Mapping Class Group Dynamics on Surface Group Representations

0.38 MB·English

by  William M. Goldman

MAPPING CLASS GROUP DYNAMICS ON SURFACE
GROUP REPRESENTATIONS
6
0
0
WILLIAM M. GOLDMAN
2

n
a Abstract. Deformation spaces Hom(π,G)/G of representations
J of the fundamental group π of a surface Σ in a Lie group G ad-

8 mitnaturalactionsofthemappingclassgroupModΣ,preservinga
2
Poissonstructure. WhenGiscompact,the actionsareergodic. In

contrast if G is noncompact semisimple, the associated deforma-
]
T tionspacecontainsopensubsetscontainingtheFricke-Teichmu¨ller
G space upon which ModΣ acts properly. Properness of the ModΣ-
. action relates to (possibly singular) locally homogeneous geomet-
h
ric structures on Σ. We summarize known results and state open
t
a questions about these actions.
m
[

3
v Contents
4
1
Introduction 2
1
9 Acknowledgments 3
0 1. Generalities 4
5
1.1. The Symplectic Structure 4
0
/ 1.2. The Complex Case 5
h
t 1.3. Singularities of the deformation space 6
a
1.4. Surfaces with boundary 9
m
1.5. Examples of relative SL(2,C)-character varieties 10
:
v
2. Compact Groups 13
i
X 2.1. Ergodicity 13
r 2.2. The unitary representation 14
a
2.3. Holomorphic objects 15
2.4. Automorphisms of free groups 16
2.5. Topological dynamics 16
Date: February 2, 2008.
1991 Mathematics Subject Classification. Primary: 57M50; Secondary: 58E20,
53C24.
Key words and phrases. Mapping class group, Riemann surface, fundamental
group, representation variety, harmonic map, Teichmu¨ller space, quasi-Fuchsian
group, real projective structure, moduli space of vector bundles, hyperbolic
manifold.
Goldman supported in part by NSF grants DMS-0103889and DMS-0405605.
1
2 WILLIAMM. GOLDMAN
2.6. Individual elements 17
3. Noncompact Groups and Uniformizations 17
3.1. Fricke-Teichmu¨ller space 18
3.2. Other components and the Euler class 19
3.3. The one-holed torus 19
3.4. Hyperbolic 3-manifolds 22
3.5. Convex Projective Structures and Hitchin representations 24
3.6. The energy of harmonic maps 24
3.7. Singular uniformizations and complex projective structures 25
3.8. Complex projective structures 27
References 28
Introduction
A natural object associated to a topological surface Σ is the defor-
mation space of representations of its fundamental group π = π (Σ)
1
in a Lie group G. These spaces admit natural actions of the mapping
class group Mod of Σ, and therefore determine linear representations
Σ
of Mod .
Σ
The purpose of this paper is to survey recent results on the dynamics
of these actions, and speculate on future directions in this subject.
The prototypes ofthistheory aretwo ofthe most basicspaces inRie-
mann surface theory: the Jacobian and the Fricke-Teichmu¨ller space.
The Jacobian Jac(M) of a Riemann surface M homeomorphic to Σ
identifies with the deformation space Hom(π,G)/G when G is the cir-
cle U(1). The Jacobian parametrizes topologically trivial holomorphic
complex line bundles over M, but its topological type (and symplectic
structure) are invariants of the underlying topological surface Σ. The
action of Mod is the action of the integral symplectic group Sp(2g,Z)
Σ
on the torus R2g/Z2g, which is a measure-preserving chaotic (ergodic)
action.
In contrast, the Teichmu¨ller space T (Fricke space if ∂Σ 6= ∅) is
Σ
comprised of equivalence classes of marked conformal structures on Σ.
A marked conformal structure is a pair (M,f) where f is a home-
omorphism and M is a Riemann surface. Marked conformal struc-
tures (f ,M ) and (f ,M ) are equivalent if there is a biholomorphism
1 1 2 2
h
M −→ M such that h◦f is homotopic to f . Denote the equivalence
1 2 1 2
class of a marked conformal structure (f,M) by
hf,Mi ∈ T .
Σ
SURFACE GROUP REPRESENTATIONS 3
A marking f determines a representation of the fundamental group:
π = π (Σ) −f→∗ π (M) ⊂ Aut(M˜).
1 1
By the uniformization theorem (at least when χ(Σ) < 0), these iden-
tifywithmarked hyperbolic structures onΣ, which inturnidentify with
conjugacy classes of discrete embeddings of the fundamental group π in
the group G = PGL(2,R) of isometries of the hyperbolic plane. These
classes form a connected component of Hom(π,G)/G, which is homeo-
morphic to a cell of dimension −3χ(Σ) [35]. The mapping class group
Mod acts properly on T . The quotient orbifold
Σ Σ
M := T /Mod
Σ Σ Σ
is the Riemann moduli space, consisting of biholomorphism classes of
(unmarked) conformal structures on Σ.
Summarizing:
• When G is compact, Hom(π,G)/G has nontrivial homotopy
type, and the action of the mapping class group exhibits non-
trivial dynamics;
• WhenG = PGL(2,R)(ormoregenerallyanoncompactsemisim-
pleLiegroup),Hom(π,G)/Gcontainsopensets(likeTeichmu¨ller
space) which are contractible and admit a proper Mod -action.
Σ
Often these open sets correspond to locally homogeneous geo-
metric structures uniformizing Σ.
Thus dynamically complicated mapping classgroupactionsaccompany
nontrivial homotopy type of the deformation space. In general the
dynamics exhibits properties of these two extreme cases, as will be
described in this paper.
Acknowledgments. This paper is an expanded version of a lecture
presented at the Special Session “Dynamics of Mapping Class Group
Actions”attheAnnualMeetingoftheAmericanMathematicalSociety,
January 6-11, 2005, in Atlanta, Georgia. I am grateful to Richard
Brown for organizing this workshop, and the opportunity to lecture
on this subject. I am also grateful to Benson Farb for encouraging
me to write this paper, and to Jørgen Andersen, David Dumas, Lisa
Jeffrey, Misha Kapovich, Franc¸ois Labourie, Dan Margalit, Howard
Masur, Walter Neumann, Juan Souto, Pete Storm, Ser Tan, Richard
Wentworth, Anna Wienhard and Eugene Xia for several suggestions
and helpful comments. I wish to thank the referee for a careful reading
of the paper and many useful suggestions.
4 WILLIAMM. GOLDMAN
1. Generalities
Let π be a finitely generated group and G a real algebraic Lie group.
The set Hom(π,G) of homomorphisms π −→ G has the natural struc-
ture of an affine algebraic set. The group
Aut(π)×Aut(G)
acts on Hom(π,G) by left- and right- composition, preserving the al-
gebraic structure: if α ∈ Aut(π) and h ∈ Aut(G) are automorphisms,
thentheactionof(α,h)onρ ∈ Hom(π,G)isthecompositionh◦ρ◦α−1:
α−1 ρ h
π −−→ π −→ G −→ G
The deformation space is the quotient space of Hom(π,G) (with the
classical topology) by the subgroup Inn(G) of inner automorphisms of
G,andisdenotedHom(π,G)/G. Theactionoftheinnerautomorphism
ιγ determined by an element γ ∈ π equals ιρ(γ−1)(ρ). Therefore Inn(π)
acts trivially on Hom(π,G)/G and the induced action of Aut(π) on
Hom(π,G)/G factors through the quotient
Out(π) := Aut(π)/Inn(π).
When Σ is a closed orientable surface with χ(Σ) < 0, then the natural
homomorphism
π (Diff(Σ)) −→ Out(π)
0
is an isomorphism. The mapping class group Mod is the subgroup of
Σ
Out(π) corresponding to orientation-preserving diffeomorphisms of Σ.
When Σ has nonempty boundary with components ∂ Σ, this defor-
i
mation space admits a boundary restriction map
(1.1) Hom(π (Σ),G)/G −→ Hom π (∂ Σ) ,G)/G.
1 1 i
i∈πY0(∂Σ) (cid:0) (cid:1)
The fibers of the boundary restriction map are the relative character
varieties. This action of Mod preserves this map.
Σ
1.1. TheSymplectic Structure. Thesespacespossessalgebraicsym-
plectic structures, invariant under Mod . For the moment we focus
Σ
on the smooth part of Hom(π,G), which we define as follows. When
G is reductive, the subset Hom(π,G)−− consisting of representations
whose image does not lie in a parabolic subgroup of G is a smooth sub-
manifold upon which Inn(G) acts properly and freely. The quotient
Hom(π,G)−−/G is then a smooth manifold, with a Mod -invariant
Σ
symplectic structure.
SURFACE GROUP REPRESENTATIONS 5
The symplectic structure depends on a choice of a nondegenerate
Ad-invariant symmetric bilinear form B on the Lie algebra g of G and
an orientation on Σ. The composition
ρ Ad
π −→ G −→ Aut(g)
defines a π-module g . The Zariski tangent space to Hom(π,G) at
Adρ
a representation ρ is the space Z1(π,g ) of 1-cocycles. The tangent
Adρ
spacetotheorbitGρequalsthesubspaceB1(π,g )of1-coboundaries.
Adρ
These facts are due to Weil [102], see also Raghunathan [88]. If G
acts properly and freely on a neighborhood of ρ in Hom(π,G), then
Hom(π,G)/G is a manifold near [ρ] with tangent space H1(π,g ). In
Adρ
that case a nondegenerate symmetric Ad(G)-invariant bilinear form
g×g −→B R
defines a pairing of π-modules
g ×g −→B R.
Adρ Adρ
Cup product using B as coefficient pairing defines a nondegenerate
skew-symmetric pairing
H1(π,g )×H1(π,g ) −B−∗−(∪→) H2(π,R) ∼= R
Adρ Adρ
on each tangent space
T Hom(π,G)/G ∼= H1(π,g ).
[ρ] Adρ
Here the isomorphism H2(π,R) ∼= R arises from the orientation on Σ.
The resulting exterior 2-formωB is closed [36], and defines a symplectic
structure on the smooth part Hom(π,G)−−/G of Hom(π,G)/G. This
topological definition makes it apparent that ωB is ModΣ-invariant. In
particular the action preserves the measure µ defined by ωB. When
G is compact, the total measure is finite (Jeffrey-Weitsman [61, 62],
Huebschmann [60]).
1.2. The Complex Case. WhenGisacomplexLiegroup,Hom(π,G)
has a complex algebraic structure preserved by the Aut(π) ×Aut(G)-
action. WhenGisacomplexsemisimple Liegroup,theaboveconstruc-
tion, applied to a nondegenerate Ad-invariant complex-bilinear form
g×g −→B C,
determines a complex-symplectic structure on Hom(π,G)−−/G, that
is, a closed nondegenerate holomorphic (2,0)-form. This complex-
symplectic structure is evidently Mod -invariant. For a discussion of
Σ
this structure when G = SL(2,C), see [45].
6 WILLIAMM. GOLDMAN
The choice ofamarked conformalstructure onΣdetermines ahyper-
K¨ahlerstructure onHom(π,G)/Gsubordinatetothiscomplex-symplectic
structure.
A complex-symplectic structure on a 4m-dimensional real manifold
V is given by an integrable almost complex structure J and a closed
nondegenerate skew-symmetric bilinear form
TM ×TM −→Ω C
which is complex-bilinear with respect to J. Alternatively, it is defined
by a reduction of the structure group of the tangent bundle TV from
GL(4m,R) to the subgroup
Sp(2m,C) ⊂ GL(4m,R).
A hyper-K¨ahler structure further reduces the structure group of
the tangent bundle from Sp(2m,C) to its maximal compact subgroup
Sp(2m) ⊂ Sp(2m,C). All of these structures are required to satisfy
certain integrability conditions. A hyper-K¨ahler structure subordinate
to a complex-symplectic structure (Ω,J) is defined by a Riemannian
metric g and integrable almost complex structures I,K such that:
• g is K¨ahlerian with respect to each of I,J,K,
• the complex structures I,J,K satisfy the quaternion identities,
• Ω(X,Y) = −g(IX,Y)+ig(KX) for X,Y ∈ TM.
Goldman-Xia [54], §5 describes this structure in detail when G =
GL(1,C).
From this we can associate to every point in Teichmu¨ller space T
Σ
a compatible hyper-K¨ahler structure on the complex-symplectic space
Hom(π,G)−−/G. However the hyper-K¨ahler structures are not Mod -
Σ
invariant.
1.3. Singularities of the deformation space. In general the spaces
Hom(π,G) and Hom(π,G)/G are not manifolds, but their local struc-
ture admits a very explicit cohomological description. For convenience
assume that G is reductive algebraic and that ρ is a reductive represen-
tation, that is, its image ρ(π) is Zariski dense in a reductive subgroup
of G. For ρ ∈ Hom(π,G), denote the centralizer of ρ(π) by Z(ρ) and
the center of G by Z.
A representation ρ ∈ Hom(π,G) is a singular point of Hom(π,G) if
and only if
dim(Z(ρ)/Z) > 0.
Equivalently, the isotropy group of Inn(G) at ρ is not discrete, that is,
the action of Inn(G) at ρ is not locally free.
SURFACE GROUP REPRESENTATIONS 7
The Zariski tangent space T Hom(π,G) equals the space Z1(π;g )
ρ Adρ
of g -valued 1-cocycles on π. The tangent space to the orbit G ·
Adρ
ρ equals the subspace B1(π;g ) of coboundaries. Thus the Zariski
Adρ
normal space atρtotheorbitG·ρinHom(π,G)equals thecohomology
group H1(π;g ).
Adρ
Here is a heuristic interpretation. Consider an analytic path ρ ∈
t
Hom(π,G) with ρ = ρ. Expand it as a power series in t:
0
(1.2) ρ (x) = exp u (x)t+u (x)t2 +u (x)t3 +... ρ(x)
t 0 2 3
(cid:0) (cid:1)
where
π −u→n g
for n ≥ 0. The condition
(1.3) ρ (xy) = ρ (x)ρ (y)
t t t
implies that the tangent vector u = u satisfies the cocycle condition
0
(1.4) u(xy) = u(x)+Adρ(x)u(y),
(the linearization of (1.3). The vector space of solutions of (1.4) is the
space Z1(π;g ) of g -valued 1-cocycles of π.
Adρ Adρ
The Zariski tangent space to the orbit G · ρ equals the subspace
B1(π,g ) ⊂ Z1(π,g ) consisting of 1-coboundaries. Suppose that a
Adρ Adρ
path ρ in Hom(π,G) is induced by a conjugation by a path g
t t
ρ (x) = g ρ(x)g−1,
t t t
where g admits a power series expansion
t
g = exp(v t+v t2 +...),
t 1 2
where v ,v ,··· ∈ g. Thus the tangent vector to ρ is tangent to the
1 2 t
orbit G·ρ. Expanding the power series, this tangent vector equals
u(x) = v −Adρ(x)v ,
1 1
that is, u = δv ∈ B1(π;g ) is a coboundary.
1 Adρ
Letu ∈ T Hom(π,G) = Z1(π;g )beatangentvectortoHom(π,G)
ρ Adρ
at ρ. We give necessary and sufficient conditions that u be tangent to
an analytic path of representations.
Solving the equation (1.3) to second order gives:
1
(1.5) u (x)−u (xy)+Adρ(x)u (y) = [u(x),Adρ(x)u(y)].
2 2 2
2
Namely, the function,
π ×π −→ g
1
(1.6) (x,y) 7−→ [u(x),Adρ(x)u(y)]
2
8 WILLIAMM. GOLDMAN
is a g -valued 2-cochain on π, This 2-cochain is the coboundary δu
Adρ 2
of the 1-cochain π −u→2 g . Similarly there are conditions on the
Adρ
coboundary of u in terms of the lower terms in the power series ex-
n
pansion (1.2).
The operation (1.6) has a cohomological interpretation as follows. π
acts on g by Lie algebra automorphisms, so that Lie bracket defines a
pairing of π-modules
[,]
g ×g −→ g .
Adρ Adρ Adρ
The Lie algebra of Z(ρ) equals H0(π;g ). The linearization of the
Adρ
action of Z(ρ) is given by the cup product on H1(π;g ) with [,] as
Adρ
coefficient pairing:
H0(π;g )×H1(π;g ) [−,]∗→(∪) H1(π;g ).
Adρ Adρ Adρ
Now consider the cup product of 1-dimensional classes. The bilinear
form
H1(π;g )×H1(π;g ) [−,]∗→(∪) H2(π;g ).
Adρ Adρ Adρ
is symmetric; let Q be the associated quadratic form.
ρ
Suppose u is tangent to an analytic path. Solving (1.2) to second
order (as in (1.5) and (1.6)) implies that
[,] (∪)([u],[u]]) = δu ,
∗ 2
that is,
(1.7) Q ([u]) = 0.
ρ
Under the above hypotheses, the necessary condition (1.7) is also
sufficient. In fact, by Goldman-Millson [50], ρ has a neighborhood N
in Hom(π,G) analytically equivalent to a neighborhood of 0 of the cone
C in Z1(π;g ) defined by the homogeneous quadratic function
ρ Adρ
Z1(π;g ) −→ H2(π;g )
Adρ Adρ
u 7−→ Q ([u]).
ρ
Then the germ of Hom(π,G)/G at [ρ] is the quotient of this cone by
the isotropy group Z(ρ). (These spaces are special cases of symplectic
stratified spaces of Sjamaar-Lerman [79].)
An explicit exponential mapping
Exp
N −−−−ρ→ Hom(π,G)
was constructed by Goldman-Millson [49] using the Green’s operator
of a Riemann surface M homeomorphic to Σ.
The subtlety of these constructions is underscored by the following
false argument, which seemingly proves that the Torelli subgroup of
SURFACE GROUP REPRESENTATIONS 9
Mod acts identically on the whole component of Hom(π,G)/G con-
Σ
taining the trivial representation. This is easily seen to be false, for G
semisimple.
Here is the fallacious argument. The trivial representation ρ is fixed
0
by all of Mod . Thus Mod acts on the analytic germ of Hom(π,G)/G
Σ Σ
at ρ . At ρ , the coefficient module g is trivial, and the tangent
0 0 Adρ
space corresponds to ordinary (untwisted) cohomology:
T Hom(π,G) = Z1(π;g) = Z1(π)⊗g.
ρ0
The quadratic form is just the usual cup-product pairing, so any ho-
mologically trivial automorphism φ fixes the quadratic cone N point-
wise. By Goldman-Millson [50], the analytic germ of Hom(π,G) at ρ
0
is equivalent to the quadratic cone N. Therefore [φ] acts trivially on
an open neighborhood of ρ in Hom(π,G). By analytic continuation, [φ]
acts trivially on the whole component of Hom(π,G) containing ρ.
The fallacy arises because the identification Exp of a neighborhood
ρ
N in the quadratic cone with the germ of Hom(π,G) at ρ depends on
a choice of Riemann surface M. Each point hf,Mi ∈ T determines
Σ
an exponential map Exp from the germ of the quadratic cone to
ρ,hf,Mi
Hom(π,G), and these are not invariant under Mod . In particular, no
Σ
family of isomorphisms of the analytic germ of Hom(π,G) at ρ with
0
the quadratic cone N is Mod -invariant.
Σ
Problem 1.1. Investigate the dependence of Exp on the marked
ρ,hf,Mi
Riemann surface hf,Mi.
1.4. Surfaces with boundary. When Σ has nonempty boundary, an
Ad-invariantinner productBongandanorientationonΣdetermines a
Poisson structure (Fock-Rosly [32], Guruprasad-Huebschmann-Jeffrey-
Weinstein [55]). The symplectic leaves of this Poisson structure are the
level sets of the boundary restriction map (1.1).
For each component ∂ Σ of ∂Σ, fix a conjugacy class C ⊂ G. The
i i
subspace
(1.8) Hom(π,G)/G ⊂ Hom(π,G)/G
(C1,...,Cb)
consisting of [ρ] such that
(1.9) ρ(∂ Σ) ⊂ C
i i
has a symplectic structure. (To simplify the discussion we assume
that it is a smooth submanifold.) De Rham cohomology with twisted
coefficients in g is naturally isomorphic with group cohomology of
Adρ
π. In terms of De Rham cohomology, the tangent space at [ρ] to
10 WILLIAMM. GOLDMAN
Hom(π,G)/G identifies with
(C1,...,Cb)
Ker H1(Σ;g ) → H1(∂Σ;g )
Adρ Adρ
(cid:18) (cid:19)
∼= Image H1(Σ,∂Σ;g ) → H1(Σ;g ) .
Adρ Adρ
(cid:18) (cid:19)
The cup product pairing
H1(Σ;g )×H1(Σ,∂Σ;g ) −B−∗−(∪→) H2(Σ,∂Σ;R)
Adρ Adρ
induces a symplectic structure on Hom(π,G)/G .
(C1,...,Cb)
Given a (possibly singular) foliation F of a manifold X by symplectic
manifolds, the Poisson structure is defined as follows. For functions
f,g ∈ C∞(X), their Poisson bracket is a function {f,g} on X defined
as follows. Let x ∈ X and let L be the leaf of F containing x. Define
x
the value of {f,g} at x as the Poisson bracket
{f| ,g| } ,
Lx Lx Lx
where {,} denotes the Poisson bracket operation on the symplectic
Lx
manifold L , and f| ,g| ∈ C∞(L ), are the restrictions of f,g to
x Lx Lx x
L .
x
The examples below exhibit exterior bivector fields ξ representing
the Poisson structure. If f,g ∈ C∞(X), their Poisson bracket {f,g}
is expressed as an interior product of ξ with the exterior derivatives of
f,g:
{f,g} = ξ ·(df ⊗dg).
In local coordinates (x1,...,xn), write
∂ ∂
ξ = ξi,j ∧
∂x ∂x
Xi,j i j
with ξi,j = −ξj,i. Then
∂ ∂ ∂f ∂g
{f,g} = ξi,j ∧ · dxi ⊗ dxj
(cid:18) ∂x ∂x (cid:19) (cid:18)∂x ∂x (cid:19)
Xi,j i j i j
∂f ∂g ∂f ∂g
= ξi,j −
(cid:18)∂x ∂x ∂x ∂x (cid:19)
Xi,j i j j i
1.5. Examples of relative SL(2,C)-character varieties. We give
a few explicit examples, when G = SL(2,C), and Σ is a three-holed
or four-holed sphere, or a one-holed or two-holed torus. Since generic
conjugacy classes in SL(2,C) are determined by the trace function
SL(2,C) −→tr C