In this talk I will present some new developments in subgroup growth. In particular, I will present an (almost) complete solution to a long-standing open question of Lubtozky and Segal about the possible types of subgroup growth of pro-$p$ groups. The solution requires finding the subgroup growth of the Grigorchuk group and the Gupta-Sidki groups and the use of a new notion, namely, orbit growth (joint with Jan-Christoph Schlage-Puchta).
I will also present joint work with Benjamin Klopsch and Jan-Christoph Schlage-Puchta where we showed that the subgroup growth of the Nottingham group (for large enough $p$) is at most $n^{1/8\log n+o(\log n)}$. Thus, answering a long-standing open question of Mann about the minimal subgroup growth of non-$p$-adic-analytic pro-$p$ groups.
In addition, I will present joint results with Jan-Christoph Schlage-Puchta about normal subgroup growth and characteristic subgroup growth.
The talk will be self-contained and I will try to pose as many challenging problems as possible.
In this talk I will present some results on (partial) equality and word problems for finitely generated groups. In particular, I will focus on (un)solvability of generic versions of these two problems with respect to classical and Banach densities. This talk is based on joint work with Matteo Cavaleri.
Every finitely generated pro-$p$ group $G$ comes equipped with a metric space structure induced by a previously chosen filtration series. In this talk, we confine ourselves to the Zassenhaus series, the Frattini series and the iterated p-power series.
Given such a metric, the "size" of subgroups of $G$ can be studied via their Hausdorff dimensions, a notion introduced in fractal geometry. The study of Hausdorff dimension in the context of pro-$p$ (and more generally, profinite) groups was initiated more than twenty years ago by Barnea and Shalev. In that paper, the authors asked whether $p$-adic analytic pro-$p$ groups can be distinguished or not via their Hausdorff spectra within the category of finitely generated pro-$p$ groups.
After introducing the aforementioned notions and results, we will talk about the joint work with Alejandra Garrido and Benjamin Klopsch concerning the Hausdorff spectra of pro-$p$ groups of positive rank gradient, a project initiated to answer a question by Barnea and Shalev.
It is well known that every finite simple group can be generated by just two elements. Moreover, two arbitrary elements are very likely to generate the entire group. For example, Guralnick and Kantor proved that every nontrivial element of a finite simple group belongs to a generating pair. Groups with the latter property are said to be 3/2-generated. In this talk, I will discuss recent results on two invariants, the spread and the uniform domination number, which generalise 3/2-generation. In this work, probabilistic techniques play a key role and we find a connection with bases of permutation groups. The talk features joint work with Tim Burness.
A recent article by Calegari, Garoufalidis and Zagier gives an elegant explicit formula for a Chern class in K-theory in terms of the `cyclic quantum dilogarithm'. They use the formula to prove one direction of Nahm's conjecture on modularity of certain hypergeometric series. I will discuss some Galois-theoretic problems arising from their construction and show how the attempt to resolve them leads to intriguing identities – some proved, some conjectural – involving cyclic quantum dilogarithms. (The questions and results I will discuss in the talk will involve only basic Galois theory. K-theory will play no explicit role.)
A transitive permutation group $G$ on a set $X$ is a Jordan permutation group if $X$ has a proper subset $Y$ with $\lvert Y\rvert > 1$ such that the pointwise stabiliser in $G$ of the complement of $Y$ is transitive on $Y$ (and we also assume that if this complement has finite size $k$ then $G$ is not $(k+1)$-transitive). Finite primitive Jordan groups were classified in the 1980s as a consequence of the classification of finite $2$-transitive groups. There is a rich supply of infinite primitive Jordan groups, explored by Adeleke and Neumann in the 1990s. Classification results (in a loose sense) were obtained in the 1990s by Adeleke and Neumann (under the assumption that the action on some Jordan set is primitive) and in general by Adeleke and Macpherson. In this talk I will describe the classification and some consequences (eg proofs that certain permutation groups are `maximal-closed' in the full symmetric group on a countable set). I will also describe the recent PhD thesis of Asma Almazaydeh, who uses model-theoretic Fraisse amalgamation to construct a mysterious treelike structure whose automorphism group is a Jordan group.
A topology on a group $G$ is called a group topology if the functions $G\times G \to G$ defined by $(x,y) \mapsto xy$ and $G \to G$ defined by $x \mapsto x^{-1}$ are continuous. For example, if $\mathbb N$ is discrete and $\mathbb{N}^{\mathbb N}$ has the product topology, then the subspace topology on the symmetric group $\operatorname{Sym}(\mathbb{N})$ is a group topology on $\operatorname{Sym}(\mathbb{N})$; called the topology of pointwise convergence. A topology is Polish if it is completely metrizable and separable. Gaughan proved that any Hausdorff group topology on $\operatorname{Sym}(\mathbb{N})$ contains the topology of pointwise convergence, and consequently the pointwise topology is the unique Polish group topology on $\operatorname{Sym}(\mathbb{N})$. In other words, in some sense, the algebraic properties of $\operatorname{Sym}(\mathbb{N})$ can be reconstructed from the topology, and vice versa. The uniqueness and non-uniqueness of Polish group topologies has been the subject of intensive research for many years.
Motivated by the corresponding results for groups, in this talk I will discuss recent work with several co-authors where we consider analogous problems in the context of semigroups.
A group $G$ has restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. We study profinite groups with restricted centralizers of word-values. We show that if $w$ is a multilinear commutator word and $G$ a profinite group with restricted centralizers of $w$-values, then the verbal subgroup $w(G)$ is abelian-by-finite. We also discuss finite groups $G$ in which $w(G)$ is non-trivial and all centralizers of non-trivial $w$-values have order at most $m$, where $w$ is some fixed word. When $w$ is a multilinear commutator word or some word of Engel type, we obtain that the order of $G$ is bounded in terms of $w$ and $m$ only.
The normal subgroup growth of a group captures the asymptotic behaviour of the lattice of its normal subgroups of finite index. The group's normal zeta function encodes the distribution of these subgroups quantiatively.
Finitely generated nilpotent groups are arithmetic subgroups of unipotent algebraic groups. It therefore comes as no surprise that questions about their subgroup structure are best thought about, and answered, in number-theoretic and, occasionally, algebro-geometric terms.
I will explain a notion of uniformity for normal zeta functions, specifically for free nilpotent groups, and present a recent uniformity result for free nilpotent groups of class 2, answering a conjecture of Grunewald, Segal, and Smith in this case.
My talk is based on joint work with Angela Carnevale and Michael Schein.