Geometric structures,

hyperbolic geometry and related topics


May 15-19, 2017  /  Room 1503(5F), KIAS

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Speaker: Caterina Campagnolo (Karlsruhe Institute of Technology)
Title: Simplicial volume of surface bundles
Abstract: This series of lectures is concerned with the use of simplicial volume in the study of surface bundles over surfaces. This homotopy invariant was introduced by Gromov in 1982 in his seminal paper "Volume and bounded cohomology" and has proven to have deep connections with the geometry of manifolds.

Surface bundles over surfaces constitute a rich family of 4-manifolds. While they have been much studied in the past decades, they still remain quite mysterious. The question of their hyperbolicity, for example, is not yet answered. Another question is the best possible ratio between the signature and Euler characteristic of these bundles. The known examples are far below the proven bounds, so there is still work to be done in order to fill the gap. In this context, simplicial volume can act as a bridge between the two other invariants, as, due in particular to work of M. Hoster, D. Kotschick and later M. Bucher, it relates to the Euler characteristic of surface bundles over surfaces.

We will begin by giving the necessary background on bounded cohomology and its relationship with simplicial volume, then present joint results with Bucher providing new inequalities between the three aforementioned invariants of surface bundles over surfaces. Finally we will discuss how bounded cohomology and simplicial volume could help in the hyperbolicity question.

Speaker: Vincent Emery (Universität Bern)

Title: On the volumes of hyperbolic lattices (I, II, and III)
Abstract: I will give a survey about known results and open problems about arithmetic aspects of the volumes of lattices in PO(n,1), focusing mostly on the case of n>3. I plan to cover the subject according to the following subdivision: arithmetic lattices (first talk), quasi-arithmetic lattices (second talk), non-quasi-arithmetic lattices
(third talk).

Speaker: Joan Porti (Universitat Autònoma de Barcelona)
Title: Reidemeister torsion for hyperbolic three-manifolds
Abstract: In the first lecture I will define Reidemeister torsion, with several examples, and give some of its properties.
In the second lecture I will discuss flat bundles, analytic torsion and Weil's rigidity theorem. With these tools I shall define a Reidemeister torsion for hyperbolic three manifolds.
In the third lecture, I will study its behaviour for sequences of manifolds.