La Llagonne workgroup on l2 invariants

The la Llagonne workgroup is a yearly activity organised by IMT since more than 20 years traditionally in the location of la Llagonne but this year in Matemale (not far from La Llagonne), in the middle of the Pyrenees. Each year a scientific theme is selected and decomposed into talks which are then distributed among the participants.The level of the talks is supposed to be increasing from basic ones given by younger students (M2 level students are encouraged to participate) to more advanced ones given by experts. The ambience is informal (we will share 3-bed rooms in the "Centre de vacances La Capcinoise") and funding for 2/3 of the lodging expenses will be available for younger participants. The number of participants will be limited to 40 people.

This year's theme is

L2 invariants of knots and three manifolds

and the organisers are T. Fiedler, J. Raimbault and F. Costantino. The activity will start on monday the 27th of march and end on friday the 31st march 2017.

Here below you will find a temptative list of the talks we'd like to organise.If you plan to attend the workgroup please tell us which of the talks below (if any) you would be able to give.

1. Classical invariants

1.a Reidemeister torsion (following [16], [2])

1.b Whitehead torsion (following [12])

1.c Alexander polynomials for knots, 3–manifolds and beyond (following [16])

1.d Simplicial and hyperbolic volumes (following [1], [11] chapter 14.1, [14])

2. L2-generalisations

2.a Group von Neumann algebras and dimension functions (following [11] chapter 1.1, chapter 6.1)

2.b L2-Betti numbers and the spectral theory of the combinatorial Laplacian (following [11] chapters 1.2 and 2.1, [5])

2.c Novikov-Shubin invariants (following [13], [11, Chapter 2])

2.d The Fuglede-Kadison determinant and L2-torsion (following [11] chap- ter 3, [9])

2.e L2-invariants of 3–manifolds (following [11, Chapter 4]).

3. Applications

3.a L2-Alexander torsion for knots and 3–manifolds (following [4], [11] chapter 4.3)

3.b Lück’s theorem about the L2-Betti numbers and conjectures (following [10], [11] chapter 13)

3.c Growth of homology torsion in finite abelian coverings and Mahler measure (following [7], [2])

3.d Growth of homology torsion in finite coverings and the hyperbolic volume (following [6], [2])

4. Heat kernels and analytic computation of L2-invariants

4.a Heat kernels on compact manifolds, de Rham–Hodge theory (following [15])

4.b Decay of heat kernels on cocompact manifolds, analytic L2-Betti numbers and analytic Novikov-Shubin invariants (following [11] chapters 1.3, 1.4 and 2.3)

4.c Analytic torsion and L2-analytic torsion, statement of the Cheeger–Müller theorem (following [15], [3] and [11, Chapter 3.5.3]).

4.d Calculation of the L2-torsion for 3-manifolds (following [11] chapters 3.3 and 3.4, [8])


  1. [1]  R. Benedetti and C. Petronio. Lectures on hyperbolic geometry. Springer, 1992.

  2. [2]  Nicolas Bergeron and Akshay Venkatesh. The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu, 12(2):391– 447, 2013.

  3. [3]  Jeff Cheeger. Analytic torsion and the heat equation. Ann. of Math. (2), 109(2):259–322, 1979.

    [4]  J. Dubois, S. Friedl, and W. Lück. The L2-alexander torsion of 3-manifolds. arxiv preprint, 2014.

    [5]  Beno Eckmann. Introduction to l2-methods in topology: Reduced l2- homology, harmonic chains, l2-betti numbers. Israel J. math., 117, 2000.

    [6]  T. Le. Growth of homology torsion in finite coverings and hyperbolic volume. ArXiv e-prints, December 2014.

    [7]  Thang Le. Homology torsion growth and Mahler measure. Comment. Math. Helv., 89(3):719–757, 2014.

    [8]  John Lott. Heat kernels on covering spaces and topological invariants. J. Differential Geom., 35(2):471–510, 1992.

    [9]  W. Lück. L2-torsion and 3-manifolds. Lecture Notes Geom. Topol., 1992.

    [10]  W. Lück. Approximating L2-invariants by their finite-dimensional analogues. Geom. Funct. Anal., 4(4):455–481, 1994.

    [11]  Wolfgang Lück. L2-invariants: theory and applications to geometry and K-theory, volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2002.

    [12]  J. Milnor. Whitehead torsion. Bull.Amer. Math. Soc., 1966.

    [13]  S. Novikov and M. Shubin. Morse inequalities and von neumann invariants of non-simply connected manifolds. Uspekhi Mat. Nauk, 1986.

    [14]  John G. Ratcliffe. Foundations of hyperbolic manifolds, volume 149 of Graduate Texts in Mathematics. Springer, New York, second edition, 2006.

    [15]  D. B. Ray and I. M. Singer. R-torsion and the Laplacian on Riemannian manifolds. Advances in Math., 7:145–210, 1971.

    [16] Vladimir Turaev. Introduction to combinatorial torsions. Lectures in Mathematics ETH Zu ̈rich. Birkh ̈auser Verlag, Basel, 2001. Notes taken by Felix Schlenk.



    Participants (Preliminary List)

    Barraud Jean-François
    Belletti Giulio
    Ben aribi Fathi
    Bénard Léo
    Bertuol Florian
    Bonandrini Céline
    Costantino Francesco
    Fiedler Thomas
    Finski Siarhei
    Fraczyk Mikolaj
    Gonzalez Pagotto Pablo
    Henneke Fabian
    Herrmann Gerrit
    Hok Jean-Marc
    Ioos Louis
    Kohli Ben-Michael
    Laroche Clément
    Le Thang
    LE Van Tu
    Le Quentrec Etienne
    Lecuire Cyril
    López Daniel
    Martel Jules
    meigniez gael
    Raimbault Jean
    Verchinine Vladimir

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