It is a classical fact that every finitely generated group embeds as a subgroup of a finitely generated simple group. In the 90's Bridson proved that if one relaxes "simple" to "no proper finite index subgroups" then such an embedding can be done in a quasi-isometric way. In joint work with Jim Belk, we prove that this is true even keeping the word "simple": every finitely generated group quasi-isometrically embeds as a subgroup of a finitely generated simple group. The simple groups we construct are "twisted" variants of Brin-Thompson groups. Certain of these twisted Brin-Thompson groups also provide examples of groups with interesting finiteness properties, and using them we can produce the second known family of simple groups with arbitrary finiteness properties (the first being due to Skipper-Witzel-Z).

I will talk about a notion of curvature for graphs introduced by Schmuckenschläger which is defined as an analogue of Ricci curvature. This quantity can be computed explicitly for various graphs and allows to find bounds on the spectral gap of the graph and isoperimetric-type inequalities. I will present some general results on the computation of the discrete Ricci curvature of any locally finite graph. I will then focus on graphs associated with Coxeter groups: Bruhat graphs, weak order graphs and Hasse diagrams of the Bruhat order.

The group of $\mathcal{C}^1$ diffeomorphisms of sparse Cantor subsets of a manifold are countable and discrete. We can obtain this way Thompson's groups and Brin's higher dimensional generalizations. Postponed to 06.01.2021. Moreover, braided Thompson groups arise as smooth ($\mathcal{C}^2$) mapping class groups of sparse Cantor sphere pairs. By considering higher genus surfaces instead of spheres one still obtains finitely presented groups. (joint works with Y.Neretin and J.Aramayona)

This talk will describe connections between structural results about the finite simple groups and the complexity of some difficult permutation group problems.

The first part of the talk will discuss the base size of a permutation group, an invariant which determines the complexity of many permutation group algorithms. We will present a new, optimal, bound on the base size of the primitive groups that are not large base. After this, we will discuss some group-theoretic questions for which there is no known polynomial time solution. In particular, we shall present a new approach to computing the normaliser of a primitive group $G$ in an arbitrary subgroup $H$ of $S_n$. Our method runs in quasipolynomial time $O(2^{\log^3 n})$, whereas the previous best known algorithm required time $O(2^n)$.

This is partly joint work with Mariapia Moscatiello (Padova), and partly with Sergio Siccha (Siegen).

In a joint work with Markus Szymik a few years ago, we computed the homology of the Higman-Thompson groups V_{n,r} by first showing that all the homology is stable, in the sense of classical homological stability, and then computing the stable homology. I’ll explain some of the ingredients that are used in the proof.

The Basilica group is a well-studied example of a group of automorphisms of the dyadic rooted tree. We will consider the basic algebra for such automorphisms, prove some lemmata on the Basilica group, and learn why you might want to study this group. Afterwards, we will see a generalisation of "the" Basilica group to "a" Basilica group and study some inheritance properties of this construction. This is joint work with Karthika Rajeev.

Artin groups are a generalization of braid groups, and arise as the fundamental groups of configuration spaces associated with Coxeter groups. A long-standing open problem, called the $K(\pi,1)$ conjecture, states that the higher homotopy groups of these configuration spaces are trivial. For finite Coxeter groups, this was proved by Deligne in 1972. In the first part of this talk I will introduce Coxeter groups, Artin groups, and the $K(\pi,1)$ conjecture. Then I will outline a recent proof of the $K(\pi,1)$ conjecture in the affine case, which is a joint work with Mario Salvetti.

The group of $\mathcal{C}^1$ diffeomorphisms of sparse Cantor subsets of a manifold are countable and discrete. We can obtain this way Thompson's groups and Brin's higher dimensional generalizations. Moreover, braided Thompson groups arise as smooth ($\mathcal{C}^2$) mapping class groups of sparse Cantor sphere pairs. By considering higher genus surfaces instead of spheres one still obtains finitely presented groups. (joint works with Y.Neretin and J.Aramayona)

Let $k$ be any field. Let $G$ be a connected reductive algebraic $k$-group. Associated to $G$ is an invariant that is called the index of $G$. Tits showed that, up to $k$-anisotropy, the $k$-isogeny class of $G$ is uniquely determined by its index. Moreover, for the cases where $G$ is absolutely simple, Tits classified all possibilities for the index of $G$.

Let $H$ be a connected reductive $k$-subgroup of maximal rank in $G$. We introduce an invariant of the pair $H < G$ called the embedding of indices of $H < G$. This consists of the index of $H$ and the index of $G$ along with an embedding map that satisfies certain compatibility conditions. We show that, up to $k$-anisotropy, the $G(k)$-conjugacy class of $H$ in $G$ is uniquely determined by its embedding of indices. Moreover, for the cases where $G$ is absolutely simple of exceptional type and $H$ is maximal connected in $G$, we classify all possibilities for the embedding of indices of $H < G$. Finally, we establish some existence results. In particular, we consider which embedding $s$ of indices exist when $k$ has cohomological dimension $1$ (resp.$ k=R$, $k$ is $p$-adic).

This talk concerns a joint work with R. Blasco-Garcia and C. Martinez-Perez. $\\$ An $\textit{Artin group}$ is a group defined by a presentation with relations of the form $s t s \cdots = t s t \cdots$, where the words on the left hand side and on the right hand side have the same length. $\\$ There are few results proved for all Artin groups, their study being rather done by more or less extended families. $\\$ One of the most studied and understood families is that of $\textit{right-angled Artin groups}$, admitting in their definition only commuting relations. $\\$ The general question that motivated our work is to determine properties of right-angled Artin groups that can be extended to other Artin groups, and to which Artin groups. $\\$ The property that interests us more particularly in this talk is poly-freeness, and the Artin groups are the even Artin groups. $\\$ A group $G$ is $\textit{poly-free}$ if there is a finite sequence of subgroups \[ G_0 = \{1 \} \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G\,, \] such that $G_i / G_{i-1}$ is free for all $i \in \{1, \dots, n \}$. An Artin group is $\textit{even}$ if it is defined by relations of the form $(st)^k = (ts)^k$ with $k \in \mathbb{N}$. $\\$ We know since the 90s that right-angled Artin groups are poly-free. $\\$ We will show that some even Artin groups are also poly-free, but not all of them, although we think they all are poly-free.

This talk concerns the Hurwitz action on reflection factorizations in the infinite family G(m, p, n) of complex reflection groups. We use combinatorial and graph-theoretic techniques to completely characterize when two minimum-length factorizations of an arbitrary element in G(m, p, n) belong to the same Hurwitz orbit. Similar techniques can also be used to characterize the Hurwitz orbits of longer-than-minimum factorizations. This work is joint with Jiayuan Wang.

Loosely continuing a theme from a recent talk in the series, this talk begins with a discussion of several problems in computational algebra that are, to a greater or lesser extent, thought to be difficult. Although quite different at surface level, each hides a computational problem involving tensors, which we broadly interpret to mean grids of data.

The second part of the talk introduces algebraic tools to attack these tensor problems. In some cases this leads to a satisfactory resolution of the original problem; one example is a polynomial-time algorithm to construct the intersection of classical groups. In other cases an efficient general solution is currently out of reach but the algebraic perspective nevertheless provides powerful tools that can be applied, in special cases, with striking success; an example of this is a polynomial-time isomorphism test for $p$-groups of genus 2.

The talk concludes by discussing obstacles to further progress and, time permitting, recent and emerging ideas to remove some of these barriers. It features joint work and ongoing projects with J.F. Maglione, E.A. O'Brien, and J.B. Wilson.