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Geometric topology of moduli spaces of curves Lizhen Ji University of Michigan UC Irvine January 16, 2012

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Page 1: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Geometric topology of

moduli spaces of curves

Lizhen Ji

University of Michigan

UC Irvine

January 16, 2012

Page 2: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Happy Birthday,

Peter!

Page 3: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Introduction.

Let Σg be a compact Riemann surface of genus

g,

and Mg be the moduli space of compact Rie-

mann surfaces Σg.

Fact. Mg is a noncompact quasi-projective

variety.

(Noncompactness follows from degeneration of

Riemann surfaces, and the quasi-projectivity

was first proved by Baily, 1960, via Satake

compactification of the Siegel modular vari-

ety.)

Page 4: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Mg has been studied extensively, for example in

algebraic geometry and mathematical physics.

It admits a compactication, the Deligne-Mumford

compactification, a projective variety, by adding

stable Riemann surfaces.

It also admits another ompactification, Satake

compactification, induced from the Satake com-

pactification of the Siegel modular variety (used

in the earlier result of Baily).

Page 5: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Fact. Mg is a complex orbifold and also

admits many natural metrics, for example, the

Teichmuller metric (a Finsler metric) and Weil-

Petersson metric (a Kahler metric).

It also admits many other metrics. Recently, a

lot of work has been done on metrics of Mg,

for example, in a series of papers of Liu-Sun-

Yau.

Conclusion. Mg should be an interesting space

(orbifold) in Geometry, topology and Anal-

ysis.

Page 6: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

The Goal of this talk is to understand Mg

from points of view of geometric topology

and geometric analysis.

All results work for the more general moduli

space Mg,n of Riemann surfaces of genus g

with n punctures. For simplicity, we will con-

centrate on the moduli space Mg of the com-

pact Riemann surfaces.

We usually assume that g ≥ 2 so that each

Riemann surface Σg admits a canonical metric

conformal to the complex structure, and Σg

is considered as a hyperbolic metric (so that

hyperbolic geometry can be used.)

Page 7: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

One effective way to studyMg is to realizeMg

as a quotient of the Teichmuller space Tg by

the mapping class group Modg.

Mg = Modg\Tg

The action of Modg on Tg is also crucial for

many problems about Modg, for example, clas-

sification of its elements, its cohomological prop-

erties, large scale geometric properties.

We will also mention some properties of Tg and

Modg below.

Page 8: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Plan of the rest of the talk

1. Basic definitions

2. Simplicial volume of Mg

3. Duality property of Modg and its cohomo-

logical dimension

4. Spine of Teichmuller space

5. Lp-cohomology of Mg.

6. Spectral theory of Weil-Petersson metric

Page 9: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

1. Basic Definition

Let Sg be a compact oriented surface of genus

g.

A marked compact Riemann surface is a com-

pact Riemann surface Σg together with a dif-

feomorphism ϕ : Σg → Sg.

Two marked Riemann surfaces (Σg, ϕ), (Σ′g, ϕ′)

are called equivalent if there exists a biholo-

morphic map h : Σg → Σ′g such that the two

maps h ◦ ϕ′, ϕ : Σg → Sg are isotropic (or ho-

motopy equivalent in this case).

The Teichmuller space

Tg = {(Σg, ϕ)}/ ∼.

Page 10: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Fact. Tg is a complex manifold diffeomorphic

to R6g−6. In particular, it is contractible.

It is easy to see that is real analytically dif-

feomorphic to R6g−6 by Fenchel-Nielsen coor-

dinates.

Homeomorphic to R6g−6 was proved by Te-

ichmuller using optimal (least distorted quasi-

conformal maps).

Complex structures was outlined by Teichmuller,

proved by Rauch and Ahlfors, and a simpler

proof by Bers later.

Page 11: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Let Diff+(Sg) be the group of all orientation

preserving diffeomorphisms of Sg, and Diff0(Sg)

the identity component (a normal subgroup).

The quotient group Diff+(Sg)/Diff0(Sg) is called

the mapping class group.

Modg acts on Tg by changing the markings,

and the quotient

Modg\Tg =Mg

(divide out all markings on the surface Sg)

Page 12: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Fact. Modg acts holomorphically and properly

on Tg.

Corollary. Mg is an orbifold.

Modg provides one effective way to understand

Mg.

In many situations, for example, in geometric

group theory, Modg is the main interesting

object, and its action on Tg provides an effec-

tive way to understand its properties. (Hence

Tg and Mg are secondary.)

Page 13: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Example and Analogy

When g = 1,

Tg = H2 = {x+ iy | x ∈ R, y > 0}, the Poincare

upper half plane,

Modg = SL(2,Z).

M1 = SL(2,Z)\H2, the modular curve, a lo-

cally symmetric space.

This suggests thatMg is an analogue of locally

symmetric space Γ\X, where X = G/K is a

symmetric space of noncompact type, and Γ ⊂G is an arithmetic subgroup.

G is a semisimple Lie group, K ⊂ G is a maxi-

mal compact subgroup.

Page 14: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

The correspondence is that

Teichmuller space Tg is an analogue of sym-

metric space X = G/K.

Mapping class group Modg is an analogue of

arithmetic groups Γ

and the moduli space Mg is an analogue of

a locally symmetric space Γ\X.

Page 15: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

This point of view has motivated many prob-

lems about Mg and Modg and their solutions

by considering problems about Γ\X and arith-

metic subgroups Γ.

Many results on homological properties and ge-

ometric group theoretic properties of the map-

ping class group Modg are motivated by cor-

responding results of arithmetic subgroups

of semisimple Lie groups.

Page 16: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Simplicial volume introduced by Gromov.

Let Mn be a compact oriented manifold. Then

it has a fundamental class [M ] ∈ Hn(M,Z).

Denote the image of [M ] under the map

Hn(M,Z)→ Hn(M,R)

also by [M ].

For every n-cycle c =∑σ aσσ, define its `1-

norm

|c|1 =∑σ |aσ|.

Page 17: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

The simplicial volume of M is defined by

|M | = inf{|c|1 | c is an R-cycle representing [M ]}.

If M is compact, |M | is finite. But it could be

zero.

Page 18: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

It follows from definition

Fact: If f : M → N is a map of degree d, then

d|N | ≤ |M |.

Corollary. If M admits a self-map of degree

not equal to ±1, then |M | = 0.

Example. Spheres Sn, tori Zn\Rn have such

self-maps and hence have zero simplicial vol-

ume.

Page 19: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

If M is an oriented noncompact manifold, then

there is a fundamental class [M ]lf in the lo-

cally finite homology group Hlfn (M,Z) and also

in Hlfn (M,R).

Using locally finite cycles, we can also define

the simplicial volume |M | of M as before.

In this case |M | could be ∞. Of course, it

could also be zero.

Page 20: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

If M is an orbifold and admits a finite smooth

cover, N →M , then we can define

|M |orb = |N |/d, where d is the degree of the

covering N →M .

It is independent of smooth covers.

If follows from the fact that the simplicial vol-

ume behaves multiplicatively.

One can show |M |orb ≥ |M |.

It could happen that |M |orb > 0 but |M | = 0.

(Note that once a space has a fundamental

class, its simplicial volume can be defined. In

particular, any algebraic variety and orbifold

has a simplicial volume.)

Page 21: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Motivations for simplicial volume.

The simplicial volume |M | is a homotopy in-

variant of M (since any homotopy equivalence

is of degree 1).

When the Euler characteristic is zero, the sim-

plicial volume gives a possible nonzero homo-

topy invariant.

Page 22: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

It also gives a lower bound for the minimal

volume of a manifold

min− vol(M) = inf{vol(M, g) | g is complete,

−1 ≤ Kg ≤ 1}.

min− vol(M) ≥ c(n)|M |, n = dimM .

Page 23: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

It is not easy to compute |M |. A natural prob-

lem is to decide whether it vanishes or not.

As mentioned before, if a compact manifold

admits a nontrivial automorphism of degree

greater than 1, then its simplicial volume is

zero.

Example, the torus and the sphere.

If a noncompact manifold admits a nontrivial

automorphism of degree greater than 1, then

its simplicial volume is either infinity and 0.

For Rn, it is equal to 0. (Due to Gromov).

Page 24: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Here is a brief summary of known results.

(1) (Thurston-Gromov) If M is a complete hy-

perbolic manifold of finite area, then |M | >0. (Also true if the curvature is negatively

pinched.)

(2) (LaFont-Schmidt, conjectured by Gromov).

If M is a compact locally symmetric space of

noncompact type, then |M | > 0.

(3) (Loh-Sauer) If M is a locally symmetric

space of Q-rank at least 3 (hence noncom-

pact), then |M | = 0.

For some locally symmetric spaces of Q-rank

1, |M | > 0.

Not known for Q-rank 2 locally symmetric spaces.

Page 25: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Given the dictionary betweenMg and Γ\X, the

next result is natural.

Theorem. The simplicial volume of Mg,

|Mg|orb = 0 if and only if g ≥ 2.

When g = 1, M1 admits finite smooth covers

given by hyperbolic manifolds of finite area,

and hence |Mg|orb > 0.

A simple known corollary:

Corollary. When g ≥ 2, Mg does not admit a

negatively pinched complete Riemannian met-

ric.

Page 26: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

What is needed for the proof?

The simplicial volume of a compact manifold

only depends on the fundamental group of the

manifold. If it is amenable, then it is zero.

For the vanishing result of noncompact man-

ifolds, we use covers which are amenable and

amenable at infinity. Such covers are con-

structed using good models of classifying spaces.

For a torsion-free finite index subgroup Γ ⊂Modg, one model of the universal covering space

EΓ of the classifying space BΓ of Γ is given

by a Borel-Serre partial compactification of

Tg, and an analogue of Solomon-Tits theo-

rem on the topology of Tits buildings for

curve complex of the surface Sg is needed to

controlled the topology of the Borel-Serre par-

tial compactification at infinity.

Page 27: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Cohomological properties

The Borel-Serre type compactification and the

Solomon-Tits theorem for curve complex have

also applications to Modg.

A discrete group Γ is called a Poincare duality

group of dimension n if for every ZΓ-module

A, there is an isomorphism

Hi(Γ, A) ∼= Hn−i(Γ, A).

Γ is called a Generalized Poincare duality

group of dimension n if there is a ZΓ-module

D, called the dualizing module, such that for

every ZΓ-module A, there is an isomorphism

Hi(Γ, A) ∼= Hn−i(Γ, D ⊗A).

If D is trivial, then Γ is a Poincare duality

group.

Page 28: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Duality groups must be torsion-free. By pas-

ing to torsion-free finite index subgroups, we

can define virtial (generalized) Poincare dual-

ity group.

If Γ is a virtual generalized Poincare duality

group of dimension n, then its virtual coho-

mological dimension is equal to n.

Page 29: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Theorem. (Harer) Modg is a virtual general-

ized Poincare duality group of dimension 4g−5.

Theorem. (Ivanov-J) Modg is a not virtual

Poincare duality group.

Motivated by Borel-Serre results on arithmetic

groups Γ ⊂ G:

Γ is a virtual generalized Poincare duality group,

and it is a virtual Poincare duality group if and

only if Γ\X is compact.

The Borel-Serre partial compactificastion of

X = G/K, and the Solomon-Tits theorem on

the rational Tits building of G are used.

By applying a duality theorem to the partial

Borel-Serre compactification of X.

Page 30: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

The boundary components of the Borel-Serrepartial compactification of X are parametrizedby rational parabolic subgroup of G, or equiva-lently by simplices of the rational Tits buildingof G.

A Borel-Serre partical compactification of Te-ichmuller space was introduced by William Har-vey, and its boundary components are parametrizedby simplices of the curve complex C(Sg) of asurface Sg.

Briefly, each vertex of C(Sg) corresponds to ahomotopy class of essential simple closed curvein Sg.

Several vertices form the vertices of a simplexif they admit disjoint simple closed curves.

Disjoint simple closed curves (or geodesics)can be pinched simultaneously. Hence, sim-plices can parametrize boundary components.

Page 31: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Harer showed C(Sg) is homotopy equivalent to

a bouquet of spheres.

By a similar proof of the Borel-Serre result,

we can show that Modg is a virtual generalized

Poincare duality group.

The following question was open (or raised).

Question. How many spheres in the bouquet?

Theorem. (Ivanov-J) It contains infinitely many

spheres, corresponding to the fact that Modgis not a virtual Poincare duality group.

Solomon-Tits Theorem shows that rational Tits

building is homotopy to a bouquet of infinitely

many spheres.

Page 32: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Universal spaces for proper actions of a dis-crete group Γ.

EΓ is a space where Γ acts properly, and forevery finite subgroup F ⊂ Γ, the set of fixedpoin set EΓF is nonempty and contractible.

EΓ exists and is unique up to homotopy equiv-alence.

If Γ is torsion-free, then EΓ is equal to EΓ,the universal covering space of the classifyingspace BΓ of Γ.

π1(BΓ) = Γ, πi(BΓ) = {1} when i ≥ 2.

Cohomology of Γ can be computed by coho-mology of BΓ.

Fact. If Γ contains torsion elements, thenthere are non zero cohomology groups in arbi-trarily large degree.

Page 33: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

This implies that if Γ contains torsion elements,

then dimBΓ = ∞ for every model. Hence

dimEΓ =∞.

Find good models of EΓ, which could have a

chance of finite dimension.

Want models of EΓ such that Γ\EΓ is com-

pact, or a finite CW-complex, called a cofi-

nite model.

Also want to have small dimEΓ.

The smallest dimension is the virtual coho-

mology dimension of Γ.

Such good models are important for proof of

Baum-Connes conjecture, Novikov conjecture,

...

Page 34: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Proposition. Tg is a universal space for proper

actions of Modg.

For any finite subgroup F ⊂Modg, the nonemp-

tyness of the fixed point set (Tg)F is the posi-

tive solution of the Nielsen realization problem.

The contractibility of the fixed point set can

be proved using earthquake paths.

Or we can use W-P metric and its geodesic

convexity.

Page 35: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

But Modg\Tg is noncompact. Hence Tg is not

a cofinite model.

Borrow ideas from locally symmetric spaces

X = G/K is a universal space for Γ since X

is a simply connected and nonpositively curved

manifold.

Assume Γ\X is noncompact.

The Borel-Serre compacification of X is a can-

didate.

Theorem (J) The Borel-Serre compacification

of X is indeed a cofinite model of EΓ.

Another method is to consider a truncated sub-

manifold.

Page 36: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

The analogue of Borel-Serre of Tg was outlined

by Harvey.

As mentioned before, in this work, he intro-

duced the curve complex, which turns out to

be fundamental in many applications.

Later Ivanov gave a construction of the Borel-

Serre partial compactification. But it is diffi-

cult to use.

Instead we use a subspace of Tg.

For small ε > 0, define the thick part Tg(ε)consisiting of hyperbolic surfaces without geodesics

shorter than ε.

Page 37: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Fact. Tg(ε) is a manifold with corners, stable

under Modg with a compact quotient.

Theorem. (J-Wolpert) There is a Modg-

equivariant deformation retraction of Tg to Tg(ε).

Hence Tg(ε) is a cofinite model of Tg(ε).

Page 38: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Problem. Find equivariant deformation re-

traction of Tg of dimension as small as possi-

ble.

These are called spines of Tg.

The smallest possible dimension is 4g − 5, vcd

of Modg.

Thurston’s attempt to construct spines of Tgin 1985.

Remark. If n > 0, an explicit spine of Tg,nof the smallest possible dimension is known,

due to Mumford, Thurston, Penner, Bowditch-

Epstein.

Page 39: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Even though things should be easier for sym-

metric spaces, spines of symmetric spaces of

the smallest possible dimension with respect to

an arithmetic subgroup is not known in gen-

eral, except for a few cases.

There are some work of Soule and Ash for lin-

ear symmetric spaces,

i.e., which are homothety section of symmetric

cones such as GL+(n,R)/SO(n), the space of

positive definite quadratic forms.

Page 40: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Define R ⊂ Tg to consist of hyperbolic surfaces

where at least two shortest closed geodesics

intersect.

This a closed real-analytic subset stable under

Modg with a compact quotient.

Theorem (J) There is an equivariant defor-

mation retraction of Tg to R. In particular, R

is a cofinite model of EModg.

R is a spine of positive codimension. But Tg(ε)is of codimension 0.

This is the first spine of positive codimension.

Problem. Get spines of smaller dimension.

Work in progress, at least get codimension 2.

Page 41: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Borrow idea from symmetric space SL(n,R)/SO(n)of unimodular lattices of Rn.

There is a well-rounded deformation retractionto well-rounded lattices.

Given a lattice Λ ⊂ Rn, define

m(Λ) = inf{||v|| | v ∈ Λ− {0}},

M(Λ) = {v ∈ Λ | ||v|| = m(Λ)}, the set ofshoretest vector vectors.

If M(Λ) spans Λ, Λ is called a well-rounded

lattice.

Deformation procedure: successive scalingup shortest vectors at the same time. i.e., scal-ing up the linear subspace spanned by M(Λ)and scaling down the orthogonal complementto keep the lattice unimodular, until a well-rounded lattice is reached.

Page 42: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Procedure to deform Tg to the spine R of

positive codimension.

Simultaneously increase the length of all sys-

toles (shortest geodesics) until they intersect.

Using gradient flow of the length functions.

In the paper of J-Wolpert, the deformation de-

pends on a partition of unity and is not canon-

ical. This procedure gives a canonical defor-

mation retraction of Tg to its thick part Tg(ε).

Page 43: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

L2-cohomology of noncompact Riemannian

manifold (M,ds2).

Hi(2)(M) defined by the complex of L2-differential

forms.

(If M is compact, then the L2-cohomology is

the de Rham cohomology.)

Similarly, we can define Lp-cohomlogy groups

They only depend on the quasi-isometry class

Page 44: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

For general noncompact Riemannian manifolds,it is not easy to determine Lp-cohomology.

De Rham theorem gives a topological interpre-tation of the L2-cohomology groups.

In general, we also want a topological interpre-tation.

In this case, locally symmetric spaces of finitevolumes are probably the simplest ones.

Let Γ\X be a noncompact arithmetic locallysymmetric varieties (i.e., arithmetic locally Her-mitian symmetric space).

It has a Baily-Borel compactification, Γ\XBB,

a normal projective variety.

example. When Γ\X is a hyperbolic surface,then it is obtained by adding finitely many points,one for each end.

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Another example is the Siegel modular variety

An, adding An−1, · · · , A1, A0.

Fact. It is a projective variety defined over a

number field.

It admits the intersection cohomology group

with respect to the middle perversity IHi(Γ\XBB).

The key point of the intersection cohomology

is that it satisfies the Poincare duality property.

Page 46: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Zucker conjecture, solution by Looijenga, Saper-

Stern

Theorem (Looijenga, Saper-Stern). The L2-

cohomology of Γ\X is canonically isomorphic

to the intersection cohomology of Γ\XBB:

Hi(p)(Γ\X) ∼= IHi(Γ\XBB

).

Page 47: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Zucker’s result on Lp-cohomology of locally

symmetric spaces

Reductive Borel-Serre compactification Γ\XRBS

of arithmetic locally symmetric spaces Γ\X.

The singularities of Γ\XBBare big.

Γ\XRBSis mapped onto Γ\XBB

and resolve

the singularities topologically to certain extent.

(The link of the singularities of Γ\XRBSis less

complicated.)

Even if Γ\X is a Hermitian locally symmetric

space, Γ\XRBSis not a complex space. This

was introduced by Zucker in the study of L2-

cohomology of Γ\X.

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The reason why is called reductive Borel-Serre

compactification is that it is a reduction of

the Borel-Serre compactification of Γ\X men-

tioned above.

The fibers of the map Γ\XBS → Γ\XRBSare

nilmanifolds, and the boundary of Γ\XRBScon-

sists of reductive locally symmetric spaces.

Example. Borel-Serre compactifiation for hy-

perbolic surfaces obtained by adding a circle to

each end, and for the reductive Borel-Serre

compactification, add one point.

Theorem. (Zucker) For p� 0, The Lp-cohomology

Hi(p)(Γ\X) is isomorphic to Hi(Γ\XRBS

).

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Generalizations to Mg.

Let MgDM be the Deligne-Mumford compact-

ification, an orbifold.

As mentioned before, Lp-cohomology groups

only depend on the quasi-isometry class of the

metrics.

On Teichmuller space and moduli space, there

two different kinds of metrics.

One is the Weil-Petersson metric, and another

is Teichmuller metrics and other complete met-

rics quasi-isometric to it.

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We recall Weil-Petersson metric

It is a metric defined on Tg, invariant with re-

spect to Modg.

For any point p = (Σg, ϕ) ∈ Tg, its cotan-

gent space T ∗pTg is the space of holomorphic

quadratic forms Q(Σg).

For f, g ∈ Q(Σ), define an inner product

〈f, g〉 =∫Σgfg(ds2)−1,

where ds2 is the area form of the hyperbolic

metric on Σg.

This is the Weil-Petersson metric ωWP .

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Fact: (1) W-P metric ωWP is invariant under

Modg.

(2) It is a Kahler metric and has negative cur-

vature.

(3) It is incomplete but is geodesically convex

(every two points are connected by a unique

geodesic).

(4) The voloume of Mg in ωWP is finite.

(5) The completion ofMg with respect to ωWP

is the Deligne-Mumford compactification.

Page 52: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Theorem [J-Zucker]

(1) For any Riemannian metric quasi-isometric

to the Teichmuller metric, and any p < +∞,

Hi(p)(Mg) ∼= IHi(Mg

DM) = Hi(MgDM).

(2) For any metric quasi-isometric to Weil-

Petersson metric, when p ≥ 4/3,

Hi(p)(Mg) ∼= Hi(Mg

DM),

when p < 4/3,

Hi(p)(Mg) ∼= Hi(Mg).

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(Note that Lp-cohomology groups only depend

on the quasi-isometry class of the metric.)

This depends on the asymptotics of the W-P

metric near the boundary divisor.

This result shows a rank-1 phenomenon of Mg

since when the rank of a Hermitian locally sym-

metric space Γ\X > 1, Γ\XRBSis different

from Γ\XBB.

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Spectral theory

Since this is a conference on geometric anal-ysis, I will conclude a result on the spectraltheory of the W-P metric of Mg.

Before studying spectral theory of (Mg, ωWP ),we recall some basic facts about self-adjointextension of the Laplace operator.

For any smooth Riemannian manifold M , itsLaplacian ∆ is a symmetric operator with thedomain C∞0 (M).

Fact. If M is a complete Riemannian mani-fold, then ∆ admits a unique self-adjoint ex-tension to L2(M). (∆ is also called essentiallyself-adjoint).

(First proved by Gaffney, later Yau, Chernoff)

The completeness of M is important.

Page 55: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

Fact. If M is a compact Riemannian mani-

fold, then ∆ has only a discrete spectrum, and

its counting function satisfies the Weyl asymp-

totic law.

Let λ1 ≤ λ2 ≤ · · · be its eigenvalues.

N(λ) = |{λi ≤ λ}|.

Then as λ→ +∞,

N(λ) ∼ c vol(M)λn2 , c depends on the dimen-

sion.

In the above result, the compactness of M is

crucial.

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But for (Mg, ωWP ), both the completeness and

compactnese conditions fail.

Theorem (J-Mazzeo-Muller-Vasy). (1) The

Laplacian ∆WP of the W-P metric of Mg is

essentially self-adjoint.

(2) ∆WP has only a discrete spectrum and

its counting function satisfies the Weyl asymp-

totic law.

In some sense, (Mg, ωWP ) behaves like a com-

pact Riemannian manifold.

We do have a compact orbifold, the Deligne-

Mumford compactification, but the metric ωWP

is singular near the divisors.

Page 57: Geometric topology of moduli spaces of curvesscgas/scgas-2012/Talks/Ji.pdf · 2. Simplicial volume of Mg 3. Duality property of Modg and its cohomo-logical dimension 4. Spine of Teichmuller

The spectral theory of Teichmuller metric is

probably not easy. For eample, there are dif-

ferent definitions of the Laplacian operator.

Mg admits many natural complete metrics which

are quasi-isometric to the Teichmuller metric.

For example, the Bergman metric, Kahler-Einstein

metric, the Ricci metric, McMullen metric.

It is easy to see that the spectrum of the Lapla-

cian of these metrics is not discrete.

It should be continuous.

A natural problem is to understand the spec-

trum and generalized eigenfunctions of the con-

tinuous spectrum.

For example, geometric scattering theory.