Search arXivSearch

arXiv · 2307.11564

Infinite Permutation Groups

Abstract

This is a transcript of a lecture course on Infinite Permutation Groups given by Peter M. Neumann (1940-2020) in Oxford during the academic year 1988-1989. The field of Infinite Permutation Groups only emerged as an independent field of study in the 1980's. Most of the results described in these notes were at the time of the lectures brand new and had either just recently appeared in print or had not appeared formally. A large part of the results described is either due to Peter himself or heavily influenced by him. These notes offer Peter's personal take on a field that he was instrumental in creating and in many cases ideas and questions that can not be found in the published literature.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Peter M. Neumann. 2023-07-24. Infinite Permutation Groups. https://arxiv.org/abs/2307.11564

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Word Measures on Wreath Products II

Every word $w$ in $F_r$, the free group of rank $r$, induces a probability measure (the $w$-measure) on every finite group $G$, by substitution of random $G$-elements in the letters. This measure is determined by its Fourier coefficients: the $w$-expectations $E_w[χ]$ of the irreducible characters of $G$. For every finite group $G$, every stable character $χ$ of $G\wr S_n$ (trace of a finitely generated $FI_G$-module), and every word $w\in F_r$, we approximate $E_w[χ]$ up to an error term of $O(n^{-π(w)})$, where $π(w)$ is the primitivity rank of $w$. This generalizes previous works by Puder, Hanany, Magee and the author. As an application we show that random Schreier graphs of representation-stable actions of $G\wr S_n$ are close-to-optimal expanders. The paper reveals a surprising relation between stable representation theory of wreath products and not-necessarily connected Stallings core graphs.

math.GR

Robust quasi-isometric embeddings inapproximable by Anosov representations

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For all but finitely many $m\in \mathbb{N}$, we exhibit the first examples of non-locally rigid, Zariski dense, robust quasi-isometric embeddings of hyperbolic groups in $\mathsf{SL}_m(\mathbb{K})$ which are not limits of Anosov representations. As a consequence, we show that higher rank analogues of Sullivan's structural stabilty theorem and of the density theorem for Kleinian groups fail for Anosov representations in $\mathsf{SL}_m(\mathbb{C}), m\geq 30$.

math.GR

$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products

We provide a necessary and sufficient condition for the restricted wreath product $A\wr B$ to be $\mathcal{C}$-hereditarily conjugacy separable where $\mathcal{C}$ is an extension-closed pseudovariety of finite groups. Moreover, we prove that the Grigorchuk group is 2-hereditarily conjugacy separable. As an application, we demonstrate that the lamplighter groups and $\mathbb{Z} \wr \mathbb{Z}$ are hereditarily conjugacy separable (but not $p$-conjugacy separable for any prime $p$). This provides infinitely many new examples of solvable, non-polycyclic hereditarily conjugacy separable groups. Furthermore, we study wreath products of cyclic subgroup separable groups and the derived length of iterated wreath products of solvable groups with an abelian base group and, as an application, we give an explicit construction of non-polycyclic hereditarily conjugacy separable groups of arbitrary derived length as iterated wreath products of abelian groups.

math.GR