Schedule

Abstract: The Boone--Higman Conjecture asserts that a finitely generated group has solvable word problem if and only if it embeds in a finitely presented simple group. While it has recently been confirmed for many interesting classes of groups, including hyperbolic groups and virtually special groups, it remains wide open in general. In this talk we explain a method for proving the Boone--Higman Conjecture for groups acting on locally finite trees, which may offer a route for proving it for many new classes of groups. We illustrate our method by showing the Boone--Higman Conjecture for all (finitely generated free)-by-cyclic groups and all Baumslag--Solitar groups, solving it in two cases that have been raised explicitly by Belk, Bleak, Matucci and Zaremsky. This is joint work with Kai-Uwe Bux and Xiaolei Wu.

Abstract: Let w be a word in a free group. A few years ago, Magee and I discovered that the stable commutator length (scl) of w, which is a well-studied topological invariant, can also be defined in terms of certain Fourier coefficients of w-random unitary matrices. But there are very natural ways to tweak the random-matrix side of this story: one may consider, for example, w-random permutations or w-random orthogonal matrices, and apply the same definition to obtain other "stable" invariants of words. Are these invariants interesting? Do they have, too, alternative topological/combinatorial definitions? In a joint work with Yotam Shomroni, we show that the above result with scl is not a lonely instance but, rather, part of a pattern. In particular, we prove a similar result with Wilton's stable primitivity rank.

Abstract: A torsion-free Tarski monster is a non-abelian group all of whose proper nontrivial subgroups are infinite cyclic. These are typically constructed as limits of small cancellation quotients of torsion-free hyperbolic groups. I will present joint work with R{\'e}mi Coulon and Turbo Ho, where we relate the first-order theory of such monsters to that of the hyperbolic groups in the sequence. This produces torsion-free Tarski monsters which satisfy several remarkable properties that can be detected from their first-order theory.

Abstract: The dynamics on the circle of a group of rotations is characterized, up to (semi)conjugacy, by a classical theorem of Hölder: if a group of circle homeomorphisms is such that every element is topologically semi-conjugate to a rotation, then the whole group is topologically semi-conjugate to a group of rotations. The characterization of the dynamics of groups of Möbius transformations is more delicate, and it was achieved in the early 90s by landmark results of Tukia, Gabai, Casson-Jungreis, through the so-called convergence property. Let us say that a group of circle homeomorphisms is Möbius-like if every element is topologically semi-conjugate to a Möbius transformation. In her thesis (1994), Kovačević provided the first examples of Möbius-like groups without the convergence property. We will discuss the possible structure of Möbius-like groups, based on the thesis work of João Carnevale (2022) and a recent project in collaboration with KyeongRo Kim.

Abstract: We propose to study actions of countable groups on measure spaces such that every group element individually acts as a conservative transformation, that is, a transformation such that no set of positive measure is disjoint from all its translates. We construct such actions of the free group, using the measurable full group of a hyperfinite equivalence relation. The motivation for our work was to find interesting examples of boomerang subgroups.

Based on joint work with Y. Glasner and T. Hartnick.

Abstract: In this talk we will discuss a generalization of Sageev’s wallspace construction that allows to study the geometry of certain spaces by combinatorial properties of certain walls. Specifically, we’ll look at the interactions with hyperbolicity and focus on two applications. In CAT(0) spaces, these techniques allow to construct a “universal hyperbolic quotient”, called the curtain model, that is analogous to the curve graph of a surface. When focusing on a space that is already hyperbolic, the construction can be used to improve its fine properties, and in particular we address a conjecture of Rips and Sela and show that residually finite hyperbolic groups admit globally stable cylinders. This is joint work with Petyt and Zalloum.

Abstract: In this talk we will present  some new invariants and solve the isomorphism problem for a wide family of generalised Bauslag-Solitar groups. This is joint work with Dario Ascari and Ilya Kazachkov.