M)/L such that φt◦ρ0= ρt◦φt. Let π : (H×M)/L→H/L be the
natural projection. Then φt◦π is an H equivariant map from ((H×M)/L,ρt) to
(H/L,ρH) which is close to π as a smooth map. The sets (φt◦π)−1([h]) are just the
sets φ−1
φ−1
on φt−1(π−1[e]) is exactly ρt. Therefore φt|π−1([e])is a conjugacy between ρ0and
ρt. It is straightforward to check that φt|π−1([e])is indeed a small, smooth path in
Diff(M). The same proof applies to perturbations in place of deformations.
t ((π−1[h])) and so are an H invariant foliation of (H×M)/L. In particular
t ((π−1[e])) is a L invariant copy of M, and it is easy to check that the L action
?
Proposition 2.4. It suffices to prove Theorem 1.2 when the Levi complement of
G is not compact.
Proof. We prove that if the Levi complement is compact, then there are deforma-
tions. By this assumption, G = C⋉U where C is compact and U is unipotent.
There exist non-trivial homomorphisms σ : U→R. As in the proof of Proposition
2.2, we can use an R action ρR with non-trivial perturbations to construct a U
action with non-trivial perturbations. This yields a G action with non-trivial per-
turbations by Lemma 2.3. The G action is volume preserving if ρRis, since there is
a G invariant volume on G/U. It is easy to check that this action is mixing when
ρRis mixing. To obtain a faithful, mixing action, one proceeds as in Proposition
2.2. To verify that the deformations are non-trivial on the faithful action requires
some care, and we sketch a simple argument using entropy or Lyapunov exponents.