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 3bed 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
L^{2} 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.a Group von Neumann algebras and dimension functions (following [11] chapter 1.1, chapter 6.1)
2.b L2Betti numbers and the spectral theory of the combinatorial Laplacian (following [11] chapters 1.2 and 2.1, [5])
2.c NovikovShubin invariants (following [13], [11, Chapter 2])
2.d The FugledeKadison determinant and L2torsion (following [11] chap ter 3, [9])
2.e L2invariants of 3–manifolds (following [11, Chapter 4]).
3. Applications
3.a L2Alexander torsion for knots and 3–manifolds (following [4], [11] chapter 4.3)
3.b Lück’s theorem about the L2Betti 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 L2invariants
4.a Heat kernels on compact manifolds, de Rham–Hodge theory (following [15])
4.b Decay of heat kernels on cocompact manifolds, analytic L2Betti numbers and analytic NovikovShubin invariants (following [11] chapters 1.3, 1.4 and 2.3)
4.c Analytic torsion and L2analytic torsion, statement of the Cheeger–Müller theorem (following [15], [3] and [11, Chapter 3.5.3]).
4.d Calculation of the L2torsion for 3manifolds (following [11] chapters 3.3 and 3.4, [8])
References

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

[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] 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 L2alexander torsion of 3manifolds. arxiv preprint, 2014.
[5] Beno Eckmann. Introduction to l2methods in topology: Reduced l2 homology, harmonic chains, l2betti numbers. Israel J. math., 117, 2000.
[6] T. Le. Growth of homology torsion in finite coverings and hyperbolic volume. ArXiv eprints, 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. L2torsion and 3manifolds. Lecture Notes Geom. Topol., 1992.
[10] W. Lück. Approximating L2invariants by their finitedimensional analogues. Geom. Funct. Anal., 4(4):455–481, 1994.
[11] Wolfgang Lück. L2invariants: theory and applications to geometry and Ktheory, 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]. SpringerVerlag, 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 nonsimply 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. Rtorsion 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 JeanFrançois 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 JeanMarc Ioos Louis Kohli BenMichael Laroche Clément Le Thang LE Van Tu Le Quentrec Etienne Lecuire Cyril López Daniel Martel Jules meigniez gael Raimbault Jean Verchinine Vladimir