Universal torsion, L

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Universal torsion, L

²

-invariants, polytopes and the Thurston norm

Wolfgang Lück Bonn Germany

email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/

Münster, December, 2015

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Review of classical L

²

-invariants

LetG→X →X be aG-covering of a connected finite CW-complexX.

The cellular chain complex ofX is a finitely generated free ZG-chain complex:

· · ·−^c−−ⁿ⁻¹→M

In

ZG−→^cⁿ M

in−1

ZG−^c−−ⁿ⁻¹→ · · · The associatedL²-chain complex

C_∗⁽²⁾(X):=L²(G)⊗_Z_GC_∗(X)

has Hilbert spaces with isometric linearG-action as chain modules and boundedG-equivariant operators as differentials

· · · ^c

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−−−n−1→M

I

L²(G) ^c

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−−→n M

i

L²(G) ^c

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−−−n−1→ · · ·

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Review of classical L

²

-invariants

LetG→X →X be aG-covering of a connected finite CW-complexX.

The cellular chain complex ofX is a finitely generated free ZG-chain complex:

· · ·−^c−−ⁿ⁻¹→M

In

ZG−→^cⁿ M

in−1

ZG−^c−−ⁿ⁻¹→ · · · The associatedL²-chain complex

C_∗⁽²⁾(X):=L²(G)⊗_Z_GC_∗(X)

has Hilbert spaces with isometric linearG-action as chain modules and boundedG-equivariant operators as differentials

· · · ^c

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−−−n−1→M

I

L²(G) ^c

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−−→n M

i

L²(G) ^c

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−−−n−1→ · · ·

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Definition (L²-homology andL²-Betti numbers) Define then-thL²-homologyto be the Hilbert space

H_n⁽²⁾(X):=ker(c_n⁽²⁾)/im(c⁽²⁾_n+1).

Define then-thL²-Betti number

b⁽²⁾_n (X):=dim_N_(G) H_n⁽²⁾(X)

∈R^≥0.

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The original notion is due toAtiyahand was motivated by index theory. He defined for aG-coveringM→M of a closed

Riemannian manifold b⁽²⁾_n (M) := lim

t→∞

Z

F

tr e^−t^·∆ⁿ(x,x)

dvol_M.

IfGis finite, we have

b_n⁽²⁾(X) = 1

|G|·b_n(X).

IfG=Z, we have

b⁽²⁾_n (X) =dim_C_[_Z_]₍₀₎ C[Z]₍₀₎⊗_C_[_Z_]Hn(X;C)

∈Z.

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t→∞

Z

F

dvol_M.

b_n⁽²⁾(X) = 1

|G|·b_n(X).

IfG=Z, we have

∈Z.

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t→∞

Z

F

dvol_M.

b_n⁽²⁾(X) = 1

|G|·b_n(X).

IfG=Z, we have

∈Z.

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In the sequel3-manifoldmeans a prime connected compact orientable 3-manifold with infinite fundamental group whose boundary is empty or a union of tori and which is notS¹×D²or S¹×S².

Theorem (Lott-Lück)

For every3-manifold M all L²-Betti numbers b_n⁽²⁾(M)e vanish.

We are interested in the case where allL²-Betti numbers vanish, since then a very powerful secondary invariant comes into play, the so calledL²-torsion.

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L²-torsion can be defined analytical in terms of the spectrum of the Laplace operator, generalizinganalytic Ray-Singer torsion. It can also be defined in terms of the cellularZG-chain complex, generalizingReidemeister torsion.

The definition ofL²-torsion is based on the notion of the Fuglede-Kadison determinantwhich is a generalization of the classical determinant to the infinite-dimensional setting. It is defined for a boundedG-equivariant operatorf:L²(G)^m→L²(G)ⁿ to be the non-negative real number

det⁽²⁾(f)=exp 1

2 · Z

ln(λ)dν_f^∗_f

∈R^>0

whereν_f^∗_f is the spectral measure of the positive operatorf^∗f. IfGis finite andm=n, then det⁽²⁾(f) =|det(f)|^1/|G|.

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2 · Z

ln(λ)dν_f^∗_f

∈R^>0

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2 · Z

ln(λ)dν_f^∗_f

∈R^>0

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Definition (L²-torsion)

Suppose thatX isL²-acyclic, i.e., allL²-Betti numbersb_n⁽²⁾(X)vanish.

Let∆⁽²⁾_n :C_n⁽²⁾(X)→C_n⁽²⁾(X)be then-Laplace operatorgiven by c⁽²⁾_n+1◦ c_n⁽²⁾∗

+ c_n−1⁽²⁾ ∗

◦c⁽²⁾_n . Define theL²-torsion

ρ⁽²⁾(X):= 1 2 ·X

n≥0

(−1)ⁿ·n·ln det⁽²⁾(∆⁽²⁾_n )

∈R.

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Theorem (Lück-Schick)

Let M be a3-manifold. Let M₁, M₂, . . . , M_mbe the hyperbolic pieces in its Jaco-Shalen decomposition.

Then

ρ⁽²⁾(M) :=e − 1 6π ·

m

X

i=1

vol(M_i).

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Universal L

²

-torsion

Definition (K₁^w(ZG))

LetK₁^w(ZG)be the abelian group given by:

generators

Iff:ZG^m →ZG^mis aZG-map such that the induced bounded G-equivariantL²(G)^m →L²(G)^m map is a weak isomorphism, i.e., the dimensions of its kernel and cokernel are trivial, then it

determines a generator[f]inK₁^w(ZG).

relations

f₁ ∗ 0 f₂

= [f₁] + [f₂];

[g◦f] = [f] + [g].

DefineWh^w(G):=K₁^w(ZG)/{±g |g∈G}.

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Universal L

²

-torsion

Definition (K₁^w(ZG))

LetK₁^w(ZG)be the abelian group given by:

generators

Iff:ZG^m →ZG^mis aZG-map such that the induced bounded G-equivariantL²(G)^m →L²(G)^m map is a weak isomorphism, i.e., the dimensions of its kernel and cokernel are trivial, then it

determines a generator[f]inK₁^w(ZG).

relations

f₁ ∗ 0 f₂

= [f₁] + [f₂];

[g◦f] = [f] + [g].

DefineWh^w(G):=K₁^w(ZG)/{±g |g∈G}.

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Definition (Weak chain contraction)

Consider aZG-chain complexC∗. Aweak chain contraction(γ∗,u∗)for C∗ consists of aZG-chain mapu∗:C∗→C∗ and aZG-chain

homotopyγ∗:u∗'0∗such thatu∗⁽²⁾:C∗⁽²⁾→C∗⁽²⁾is a weak

isomorphism for alln∈Zandγn◦u_n =u_n+1◦γnholds for alln∈Z.

Definition (UniversalL²-torsion)

LetC∗ be a finite based freeZG-chain complex such that C∗⁽²⁾is L²-acyclic. Define itsuniversalL²-torsion

ρ⁽²⁾_u (C∗)∈Ke₁^w(ZG) by

ρ⁽²⁾_u (C∗) = [(uc+γ)_odd]−[u_odd], where(γ∗,u∗)is any weak chain contraction ofC∗.

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Definition (Weak chain contraction)

Consider aZG-chain complexC∗. Aweak chain contraction(γ∗,u∗)for C∗ consists of aZG-chain mapu∗:C∗→C∗ and aZG-chain

homotopyγ∗:u∗'0∗such thatu∗⁽²⁾:C∗⁽²⁾→C∗⁽²⁾is a weak

isomorphism for alln∈Zandγn◦u_n =u_n+1◦γnholds for alln∈Z.

Definition (UniversalL²-torsion)

LetC∗ be a finite based freeZG-chain complex such that C∗⁽²⁾is L²-acyclic. Define itsuniversalL²-torsion

ρ⁽²⁾_u (C∗)∈Ke₁^w(ZG) by

ρ⁽²⁾_u (C∗) = [(uc+γ)_odd]−[u_odd], where(γ∗,u∗)is any weak chain contraction ofC∗.

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AnadditiveL²-torsion invariant(A,a)consists of an abelian group Aand an assignment which associates to a finite based free ZG-chain complexC∗, for whichC∗⁽²⁾isL²-acyclic, an element a(C_∗)∈Asuch that for any based exact short sequence of such ZG-chain complexes 0→C∗ →D∗→E∗ →0 we get

a(D_∗) =a(C_∗) +a(E_∗), and we havea · · · →0→ZG−−→^±^id ZG→0→ · · ·

=0.

We call an additiveL²-torsion invariant(U,u)universalif for every additiveL²-torsion invariant(A,a)there is precisely one group homomorphismf:U →Asatisfyingf(u(C∗)) =a(C∗)for any such ZG-chain complex.

Then(K₁^w(ZG), ρ⁽²⁾_u )is the universal additiveL²-torsion invariant.

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=0.

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=0.

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The universalL²-torsion is asimple homotopy invariant.

It satisfies usefulsum formulasandproduct formulas. There are also formulas for appropriatefibrationsandS¹-actions.

IfGis finite, we rediscover essentially the classicalReidemeister torsion.

We haveρ⁽²(Sf¹) = (z−1)in Wh^w(Z)∼=Q(z^±1)^×/{±zⁿ|n∈Z}.

We haveρ⁽²(Tfⁿ) =0 forn≥2.

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Theorem (Jaco-Shalen-Johannson decomposition)

Let M be a compact connected orientable irreducible3-manifold with infinite fundamental group whose boundary is empty or toroidal. Let M₁, M₂, . . . , Mr be its pieces in the Jaco-Shalen-Johannson

decomposition. Let j_i:π1(M_i)→π1(M)be the injection induced by the inclusion M_i →M.

Then each M_i and M are L²-acyclic and we have

ρ⁽²⁾_u (M) =e

r

X

i=1

(j_i)∗ ρ⁽²⁾_u (Mf_i) .

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Many other invariants come from the universalL²-torsion by applying a homomorphismK₁^w(ZG)→Aof abelian groups.

For instance, the Fuglede-Kadison determinant defines a homomorphism

det⁽²⁾: Wh^w(ZG)→R

which maps the universalL²-torsionρ⁽²⁾_u (X)to the (classical) L²-torsionρ⁽²⁾(X).

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The fundamental square and the Atiyah Conjecture

Thefundamental squareis given by the following inclusions of rings

ZG ^//

N(G)

D(G) ^//U(G)

U(G)is thealgebra of affiliated operators. Algebraically it is just theOre localizationofN(G)with respect to the multiplicatively closed subset of non-zero divisors.

D(G)is thedivision closureofZGinU(G), i.e., the smallest subring ofU(G)containingZGsuch that every element inD(G), which is a unit inU(G), is already a unit inD(G)itself.

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The fundamental square and the Atiyah Conjecture

ZG ^//

N(G)

D(G) ^//U(G)

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The fundamental square and the Atiyah Conjecture

ZG ^//

N(G)

D(G) ^//U(G)

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IfGis finite, its is given by

ZG ^//

CG

id

QG ^//CG

IfG=Z, it is given by

Z[Z] ^//

L^∞(S¹)

Q[Z]₍₀₎ ^//L(S¹)

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IfGis finite, its is given by

ZG ^//

CG

id

QG ^//CG

IfG=Z, it is given by

Z[Z] ^//

L^∞(S¹)

Q[Z]₍₀₎ ^//L(S¹)

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IfGis elementary amenable torsionfree, thenD(G)can be identified with the Ore localization ofZGwith respect to the multiplicatively closed subset of non-zero elements.

In general the Ore localization does not exist and in these cases D(G)is the right replacement.

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Conjecture (Atiyah Conjecture for torsionfree groups)

Let G be a torsionfree group. It satisfies theAtiyah ConjectureifD(G) is a skew-field.

Fix a natural numberd ≥5. Then a finitely generated torsionfree groupGsatisfies the Atiyah Conjecture if and only if for any G-coveringM→M of a closed Riemannian manifold of dimension d we haveb⁽²⁾_n (M)∈Zfor everyn≥0.

The Atiyah Conjecture implies for a torsionfree groupGthat the rational group ring has no non-trivial zero-divisors.

Notice that the Farrell-Jones Conjecture implies for a torsionfree groupGthat the group ring over any field of characteristic zero has no non-trivial idempotents.

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Theorem (Linnell, Schick)

1 LetC be the smallest class of groups which contains all free groups, is closed under extensions with elementary amenable groups as quotients and directed unions.

Then every torsionfree group G which belongs toCsatisfies the Atiyah Conjecture, actually even overC.

2 If G is residually torsionfree elementary amenable, then it satisfies the Atiyah Conjecture.

This theorem and results byWaldhausenshow for the

fundamental groupπof a 3-manifold (with the exception of some graph manifolds) that it satisfies the Atiyah Conjecture and that Wh(π)vanishes.

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Theorem (Linnell, Schick)

1 LetC be the smallest class of groups which contains all free groups, is closed under extensions with elementary amenable groups as quotients and directed unions.

Then every torsionfree group G which belongs toCsatisfies the Atiyah Conjecture, actually even overC.

2 If G is residually torsionfree elementary amenable, then it satisfies the Atiyah Conjecture.

This theorem and results byWaldhausenshow for the

fundamental groupπof a 3-manifold (with the exception of some graph manifolds) that it satisfies the Atiyah Conjecture and that Wh(π)vanishes.

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Identifying K

₁^w

( Z G) and K

₁

(D(G))

Theorem (Linnell-Lück)

If G belongs toC, then the natural map

K₁^w(ZG)−^∼⁼→K₁(D(G)) is an isomorphism.

Its proof is based on identifyingD(G)as an appropriate Cohn localization ofZGand the investigating localization sequences in algebraicK-theory.

There is aDieudonné determinantwhich induces an isomorphism det_D:K₁(D(G))−→ D(G)^∼⁼ ^×/[D(G)^×,D(G)^×].

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Identifying K

₁^w

( Z G) and K

₁

(D(G))

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Identifying K

₁^w

( Z G) and K

₁

(D(G))

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Identifying K

₁^w

( Z G) and K

₁

(D(G))

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In particular we get forG=Z

K₁^w(Z[Z])∼=Q(z^±1)^×

It turns out that in the caseG=Zthe universal torsion is the same as theAlexander polynomialof an infinite cyclic covering, as it occurs for instance in knot theory.

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Twisting L

²

-invariants

Consider aCW-complexX withπ=π₁(M). Fix an element φ∈H¹(X;Z) =hom(π;Z).

Fort∈(0,∞), letφ^∗Ct be the 1-dimensionalπ-representation given by

w·λ:=t^φ(w)·λ forw ∈π, λ∈C.

One cantwisttheL²-chain complex ofX with this representation, or, equivalently, apply the following ring homomorphism to the cellularZG-chain complex before passing to the Hilbert space completion

CG→CG, X

g∈G

λg·g 7→ X

g∈G

λ·t^φ(g)·g.

Notice that for irrationalt the relevant chain complexes do not have coefficients inQGanymore and theDeterminant Conjecture does not apply. Moreover, the Fuglede-Kadison determinant is in

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Twisting L

²

-invariants

CG→CG, X

g∈G

λg·g 7→ X

g∈G

λ·t^φ(g)·g.

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Twisting L

²

-invariants

CG→CG, X

g∈G

λg·g 7→ X

g∈G

λ·t^φ(g)·g.

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Twisting L

²

-invariants

CG→CG, X

g∈G

λg·g 7→ X

g∈G

λ·t^φ(g)·g.

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Twisting L

²

-invariants

CG→CG, X

g∈G

λg·g 7→ X

g∈G

λ·t^φ(g)·g.

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Defineφ-twistedL²-torsion function

ρ(Xe;φ): (0,∞)→R by sendingt to theCt-twistedL²-torsion.

Its value att =1 is just theL²-torsion.

On the analytic side this corresponds for closed Riemannian manifoldMto twisting with the flat line bundleMe ×_πCt →M. It is obvious that some work is necessary to show that this is a well-defined invariant since theπ-action onCt isnotisometric.

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Theorem (Lück)

Suppose thatX is Le ²-acyclic.

1 The L²torsion functionρ⁽²⁾:=ρ⁽²⁾(Xe;φ) : (0,∞)→Ris well-defined.

2 The limitslim sup_t→∞ ^ρ_ln(t⁽²⁾^(t₎⁾ andlim inf_t→0^ρ_ln(t⁽²⁾^(t₎⁾ exist and we can define thedegree ofφ

deg(X;φ)∈R to be their difference.

3 There is aφ-twisted Fuglede-Kadison determinant det⁽²⁾_tw,φ:K₁^w(ZG)→map((0,∞),R) which sendsρ⁽²⁾_u (Xe)toρ⁽²⁾(Xe;φ).

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Definition (Thurston norm)

LetMbe a 3-manifold andφ∈H¹(M;Z)be a class. Define its Thurston norm

x_M(φ)=min{χ₋(F)|F embedded surface inM dual toφ}

where

χ−(F) = X

C∈π0(M)

max{−χ(C),0}.

Thurstonshowed that this definition extends to the real vector spaceH¹(M;R)and defines aseminormon it.

IfF →M −→^p S¹is a fiber bundle with connected closed surface F 6∼=S²andφ=π₁(p), then

x_M(φ) =−χ(F).

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Definition (Thurston norm)

LetMbe a 3-manifold andφ∈H¹(M;Z)be a class. Define its Thurston norm

x_M(φ)=min{χ₋(F)|F embedded surface inM dual toφ}

where

χ−(F) = X

C∈π0(M)

max{−χ(C),0}.

Thurstonshowed that this definition extends to the real vector spaceH¹(M;R)and defines aseminormon it.

IfF →M −→^p S¹is a fiber bundle with connected closed surface F 6∼=S²andφ=π₁(p), then

x_M(φ) =−χ(F).

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Theorem (Friedl-Lück,Liu)

Let M be a3-manifold. Then for everyφ∈H¹(M;Z)we get the equality deg(M;φ) =x_M(φ).

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Polytopes

Consider a finitely generated abelian free abelian groupA. Let A_R:=R⊗_ZAbe the real vector space containingAas a spanning lattice;

ApolytopeP ⊆A_Ris a convex bounded subset which is the convex hull of a finite subsetS;

It is calledintegral, ifSis contained inA;

TheMinkowski sumof two polytopesP andQis defined by P+Q={p+q|p∈P,q ∈Q};

It iscancellative, i.e., it satisfiesP₀+Q=P₁+Q =⇒ P₀=P₁;

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Polytopes

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Polytopes

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Polytopes

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Polytopes

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Polytopes

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TheNewton polytope

N(p)⊆Rⁿ of a polynomial

p(t₁,t₂, . . . ,tn) = X

i1,...,in

a_i₁_,i₂_,...,i_n·t₁ⁱ¹t₂ⁱ²· · ·t_nⁱⁿ

innvariablest₁,t₂, . . . ,tnis defined to be the convex hull of the elements{(i₁,i₂, . . .i_n)∈Zⁿ |a_i₁_,i₂_,...,i_n 6=0};

One has

N(p·q) =N(p) +N(q).

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TheNewton polytope

N(p)⊆Rⁿ of a polynomial

p(t₁,t₂, . . . ,tn) = X

i1,...,in

a_i₁_,i₂_,...,i_n·t₁ⁱ¹t₂ⁱ²· · ·t_nⁱⁿ

innvariablest₁,t₂, . . . ,tnis defined to be the convex hull of the elements{(i₁,i₂, . . .i_n)∈Zⁿ |a_i₁_,i₂_,...,i_n 6=0};

One has

N(p·q) =N(p) +N(q).

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Definition (Polytope group)

LetP_Z(A)be the Grothendieck group of the abelian monoid of integral polytopes inA_R.

Denote byP_Z_,Wh(A)the quotient ofP_Z(A)by the canonical homomorphismA→ P_Z(A)sendingato the class of the polytope {a}.

InP_Z,Wh(A)we consider polytopes up to translation with an element inA.

Given a homomorphism of finitely generated abelian groups f:A→A⁰, we obtain a homomorphisms of abelian groups

P_Z(f) :P_Z(A)→ P_Z(A⁰), [P]7→[id_R⊗_Zf(P)];

and analogously forP (A).

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Example (A=Z)

An integral polytope inZ_Ris just an interval[m,n]form,n∈Z satisfyingm≤n.

The Minkowski sum becomes

[m₁,n₁] + [m₂,n₂] = [m₁+m₂,n₁+n₂].

One obtains isomorphisms of abelian groups

P_Z(Z) −→^∼⁼ Z² [[m,n]]7→(n−m,m).

P_Z_,Wh(Z) −→^∼⁼ Z, [[m,n]]7→n−m.

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We obtain an injection P_Z(A)→ Y

φ∈hom_Z(A,Z)

P_Z(Z), x 7→ φ(x)

φ.

It implies thatP_Z(A)is torsionfree and not divisible.

ConjecturallyP_Z(Zⁿ)is always a free abelian group.

We obtain a well-defined homomorphism of abelian groups Q[Zⁿ]₍₀₎×

→ P_Z(Zⁿ), p

q 7→[N(p)]−[N(q)].

We want to generalize it to the so called polytope homomorphism.

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φ∈hom_Z(A,Z)

P_Z(Z), x 7→ φ(x)

φ.

→ P_Z(Zⁿ), p

q 7→[N(p)]−[N(q)].

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φ∈hom_Z(A,Z)

P_Z(Z), x 7→ φ(x)

φ.

→ P_Z(Zⁿ), p

q 7→[N(p)]−[N(q)].

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φ∈hom_Z(A,Z)

P_Z(Z), x 7→ φ(x)

φ.

→ P_Z(Zⁿ), p

q 7→[N(p)]−[N(q)].

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Polytope homomorphism

Consider the projection

pr:G→H₁(G)_f :=H₁(G)/tors(H₁(G)).

LetK be its kernel.

After a choice of a set-theoretic section of pr we get isomorphisms ZK ∗H₁(G)_f −^∼⁼→ ZG;

S⁻¹ D(K)∗H₁(G)_f ^∼=

−→ D(G),

where here and in the sequelS⁻¹denotes Ore localization with respect to the multiplicative closed set of non-trivial elements.

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Polytope homomorphism

LetK be its kernel.

S⁻¹ D(K)∗H₁(G)_f ^∼=

−→ D(G),

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Polytope homomorphism

LetK be its kernel.

S⁻¹ D(K)∗H₁(G)_f ^∼=

−→ D(G),

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Givenx =P

h∈H₁(G)_f u_h·h∈ D(K)∗H₁(G)_f, define itssupport supp(x):={h∈H₁(G)_f |h∈H₁(G)_f),u_h 6=0}.

The convex hull of supp(x)defines apolytope P(x)⊆R⊗_ZH₁(G)_f =H₁(M;R).

We haveP(x ·y) =P(x) +P(y)forx,y ∈(D(K)∗H₁(G)_f. Hence we can define a homomorphism of abelian groups

P⁰:

S⁻¹ D(K)∗H₁(G)_f×

→ P_Z(H₁(G)_f),

by sendingx ·y⁻¹to[P(x)]−[P(y)].

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Givenx =P

P⁰:

S⁻¹ D(K)∗H₁(G)_f×

→ P_Z(H₁(G)_f),

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Givenx =P

P⁰:

S⁻¹ D(K)∗H₁(G)_f×

→ P_Z(H₁(G)_f),

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Givenx =P

P⁰:

S⁻¹ D(K)∗H₁(G)_f×

→ P_Z(H₁(G)_f),

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The composite

K₁^w(ZG)−→^∼⁼ K₁(D(G))−→ D(G)^∼⁼ ^{× ∼}−⁼→

S⁻¹ D(K)∗H₁(G)_f× P⁰

−→ P_Z(H₁(G)_f) factories to thepolytope homomorphism

P: Wh^w(G)→ P_Z,Wh(H₁(G)_f).

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Definition (Dual Thurston polytope)

LetMbe a 3-manifold. Define thedual Thurston polytopeto be subset ofH₁(M;R)

T(M) :={v ∈H₁(M;R)|φ(v)≤x_M(φ)for allφ∈H¹(M;R)}.

Thurstonhas shown that the dual Thurston polytope is always an integral polytope.

The Thurston seminormx_M obviously determines the dual Thurston polytope.

The converse is also true, namely, we have x_M(φ) := 1

2·sup{φ(x₀)−φ(x₁)|x₀,x₁∈T(M)}.

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2·sup{φ(x₀)−φ(x₁)|x₀,x₁∈T(M)}.

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2·sup{φ(x₀)−φ(x₁)|x₀,x₁∈T(M)}.

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2·sup{φ(x₀)−φ(x₁)|x₀,x₁∈T(M)}.

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Theorem (Friedl-Lück)

Let M be a3-manifold. Then the image of the universal L²-torsion ρ⁽²⁾_u (M)e under the polytope homomorphism

P: Wh^w(π₁(M))→ P_Z_,Wh(H₁(π₁(M))_f) is represented by the dual of the Thurston polytope.

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Higher order Alexander polynomials

Higher order Alexander polynomialswere introduced for a coveringG→M→M of a 3-manifold byHarveyandCochran, provided thatGoccurs in the rational derived series ofπ₁(M).

At least thedegreeof these polynomials is a well-defined invariant ofM andG.

We can extend this notion of degree also to the universal covering ofM and can prove the conjecture that the degree coincides with the Thurston norm.