International Young Seminar on
Bounded Cohomology and Simplicial Volume 

Abstract: We prove that any closed, non-positively curved 4-manifold with non-zero Euler characteristic has positive simplicial volume. As a consequence, Gromov's conjecture on the positivity of simplicial volume in dimension four is confirmed. This work is joint with Inkang Kim.

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Abstract: I will describe the continuous bounded cohomology of the automorphism group of a regular tree in all positive degrees, with coefficients arising from all of its irreducible continuous unitary representations. To my knowledge, this seems to be the first instance that the continuous bounded cohomology in all positive degrees in this class of coefficients has been fully determined without always being trivial.

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Abstract: In 2021, Monod introduced a novel approach to prove the bounded acyclicity of abstract groups, which he successfully applied to Thompson's group F. This idea was further developed by N.Monod, S. Nariman, and F.F. Francesco, who combined it with tools from algebraic topology to compute the bounded cohomology of transformation groups of manifolds. A key application of their work is establishing the (un)boundedness of characteristic classes in the corresponding classifying spaces. In this talk I will briefly introduce Monod's observation, and show how to combine tools from algebraic topology, dfferential topology and group theory to compute the bounded cohomology of Diff(M), especially for M=S^n with n>3.

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Abstract: A central extension 0->Z->E->G->0 is bounded if the associated Euler class is bounded. In this talk we are concerned with the following question: if N<E is a normal subgroup which does not intersect Z, when is the quotient central extension 0->Z->E/N->G/N->0 bounded? We show that this is equivalent to the fact that a given real quasimorphism on N extends to G, and survey all the cases where this is known to happen. We shall conclude with an open question in the case where G is the mapping class group of a surface; if time permits, we shall also explain how this relate to hierarchical hyperbolicity of certain quotients of mapping class groups. This is joint work with Francesco Fournier-Facio and Alessandro Sisto. 

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Abstract: In an n-manifold with negative curvature bounded away from zero, Thurston's straightening procedure permits to define a bounded n-cocycle by integrating the volume form over straight simplices. This cocycle defines an element in H_b^n(M), the bounded volume class, and it is an interesting question to understand when it vanishes. By the work of Soma, for infinite volume hyperbolic 3-manifolds its vanishing is characterised by the positivity of a geometric invariant, the Cheeger isoperimetric constant. A conjecture of Kim and Kim asserts that this also holds for a wider class of manifolds. I will give an overview of this problem and give a partial answer to the conjecture.

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Abstract: The integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices required to represent the fundamental class with integral coefficients.

In this talk, we focus on manifolds that fiber over the circle and show that their integral simplicial volume is related to the spectral radius of the map induced in homology by the monodromy of the fibration. In the special case where the fiber is an n-dimensional torus, we use this relation to connect the integral simplicial volume with the topological entropy of the monodromy.

This is joint work with Ervin Hadžiosmanović.

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Abstract: If G is the fundamental group of a closed hyperbolic manifold, it is by now classical that the kernel of the comparison map corresponding to the second group of bounded cohomology of G is infinite dimensional. And yet, not much more is known about this space, particularly with respect to its Banach structure.

In this talk I will report on some dynamical system methods developed in collaboration with Federico Rodriguez-Hertz (PSU) that permit to give an explicit description of this Banach space. As a by-product, one  obtains dynamical properties of non-trivial cohomology classes.  Time permitting, I'll go over some possible extensions of the methods to non-hyperbolic groups. 

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Abstract: The positive theory of a group is the fragment of its first order theory that can be expressed without negation. There seems to be a very large overlap between groups with the same positive theory as a non-abelian free group, and groups with large second bounded cohomology. This talk is an invitation to study the relationship between these classes in more detail, through lots of examples.

Abstract: In this talk I would like to advertise and give an account of progress on the following question: "when does a group have sufficiently many quasimorphisms to recognize whether an automorphisms is inner?" This is obviously the case for the free abelian group; it has been shown by Hartnick and Schweitzer for the non-abelian free groups; and has very recently been proved for most acylindrically hyperbolic groups (joint work with Minasyan and Sisto). Motivation for this question comes from the study of bi-invariant metrics on groups and of the thus defined groups of coarse automorphisms.

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Abstract: I will talk about a result obtained with Giuseppe Bargagnati, Alberto Casali and Marco Moraschini; its statement is in the title of the talk. The minimal volume entropy is a real invariant for smooth manifolds, related to the exponential growth rate of metric balls in their universal covers. This invariant seats in a sequence of inequalities between the minimal volume and the simplicial volume. Our result strengthens a theorem of Bucher and Neofytidis, who obtained the vanishing of simplicial volumes for the same class of manifolds.

Abstract: We prove a combination theorem for amenable open covers. Via Gromov's vanishing theorem, we obtain vanishing results for the simplicial volume of gluings and of cyclic branched coverings. Joint with Pietro Capovilla and Clara Löh.

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Abstract: We know from previous work with Italiano and Migliorini that there exists some hyperbolic 5-manifold that fibers over the circle. We show one example where the monodromy is a "pseudo-Anosov homeomorphism" of the 4-dimensional fiber, in a way that is surprisingly similar to the familiar and beautiful two-dimensional picture of Nielsen and Thurston for surfaces.

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Abstract: I will present a family of closed aspherical 4-manifolds realizing every non-negative Euler characteristic. This family of manifolds saturates Hopf-Thurston conjecture on the Euler characteristic of aspherical manifolds, as well as Gromov-Lück inequality on their geography. Along the way, I will show that every closed aspherical 3-manifold with amenable fundamental group is virtually the pi_1-injective boundary of an aspherical 4-manifold with vanishing simplicial volume, answering a question of Löh, Moraschini and Raptis up to finite covers.

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Abstract: In this talk, I will advertise the extension problem of quasimorphisms, that is: when does a quasimorphism on a subgroup extend to a quasimorphism on the whole group?

I will discuss recent progress on this problem. In particular, we can reprove the previous result of Hull and Osin with simpler proof in this case. As an application, we produce new hierarchically hyperbolic quotients of mapping class groups, answering a question of Fournier-Facio, Mangioni, and Sisto.

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