International Young Seminar on
Bounded Cohomology and Simplicial Volume 

 Abstract: The simplicial volume is a topological invariant which, roughly speaking, measures how well a manifold can be triangulated. Its link with Ricci curvature has been apparent since the beginning of the theory, when Gromov showed that compact manifolds with nonnegative Ricci curvature have vanishing simplicial volume. In this talk, we show that this vanishing result extends to the non-compact setting. We then explain how it can be used to obtain the topological classification of a class of metric spaces with nonnegative Ricci curvature in a generalized sense. These results were obtained in joint work with Mattia Magnabosco and Daniele Semola.

Slides.

 Abstract: The gnarledness of simplicial complexes is an invariant introduced by Chambers, Dotterrer, Manin, and Weinberger in their study of quantitative null-cobordism. This quantity is useful for cohomological filling estimates. In this talk, we consider algebraic versions of gnarledness for chain complexes, which lead to simplified estimates.

Slides.

 Abstract: The normal rank (or weight) of a group G is the smallest number of elements that normally generate G. This plays an important role in 3-manifold topology, but it is poorly understood. It is extremely difficult to give lower bounds apart from looking at the abelianization. The Wiegold problem asks if there is a finitely generated perfect group that has normal rank greater than one, namely the group cannot be normally generated by any single element. In a joint work with Yash Lodha, we show free products of nontrivial left-orderable groups all have normal rank greater than one, which solves the Wiegold problem. Our result also has nice implications about Dehn surgeries of 3-manifolds. I will explain the topological and dynamical ingredients in the proof of our theorem.

 Abstract: Jonathan K. Simon posed a longstanding problem regarding the algebraic constraints on the geometric genus of knots. In this talk, we present a recent positive answer to this problem: if K and K' are two knots such that there exists an epimorphism from the knot group of K to the knot group of K', then the genus of K is greater or equal to the genus of K'. We will explore this result in detail and outline how it is established via the Lewin property for 3-manifold groups.

Slides.

 Abstract: The notion of ribbon concordance between knots in S^3 was introduced by Gordon, and it was recently shown by Agol that ribbon concordance is a partial order on the set of knots. In this talk, we explain that a result of Casson--Gordon reduces the study of ribbon concordance between fibered knots to a study of compressions of surface homeomorphisms. We then sketch a proof that the simplicial volume of fibered knots is monotone under ribbon concordances, providing evidence to the more general question by Gordon without the fibered assumption. Time permitting, we mention some generalizations and related results. This is joint work with Ian Agol.

Slides.

 Abstract: The calculation of the Gromov norm of the Kähler class for three classical bounded symmetric domains (Types A, B, and C) was first carried out by Domic and Toledo in 1987. The remaining cases (including these three classical types) were treated by Clerc and Ørsted in 2003. In all cases, the norm is equal to $r\pi$, where r is the rank of the bounded symmetric domain. In this talk, we present a unified calculation for all types of bounded symmetric domains that is simpler than the approach of Clerc and Ørsted. Our method is based on fine geometric properties of bounded symmetric domains, such as the polydisc theorem, and is inspired by an earlier paper of Toledo, where he computed the norm in the case of the complex ball. Besides recovering the upper bound, our approach also yields the equality in a more direct way through an application of the Gauss–Bonnet formula.

Slides.

 Abstract: Since Cannon and Thurston, it is well known that one can naturally obtain a sphere filling Peano-like curve associated to a hyperbolic 3-manifold that fibers over the circle. Subsequently, this construction has been generalized to several settings in the realm of 3-dimensional topology. But what do these maps have in common? How can one understand their behavior? In this talk, we will introduce the concept of a CaTherine wheel as an attempt to address these questions. Furthermore, we will illustrate how we can go back: using the data of a G-invariant CaTherine wheel, one can recover a hyperbolic 3-manifold with fundamental group G and a pseudo-Anosov flow on it. This is joint work with Danny Calegari. 

Slides.

 Abstract: A real convex projective manifold is a $(\mathbb{RP}^d, PGL(d+1,R))$-structure where the developing map's image is a properly convex domain $Ω\subsetRP^d$ and the holonomy group is a discrete, torsion-free subgroup $\Gamma\le PGL(d+1,R)$ preserving $Ω$. In this talk, we will study the simplicial volume of a closed real convex projective manifold $M\cong Ω/Γ$. We will present some examples where the simplicial volume is positive. To conclude, we will relate the simplicial volume to the Euler characteristic, affirmatively answering a famous question by Gromov in this setting: if the simplicial volume of a closed aspherical manifold vanishes, must its Euler characteristic vanish as well?

 Abstract: Let M be a complete hyperbolic n-manifold. Via integration over geodesic simplices, any closed bounded differential 2-form on M defines a bounded cohomology class in H^2_b(M). It was proved by Barge and Ghys (for surfaces) and by Battista et al. (in higher dimensions) that, if $M$ is closed, then this procedure defines an injective embedding of the (infinite-dimensional) space of closed  differential 2-forms on M into  H^2_b(M). We extend this result to the case when the fundamental group of M is of the first kind, i.e. its limit set is equal to the whole boundary at infinity of hyperbolic space (this holds, for example, when M has finite volume). Our argument is different from Barge and Ghys' original one, and relies on the following fact of independent interest: an L^\infty function on the hyperbolic plane is determined by its integrals over all ideal triangles. We prove this fact by way of Fourier analysis on the hyperbolic plane. This is joint work with Gian Maria Dall'Ara and Ervin Hadzsiosmanovic. 

Slides.