Ian Leary and I recently constructed a new uncountable family of groups of type FP. I will use small cancellation theory to show that this family of groups contains uncountably many quasi-isometry classes.

In many ways, finiteness conditions for profinite groups behave analogously to those for abstract groups, but I will describe some important differences in the behaviour of the FP_1 and FP_2 conditions between the two cases. I will also talk about progress on new finiteness conditions that do not exist in the abstract case, in particular UBERG: this says that, for a profinite group G, the completed group ring $\hat{\mathbb{Z}}[[G]]$, considered as a module for itself, is generated with positive probability by some fixed number of random elements.

Warped cone is a geometric object associated with a measure preserving isometric action of a finitely generated group on a compact manifold. It encodes the geometry of the manifold, geometry of the group (Cayley graph) and the dynamics of the group. This geometric object has been introduced by J. Roe in the context of Coarse Baum-Connes conjecture (CBC conjecture). Warped cones associated with the action of amenable groups provide examples of CBC conjecture (due to Guoliang Yu) and some expander graphs/super-expander graphs can be constructed from the warped cones associated with the action of Property (T) groups (building on the works of Sullivan-Margulis). On the other hand, Measured Equivalence (ME) is an equivalence relation between two countable groups introduced by M. Gromov as a measure-theoretic analogue of quasi-isometry. If the ‘cocyles’ associated with a measured equivalence relation are bounded, the relation is called Uniform Measured Equivalence. In this lecture, we prove that if two warped cones are quasi-isometric, then the associated groups are Uniformly Measured Equivalent. As an application, we will talk about different ME-invariants which distinguish two warped cones up to quasi-isometry.

The separation profile is an invariant introduced by Benjamini, Schramm and Timàr as an obstruction to the existence of "regular maps". In this talk I will present how to get upper and lower bounds on the separation profile using isoperimetry and compression. These upper and lower bound enable us to get very narrow windows for many groups (nilpotent groups, lamplighter groups, solvable Baumslag-Solitar, and some more) as well as to prove that a solvable groups which has a regular map into a hyperbolic space is virtually nilpotent.

Joint work with Corentin Le Coz

Using tricks from L^2 homology and some (really) abstract algebra we will show how to algorithmically compute the structure of the fibred cohomology classes of free-by-cyclic groups and most 3-manifolds. (Joint with G. Gardam.)

The class of surface-by-surface groups comes as a natural extension of the class of fibered 3-manifold groups. We are interested in what properties surface-by-surface groups have in common with fibered 3-manifold groups. I will motivate the class of groups we will study and discuss surrounding results. I will then talk about recent work with S. Vidussi and G. Walsh showing that such a bundle is coherent if and only if one of the surfaces is a torus, this answers a question of Hillman.

Let G be a 2-generated group. The generating graph Γ(G) of G is the graph whose vertices are the elements of G and where two vertices g and h are adjacent if G = < g, h >. This definition can be extended to a 2-generated profinite group G, considering in this case topological generation. We prove that the set V(G) of non-isolated vertices of Γ(G) is closed in G and that, if G is prosoluble, then the graph Δ(G) obtained from Γ(G) by removing its isolated vertices is connected with diameter at most 3. However, we construct an example of a 2-generated profinite group G with the property that Δ(G) has 2^ℵ_o connected components. This implies that the so called ``swap conjecture'' does not hold for finitely generated profinite groups. We also prove that if an element of V(G) has finite degree in the graph Γ(G), then G is finite.

For G an even Artin groups of FC type, we consider kernels of characters, i.e., of maps from G to free abelian groups. In the case of discrete characters, i.e., when the free abelian group is cyclic we describe the homology groups of the kernel. We also give some partial results about the Bieri-Strebel-Renz of these kernels.

This is a joint work with Ruben Blasco and Jose Ignacio Cogolludo.

A group G of homeomorphisms of the real line is locally moving if every open interval supports a subgroup which acts on it without global fixed points. An example of such group is Thompson's group F.

In this talk, given a locally moving group G, I will investigate rigidity and flexibility properties of the possible actions of G on the line. It turns out that many locally moving groups (and in particular Thompson's groups F) admit rich (uncountable) families of ``exotic'' actions which are not semi-conjugate to their ``natural'' locally moving action. After giving some examples, I will discuss a result showing all such actions satisfy a specific type of topological dynamical behaviour. Among applications, we will see that if G is locally moving, then all its actions on the real line by C^1-diffeomorphisms must be semi-conjugate to its locally moving action, and that under some additional conditions, a locally moving action is structurally stable under small deformations.

This is a joint work with Joaquín Brum, Cristóbal Rivas and Michele Triestino.

Mackey functors for finite groups are very well understood, and, under certain conditions, have been extended to discrete infinite groups. In this talk I will give a survey on possible definitions of Mackey functors for profinite groups, due to Weigel and Dress-Siebeneicher, and will attempt to find a way to connect these different approaches. This is ongoing joint work with I. Castellano, N. Mazza and A. Vera-Gajardo.

https://drive.google.com/file/d/1T6exjTMQwpTH0z757iidq7JSqckG46lM/view?usp=sharing

Among the many properties one would wish a group to have is coherence: the property that every finitely generated subgroup is finitely presented. Among the 2-dimensional hyperbolic groups, which in some senses are `generic' groups, coherence has been observed to have an empirical connection with Euler characteristic: those groups which are known to be coherent have nonpositive Euler characteristic. In this talk I will discuss joint work with D. Kielak & R. Kropholler which makes this connection probabilistic: a random group of negative Euler characteristic is coherent with high probability.

The commutator length of an element w of the commutator subgroup of a group G is the minimal number of commutators needed to express w as a product of commutators. A more fruitful definition is obtained by stabilising the definition, yielding the notion of "stable" commutator length. In the context of free groups, Puder has recently introduced the notion of "imprimitivity rank", which can be thought of as a homotopical version of commutator length. In this talk, I’ll propose a stable version of imprimitivity rank, and state some of its properties.