Winter Semester 2021/2022

Abstract:

We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions n≤5. As a consequence, there are hyperbolic groups with finite-type subgroups that are not hyperbolic.

The main tool is Bestvina—Brady theory enriched with a combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.

Abstract:

We show that the simplicial volume of a contractible open 3-manifold M is equal to 0 if and only if M is homeomorphic to R³; otherwise it is equal to +∞. To this aim, we show that the excision of a compact connected codimension-0 submanifold with amenable and π₁-injective boundary induces isometric isomorphisms in l¹-homology in degree at least 2. In contrast, we show that in dimension n ≥ 4 there exist open contractible n-manifolds not homeomorphic to Rⁿ with vanishing simplicial volume. We also compute the spectrum of the simplicial volume of irreducible open 3-manifolds.

This is a joint work with my advisor Roberto Frigerio.

Abstract:

It is a classical result that manifolds that are total spaces of fiber bundles, whose fiber has amenable fundamental group, have vanishing simplicial volume. In this talk I will explore the opposite question, where the manifolds fiber non-trivially with arbitrary fiber over a manifold with amenable fundamental group. The main focus will lay on manifolds fibering over spheres and an application to characteristic classes. All of this is joint work with Jens Reinhold.

Abstract:

By constructing a transfer map on the level of bounded cohomology Brandenbursky and Marcinkowski showed that the group of area-preserving, isotopic-to-identity homeomorphisms on certain manifolds has infinite-dimensional bounded cohomology in degree 3. Kimura adapted the argument to the area-preserving diffeomorphisms on the 2-disk relative the boundary.

In this talk I will explain the construction of the transfer map in the language of couplings between groups and sketch how the results can be extended to higher (even) degrees.

Abstract:

In joint work with Sam Nariman, we determine the bounded cohomology of the homeomorphism groups of the circle and of the closed disc. In both cases, it is the polynomial algebra on the bounded Euler class.

Although certain homeomorphism groups were known to have trivial bounded cohomology since the 1985 work of Matsumoto and Morita, our motivation was that apparently no group had a completely understood bounded cohomology unless it was trivial.

In this lecture, I will describe the new techniques that we introduced in order to prove our results. It turns out that the case of the circle is much easier than that of the disc, but still illustrates some of the main ideas.

Abstract:

I will define two new constructions of finitely generated simple left orderable groups (in recent joint work with Hyde and Rivas). Among these examples are the first examples of finitely generated simple left orderable groups that admit a minimal action by homeomorphisms on the Torus, and the first family that admits such an action on the circle. I shall also present examples of finitely generated simple left orderable groups that are uniformly simple (these were constructed by me with Hyde in 2019). And present new examples that, somewhat surprisingly, have infinite commutator width.

Finally, I will present some new results around the second bounded cohomology of these groups (joint with Fournier-Facio).

Abstract:

In this talk, I will prove that the simplicial volume of the total space of a smooth fiber bundle with fiber being an oriented closed connected (occ) manifold of nonpositive curvature and negative Ricci curvature over an occ manifold with a closed universal covering is zero.

Furthermore, if the fiber is an occ negatively curved manifold with dimension more than 2, the simplicial volume of the total space is zero if and only if the simplicial volume of the base space is zero.

Abstract:

In an ongoing joint work with Glebsky, Lubotzky and Monod, we construct an analogue of bounded cohomology in an asymptotic setting in order to prove uniform stability of lattices in Lie groups (of rank at least two) with respect to unitary groups equipped with a metric induced by a submultiplicative norm.

The main idea is the notion of "defect diminishing", which allows us to reduce stability as a homomorphism lifting problem with abelian kernel, and relates to an asymptotic bounded cohomology H²a whose vanishing implies uniform stability. If time permits, I shall sketch a proof of the main result: namely, the vanishing of H²a(G,V) for lattices in high rank Lie groups (along the lines of the proof in Burger-Monod that bounded cohomology vanishes for such groups), implying uniform stability for this class of groups.

Abstract:

I report on recent joint work with Tobias Hartnick, establishing the stabilization of continuous bounded cohomology along many classical families of real and complex Lie groups. This is obtainedas a consequence of the "high acyclicity" of the Stiefel complex associated to any vector space endowed with a non-degenerate sesquilinear form. In this talk I will concentrate on the proof of said acyclicity statement.