Research

We survey several criteria that have been used to prove vanishing of bounded cohomology or stable commutator length for transformation groups with certain displacement properties. This is a companion to a paper where we introduce one of these criteria. We compare them, prove implications, and construct counterexamples.

We construct infinite simple characteristic quotients of free groups, answering a 1978 question of James Wiegold (Question 6.45 in the Kourovka Notebook). The construction is very flexible, and allows additional control on the quotients, and generalizations to more groups with hyperbolic features.

We provide a simple algebraic criterion for the vanishing of bounded cohomology with separable dual coefficients in all positive degrees, called the commuting cyclic conjugates condition. We apply this to compute the bounded cohomology of several important families of groups: structure-preserving diffeomorphism groups, stable braid and mapping class groups, direct limit linear groups, piecewise linear and piecewise projective groups, interval exchange transformation groups, and generic enumerated groups.

Following the recent introduction of local permutation stability, we set up a general framework for local stability, and initiate the study of local Hilbert--Schmidt (HS) stability. We prove a version of the Hadwin--Shulman character criterion for amenable groups, provide several examples, and prove that property (T) is an obstruction to local stability.

We study the problem of when a wreath product of finitely generated Hopfian groups is Hopfian. Our main result establishes a strong connection between the case in which the base group is abelian, and the direct and stable finiteness conjectures of Kaplansky in group rings.

I studied stability of metric approximations of groups, when the approximating groups are endowed with bi-invariant ultrametrics. The main case study is a p-adic analogue of Ulam stability, where unitary matrices are replaced by integral p-adic ones.

We prove Ulam stability of wreath products with infinite amenable acting group, as well as several groups of dynamical origin such as Thompson's groups F, F', T and V. The proof uses a new cohomology theory called asymptotic cohomology, introduced in this paper, and along the way we prove several new results about asymptotic cohomology. We also tackle metric approximation questions for such groups, with respect to unitary and symmetric groups.

We introduce and study property (NL), standing for "no loxodromics": a group is said to have property (NL) if it admits as few actions on hyperbolic spaces as possible (i.e. only elliptic and parabolic). We produce many examples of groups with property (NL), mainly Thompson-like groups, and prove that property (NL) is stable under several natural group constructions. Alessandro Sisto's appendix describes a method to associate to an action on a hyperbolic space, another action that is moreover cobounded, and keeps several useful properties from the original action.

A left orderable monster is a finitely generated group acting faithfully on the real line, all of whose fixed point-free actions on the line are proximal (i.e. as complicated as possible). The first examples emerged in the last few years, and are all infinitely presented and quite complicated to construct. We provide an elementary construction that moreover produces the first finitely presented examples.

We prove that for groups in a large class that includes non-elementary hyperbolic groups and non-virtually abelian RAAGs and RACGs, there is an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. The case of the free group of rank at least 3 settles a question of Miklós Abert.

We extended vanishing results on cup products of Brooks quasimorphisms of free groups to cup products of median quasimorphisms, i.e., Brooks-type quasimorphisms of group actions on CAT(0) cube complexes. Special attention is paid to groups acting on trees and right-angled Artin groups.

We exhibit groups acting properly and cocompactly on CAT(0) cube complexes, with quasi-isometric groups that do not admit any proper actions on a CAT(0) cube complex. This gives a systematic method of building counterexamples to a question of Niblo, Sageev and Wise. 

We study second bounded cohomology and quasimorphisms of braided Thompson groups such as bV, bF, or their ribbon cousins such as rV. These exhibit some interesting bounded cohomological behaviours.

We provide a simple dynamical criterion for second bounded cohomology vanishing, and apply it to settling several questions about the bounded cohomology of left-orderable groups, the spectrum of stable commutator length, and the spectrum of simplicial volume. 

We prove that binate groups are boundedly acyclic, i.e. their bounded cohomology vanishes in every positive degree. This essentially contains all previously known non-amenable examples (at the time of posting) and includes many groups of homeomorphisms. We also computed the bounded cohomology of Thompson's group T, assuming a conjecture about the bounded cohomology of Thompson's group F, which is now a theorem by Monod.

We provide the first examples of non-amenable finitely generated and finitely presented groups for which it is possible to compute bounded cohomology in all degrees: these include boundedly acyclic groups (with vanishing in every degree) and groups with large bounded cohomology (with strong non-vanishing in every degree). Moreover, we prove that vanishing of bounded cohomology is not algorithmically decidable.

I study natural notions of amenability and bounded cohomology over non-Archimedean fields (with particular attention to the field of p-adic numbers), and the way they interact. This is all done for totally disconnected locally compact groups, with the last (independent) section on bounded cohomology of topological spaces.

These are proceedings from the "What is?" style seminar that was held online during the Fall Semester of 2020. It includes 12 chapters that should serve as a gentle introduction to young researchers in the field to topics of current research interest.

The main body is based on the two papers I wrote alone during my PhD, namely Normed amenability and bounded cohomology over non-Archimedean fields and Ultrametric analogues of Ulam stability of groups. The last chapter contains a summary of the papers I wrote in collaboration during my PhD.

I study infinite sums of Brooks quasimorphisms with combinatorial methods, and provided new classes of quasimorphisms of the free group which have trivial cup product in bounded cohomology.

The goal of this paper was to fill in the details of the first four chapters of this wonderful book, and add a couple new things.