Magnus representations of the mapping class group and L 2 -torsion - PowerPoint PPT Presentation

2008. March. 29 CTQM workshop Magnus representations of the mapping class group and L 2 -torsion invariants Teruaki KITANO (Soka University)

Joint work with • M. TAKASAWA-T. Morifuji: – Interdisciplinary Infomation Sci. Vol. 9, No. 1, 2003. – Proc. Japan Academy, Vol. 79, ser. A. No. 4, 2003. – J. Math. Soc. Japan Vol. 56, No. 2, 2004. • T. Morifuji: – arXiv:0801.4429(math.GT). – In progress.

1 Plan of my talk The main subjects of my talk; • Magnus representation, • L 2 -torsion. I want to explain mainly L 2 -torsion, and in particular Fuglede-Kadison determinant which is the main tool to define it.

Plan 1. Determinant in Linear Algebra 2. Fuglede-Kadison determinant 3. Magnus representation of the mapping class group 4. L 2 -torsion 5. Nilpotent quotient and L 2 -torsion invariants 6. Results

2 Determinant in Linear Algebra For a matrix B ∈ M ( n ; C ) , • tr( B ) , det( B ) , or more generally, symmetric polynomials of the eigenvalues, • the characteristic polynomial det( tE − B ) , are fundamental quantities of B . Here • E : the identity matrix, • t : the variable of the characteristic polynomial ．

We want to define a kind of determinant over non-commutative rings, which is group rings of fundamental groups in mind.

Determinant Recall one of the definitions of the determinant. Not standard, but well known in the are of zeta function theory, dynamical systems, or spectral geometry. fundamental equality: log | det( B ) | ” = ”tr(log( B )) . We want to explain more precisely the above. It can be generalized over the group algebra.

Most simple case: A diagonal matrix.   λ 1 0 0 . . . 0 B = 0 . . . 0  .  0 0 . . . 0 λ n Here, we assume that the eigenvalues are 0 < λ 1 , . . . , λ n < 1 .

Directly we compute, log(det( B )) = log( λ 1 · · · λ n ) n X = log( λ i ) i =0 n X = log(1 + ( λ i − 1)) . i =0 Here recall the expansion of log(1 + x ) at x = 0 ∞ ( − 1) p +1 X x p . log(1 + x ) = p p =1

Then ∞ ! n ( − 1) p +1 X X ( λ i − 1) p log(det( B )) = p p =1 i =0 ∞ ! n 1 X X p (1 − λ i ) p = − i =0 p =1 n ! ∞ 1 X X (1 − λ i ) p = − . p p =1 i =0

Hence, we can get the following equality: ( ) ∞ 1 ∑ p tr (( E − B ) p ) det( B ) = exp − . p =1 or equivalently, ∞ 1 ∑ p tr (( E − B ) p ) . log det( B ) = − p =1

General case ： • Non diagonal matrix case ： – Symmetric matrix, or Hermitian matrix ． ∗ replace B to BB ∗ ( B ∗ :the adjoint matrix of B ) ． ∗ For BB ∗ , eigenvalue is changed from λ i λ i = | λ i | 2 of BB ∗ . of B to λ i ¯

• some engenvalue | λ i | > 1 ： The problem is that the convergence radius of log(1 + x ) equals 1. – For a sufficiently large constant K > 0 such that 0 < λ/K < 1 , ( ) K λ log( λ ) = log K ( λ ) = log( K ) + log . K

– replace B to 1 K B (equivalently, BB ∗ to 1 K 2 BB ∗ ) ． Summary: For any matrix B ∈ GL ( n ; C ) , ( ) ∞ − 1 1 p tr( E − 1 | det( B ) | = K 2 n exp ∑ K 2 BB ∗ ) p . 2 p =1

We extend this equality to the one in the non commutative group algebra as the definition of | det( B ) | . Our targets are group rings of fundamental group of 3-manifolds, or 2-manifolds.

3 Fuglede-Kadison determinant Origin: Theory of the von Neumann algebra. • Fuglede-Kadison:Determinant theory in finite factor, Ann. of Math. (2), 55 (1952). In this talk, we treat only group (von Neumann) algebra cases ．

Why we need the operator theory ? One reason is that C π is not a Noetherian ring. It means, for finitely generated C π -module C and its submodule D , it is not guaranteed that its quotient module C/D is finitely generated, in general. It is obstruction to handle directly the homology, or cohomology theory over the group ring.

Here we fix some notations: • π : a group. • e : the unit of π . • C π : the group algebra of π over C (a linear space over C ). • l 2 ( π ) : l 2 -completion of C π ， namely ， algebra ∑ of all infinite sums λ g g such that g ∈ π | λ g | 2 < ∞ . ∑ g ∈ π

By using the equality log | det | = tr log , if we can define tr , we can do det ． First the trace over C π is defined as follows ． Definition 3.1 C π -trace: (∑ ) tr C π λ g g = λ e ∈ C . g ∈ π This C π -trace tr C π : C π → C can be naturally extended to the trace on the matrices over C π .

For a matrix B = ( b ij ) ∈ M ( n ; C π ) , n ∑ tr C π ( B ) = tr C π ( b ii ) . i =1 By using this trace tr C π : M ( n ; C π ) → C , Fuglede-Kadison determinant is defined as follows.

Definition 3.2 Fuglede-Kadison determinant: « p ! ∞ „ − 1 1 E − BB ∗ det C π ( B ) = K 2 n exp X p tr C π 2 K 2 p =1 ∈ R > 0 . Here • K > 0 : a sufficiently large constant. • B ∗ = ( b ji ) : the adjoint matrix of B = ( b ij ) .

The adjoint matrix B ∗ is defined by • the complex conjugate of coefficients ， • antihomomorphism : ∑ ∑ λ g g − 1 . λ g g := Remark 3.3 The matrix B can be consider the operator on Hilbert space l 2 ( π ) n , and then the above adjoint matrix is just the adjoint operator in the usual sense ．

Remark 3.4 The convergence of the infinite series is not trivial. However, it is known that under some general condition of the group π , • If L 2 -betti number of B ( 1 E − K − 2 BB ∗ ) p )) (( lim p tr C π = 0 , p →∞ then it is guaranteed.

Example 3.5 In the case of • a free group of a finite rank, • a nilpotent group, • an amenable group, • a hyperbolic group, the Fugkede-Kadioson determinant converges if L 2 -betti number is vanishing.

4 Magnus representation • Σ g, 1 : oriented compact surface of a genus g ≥ 1 with 1 boundary component. • ∗ ∈ ∂ Σ g, 1 : a base point of Σ g, 1 . • M g, 1 = π 0 ( Diff + (Σ g, 1 , ∂ Σ g, 1 )) : the mapping class group of Σ g, 1 .

• Γ = π 1 (Σ g, 1 , ∗ ) : free group of rank 2 g . • � x 1 , . . . , x 2 g � : a generating system of Γ . • ϕ ∗ ∈ Aut (Γ) : the induced automorphism by ϕ ∈ M g, 1 .

Proposition 4.1 (Dehn-Nielsen-Zieschang) M g, 1 ∋ ϕ �→ ϕ ∗ ∈ Aut (Γ) is injection. Under fixing generator { x 1 , . . . , x 2 g } , a mapping class ϕ can be determined by the words ϕ ∗ ( x 1 ) , . . . , ϕ ∗ ( x 2 g ) . The Magnus representation of the mapping class group is defined as follows.

Definition 4.2 Magnus representation: ( ) ∂ϕ ∗ ( x j ) r : M g, 1 ∋ ϕ �→ ∈ GL (2 g ; Z Γ) . ∂x i i,j Here • ∂/∂x 1 , . . . , ∂/∂x 2 g : Z Γ → Z Γ are the Fox’s free differentials.

• The conjugation on Z Γ is defined as follows. ∑ For any element λ g g ∈ Z Γ , g ∑ ∑ λ g g − 1 . λ g g = g g

Recall Fox’s free differentials • ∂x j = δ ij , ∂x i + γ ∂γ ′ ∂ ( γγ ′ ) = ∂γ ( γ, γ ′ ∈ π ) , • ∂x i ∂x i ∂x i • it is extended as a Z -linear map ． Remark 4.3 This map is not a homomorphism, but a crossed homomorphism. According to the practice, it is called the Magnus representation of M g, 1 .

By taking the abelianization Γ = π 1 (Σ g, 1 ) → H = H 1 (Σ g, 1 ; Z ) , the map r 2 : M g, 1 → GL (2 g ; Z H ) is obtained. If we restrict this map to the Torelli group I g, 1 = Ker {M g, 1 → Sp(2 g ; Z ) } , r 2 : I g, 1 → GL (2 g ; Z H ) is a homomorphism.

L 2 -torsion invariants 5 The characteristic polynomial of the image of the Magnus representation r ( ϕ ) ∈ GL ( n ; C Γ) can be considered as the Fuglede-Kadison determinant of tE − r ( ϕ ) . Final problem is ; How can we consider the variable t ? For the mapping class ϕ ∈ M g, 1 , we take its mapping torus W ϕ := Σ g, 1 × [0 , 1] / ( x, 1) ∼ ( ϕ ( x ) , 0) .

From here, we put π = π 1 ( W ϕ , ∗ ) . We fix a base point ∗ ∈ ∂ Σ g, 1 × { 0 } ⊂ Σ g, 1 × { 0 } ⊂ W ϕ . Now the group π has the following presentation: π = � x 1 , · · · , x 2 g , t | r 1 , . . . , r n � , where r i := tx i t − 1 ( ϕ ∗ ( x i )) − 1 ( i = 1 ... 2 g ) and t is the generator of π 1 S 1 ∼ = Z .

We can consider the variable ”t” of the characteristic polynomial as the S 1 -direction element in the fundamental group. Put together ， in the C π ∼ = C (Γ ⋊ Z ) , we can consider the characteristic polynomial, as a real number, of the image of the Magnus representation by using the Fuglede-Kadison determinant.

What is the geometric meaning ? By the theorem of L¨ uck, − 2 log det C π ( tE − r ( ϕ )) is the L 2 -torsion of the 3-manifold W ϕ for the regular representation of π .

Remark 5.1 • L 2 -torsion [Lott, L¨ uck, Carey, Mathai, ....] is a generalization of Reidemeister-Ray-Singer torsion to the torsion invariant with infinite unitary representation. • Recall that the natural linear space with actions of the group π is C π , and its natural completion is l 2 ( π ) . It is the regular representation of π . Let us denote ρ ( ϕ ) by the L 2 -torsion of W ϕ .

uck’s formula ． More precisely, we see the L¨ Applying the Fox free differentials to the relators r 1 , · · · , r 2 g of π , we obtain Fox matrix ( ∂r i ) A := ∈ M (2 g ; Z π ) . ∂x j i,j Theorem 5.2 (L¨ uck) log ρ ( ϕ ) = − 2 log det C π ( A ) .