A subgroup $H$ of the circle group $\mathbb T$ is said to be characterized by a sequence of integers $\mathbf u = (u_n)_{n\in\mathbb N}$ if $H=\{x\in\mathbb T: u_nx\to 0\}$. The first part of the talk discusses characterized subgroups of $\mathbb T$ and their relevance in several areas of Mathematics where the behavior of the sequence $(u_nx)_{n\in\mathbb N}$ as above is studied, as Topological Algebra (topologically torsion elements and characterized subgroups), Harmonic Analysis (sets of convergence of trigonometric series, thin sets) and Number Theory (uniform distribution of sequences).
Recently, generalizations of the notion of characterized subgroup of $\mathbb T$ were introduced, based on weaker notions of convergence, starting from statistical convergence and ending with $\mathcal I$-convergence for an ideal $\mathcal I$ of $\mathbb N$, due to Cartan. A sequence $(y_n)_{n\in\mathbb N}$ in $\mathbb T$ is said to $\mathcal I$-converge to a point $y\in \mathbb T$, denoted by $y_n\overset{\mathcal I}\to y$, if $\{n\in\mathbb N: y_n \not \in U\}\in \mathcal I$ for every neighborhood $U$ of $y$ in $\mathbb T$. A subgroup $H$ of the circle group $\mathbb T$ is said to be $\mathcal I$-characterized with respect to $\mathcal I$ by a sequence of integers $\mathbf u = (u_n)_{n\in\mathbb N}$ if \[H=\{x\in\mathbb T: u_nx\overset{\mathcal I}\to 0\}.\] The second part of the presentation proposes an overview on the results obtained on these new kind of characterized subgroups, with special emphasis on $\mathcal I$-characterized subgroups of $\mathbb T$.
Based on a joint work with D. Dikranjan, R. Di Santo and H. Weber.
The conjugacy growth function of a finitely generated group is a variation of the standard growth function, counting the number of conjugacy classes intersecting the $n$-ball in the Cayley graph. The asymptotic behaviour is not a commensurability invariant in general, but the conjugacy growth of finite extensions can be understood via the twisted conjugacy growth function, counting automorphism-twisted conjugacy classes. I will discuss what is known about the asymptotic and formal power series behaviour of (twisted) conjugacy growth, in particular some relatively recent results for certain groups of polynomial growth (i.e. virtually nilpotent groups).
Let $w(x,y)$ be a word in a free group. For any group $G$, $w$ induces a word map $w:G^2 \to G$. For example, the commutator word $w=xyx^{-1}y^{-1}$ induces the commutator map. If $G$ is finite, one can ask what is the probability that $w(g,h)$ is equal to the identity element $e$, for a pair $(g,h)$ of elements in $G$, chosen independently at random. In the setting of finite simple groups, Larsen and Shalev showed there exists $\epsilon(w)>0$ (depending only on $w$), such that the probability that $w(g,h)=e$ is smaller than $|G|^{-\epsilon(w)}$, whenever $G$ is large enough (depending on $w$). In this talk, I will discuss analogous questions for compact groups, with a focus on the family of unitary groups; For example, given a word $w$, and given two independent Haar-random $n$ by $n$ unitary matrices $A$ and $B$, what is the probability that $w(A,B)$ is contained in a small ball around the identity matrix?
Based on a joint work with Nir Avni and Michael Larsen.
Given a locally compact group, its set of closed subgroups can be endowed with a compact, Hausdorff topology. With this topology, it is called the Chabauty space of the group. Every group acts on its Chabauty space via conjugation. This action has connections to rigidity theory, Margulis’ normal subgroup theorem and measure preserving actions of the group via so-called Invariant Random Subgroups (IRS). I will give a gentle introduction into Chabauty spaces and IRS and state a few classical results. I will define boomerang subgroups and explain how special cases of the classical results can be proven via them. Based on joint work with Yair Glasner.
Dyer groups are a family encompassing both Coxeter groups and right-angled Artin groups. Each of these two classes of groups have natural piecewise Euclidean CAT(0) spaces associated to them: the Davis-Moussong complex for Coxeter groups and the Salvetti complex for right-angled Artin groups. In this talk I will introduce Dyer groups, give some of their properties.
Given a field $K$, to each alternating $n \times n$ matrix of linear forms in $K[y_1,\dots ,y_d]$ one can associate a group scheme $\mathrm{G}$ over $K$. In particular, when $K$ is the field of rationals and $F$ is the field of $p$ elements, the $F$-points $\mathrm{G}(F)$ of $\mathrm{G}$ form a group of order $p^{n+d}$ and so, as $p$ varies, one obtains an infinite family of $p$-groups from $\mathrm{G}$. In this talk, I will present joint work with Josh Maglione and Christopher Voll, as well as ongoing work with Eamonn O’Brien, on the geometric study of automorphisms and isomorphism types of groups associated to small values of the parameters $n$ and $d$. I will also explain the implications of our work in connection to claims made around Higman’s famous PORC conjecture.
A profinite group $G$ carries naturally the structure of a probability space, namely with respect to its normalised Haar measure. We study the probability $Q(G,k)$ that $k$ Haar-random elements generate an open subgroup in the profinite group $G$. In particular, in this talk I will introduce the probabilistic virtual rank $\mathrm{pvr}(G)$ of $G$; that is, the smallest $k$ such that $Q(G,k)=1$. We will discuss some key theorems and open problems about random generation in profinite groups, with a view toward finite direct products of hereditarily just infinite profinite groups. Classic examples of the latter type of groups are semisimple algebraic groups over non-archimedean local fields. This is joint work with Benjamin Klopsch and Davide Veronelli.
A $p$-adic analytic group is a topological group that is endowed with an analytic manifold structure over $\mathbb{Z}_p$, the ring of $p$-adic integers. This definition can be extended by considering the manifold structure over more general pro-$p$ domains, such as the power series rings $\mathbb{Z}_p[[t_1, \dots, t_m]]$ or $\mathbb{F}_p[[t_1, \dots, t_m]]$ (where $\mathbb{F}_p$ denotes the finite field of $p$ elements).
Lazard established already in the 1960s that compact $p$-adic analytic groups are linear, as they can be embedded as a closed subgroup within the group of invertible matrices over $\mathbb{Z}_p$. Nonetheless, the question of the linearity of analytic groups over more general domains remains unsolved.
In this talk, we shed some light to this question by proving that when the coefficient ring is of characteristic zero, every compact analytic group is linear. We will provide background on the problem and outline the strategy of our argument.
Joint with M. Casals-Ruiz.