arXiv subjects
Publications and source records attributed to Colin D. Reid.
Groups of almost upper triangular infinite matrices with entries indexed by integers are studied. It is shown that, when the matrices are over a finite field, these groups admit a nondiscrete totally disconnected, locally compact group topology and are topologically simple.
The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the residual closure of $H$ in $G$ is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.
We obtain a sufficient condition, given a totally disconnected, locally compact group $G$ with a topologically simple monolith $S$, to ensure that $S$ is open in $G$ and abstractly simple.
The class, denoted by $\mathscr{S}$, of totally disconnected locally compact groups which are non-discrete, compactly generated, and topologically simple contains many compelling examples. In recent years, a general theory for these groups, which studies the interaction between the compact open subgroups and the global structure, has emerged. In this article, we study the non-discrete totally disconnected locally compact groups $H$ that admit a continuous embedding with dense image into some $G\in \mathscr{S}$; that is, we consider the dense locally compact subgroups of groups $G\in \mathscr{S}$. We identify a class $\mathscr{R}$ of almost simple groups which properly contains $\mathscr{S}$ and is moreover stable under passing to a non-discrete dense locally compact subgroup. We show that $\mathscr{R}$ enjoys many of the same properties previously obtained for $\mathscr{S}$ and establish various original results for $\mathscr{R}$ that are also new for the subclass $\mathscr{S}$, notably concerning the structure of the local Sylow subgroups and the full automorphism group.
Let $X$ be a locally compact zero-dimensional space, let $S$ be an equicontinuous set of homeomorphisms such that $1 \in S = S^{-1}$, and suppose that $\overline{Gx}$ is compact for each $x \in X$, where $G = \langle S \rangle$. We show in this setting that a number of conditions are equivalent: (a) $G$ acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every compact open subset $U$ of $X$, there is $F \subseteq G$ finite such that $\bigcap_{g \in F}g(U)$ is $G$-invariant. All of these are equivalent to a notion of recurrence, which is a variation on a concept of Auslander-Glasner-Weiss. It follows in particular that the action is distal if and only if it is equicontinuous.
Let $ϕ: G \rightarrow H$ be a group homomorphism such that $H$ is a totally disconnected locally compact (t.d.l.c.) group and the image of $ϕ$ is dense. We show that all such homomorphisms arise as completions of $G$ with respect to uniformities of a particular kind. Moreover, $H$ is determined up to a compact normal subgroup by the pair $(G,ϕ^{-1}(L))$, where $L$ is a compact open subgroup of $H$. These results generalize the well-known properties of profinite completions to the locally compact setting.
Let $G$ be a totally disconnected, locally compact group and let $H$ be an equicontinuously (for example, compactly) generated group of automorphisms of $G$. We show that every distal action of $H$ on a coset space of $G$ is a SIN action, with the small invariant neighbourhoods arising from open $H$-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups. For example, it follows that for every compactly generated subgroup $K$ of $G$, there is a compactly generated open subgroup $E$ of $G$ such that $K \le E$ and such that every open subgroup of $G$ containing a finite index subgroup of $K$ contains a finite index subgroup of $E$. We also show that for a large class of closed subgroups $L$ of $G$ (including for instance all closed subgroups $L$ such that $L$ is an intersection of subnormal subgroups of open subgroups), every compactly generated open subgroup of $L$ can be realized as $L \cap O$ for an open subgroup of $G$.
We first obtain finiteness properties for the collection of closed normal subgroups of a compactly generated locally compact group. Via these properties, every compactly generated locally compact group admits an essentially chief series - i.e. a finite normal series in which each factor is compact, discrete, or a topological chief factor. Additionally, a Jordan-Hölder theorem holds for the `large' factors in an essentially chief series.
We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are 'universal' in that all (hereditarily) just infinite profinite groups arise as limits of the specified form. This is a corrected and revised version of the article: 'Inverse system characterizations of the (hereditarily) just infinite property in profinite groups', Bull. LMS vol 44, 3 (2012) 413--425.
We use the structure lattice, introduced in Part I, to undertake a systematic study of the class $\mathscr S$ consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are non-discrete. Given $G \in \mathscr S$, we show that compact open subgroups of $G$ involve finitely many isomorphism types of composition factors, and do not have any soluble normal subgroup other than the trivial one. By results of Part I, this implies that the centraliser lattice and local decomposition lattice of $G$ are Boolean algebras. We show that the $G$-action on the Stone space of those Boolean algebras is minimal, strongly proximal, and micro-supported. Building upon those results, we obtain partial answers to the following key problems: Are all groups in $\mathscr S$ abstractly simple? Can a group in $\mathscr S$ be amenable? Can a group in $\mathscr S$ be such that the contraction groups of all of its elements are trivial?
In 'The essentially chief series of a compactly generated locally compact group', an analogue of chief series for finite groups is discovered for compactly generated locally compact groups. In the present article, we show that chief factors necessarily exist in all locally compact groups with sufficiently rich topological structure. We also show that chief factors have one of seven types, and for all but one of these types, there is a decomposition into discrete groups, compact groups, and topologically simple groups. Our results for chief factors require exploring the theory developed in 'Chief factors in Polish groups' in the setting of locally compact groups. In this context, we obtain tighter restrictions on the factorization of normal compressions and the structure of quasi-products. Consequently, both (non-)amenability and elementary decomposition rank are preserved by normal compressions.
Let G be a totally disconnected, locally compact group. A closed subgroup of G is locally normal if its normaliser is open in G. We begin an investigation of the structure of the family of closed locally normal subgroups of G. Modulo commensurability, this family forms a modular lattice, called the structure lattice of G. We show that G admits a canonical maximal quotient H for which the quasi-centre and the abelian locally normal subgroups are trivial. In this situation the structure lattice of H has a canonical subset called the centraliser lattice, forming a Boolean algebra whose elements correspond to centralisers of locally normal subgroups. If H is second-countable and acts faithfully on its centraliser lattice, we show that the topology of H is determined by its algebraic structure (and thus invariant by every abstract group automomorphism), and also that the action on the Stone space of the centraliser lattice is universal for a class of actions on profinite spaces. Most of the material is developed in the more general framework of Hecke pairs.
Let $G$ be a totally disconnected, locally compact group and let $H$ be a virtually flat (for example, polycyclic) group of automorphisms of $G$. We study the structure of, and relationships between, various subgroups of $G$ defined by the dynamics of $H$. In particular, we consider the following four subgroups: the intersection of all tidy subgroups for $H$ on $G$ (in the case that $H$ is flat); the intersection of all $H$-invariant open subgroups of $G$; the smallest closed $H$-invariant subgroup $D$ such that no $H$-orbit on $G/D$ accumulates at the trivial coset; and the group generated by the closures of contraction groups of elements of $H$ on $G$.
Let $G$ be a group and let $K$ be a commensurated subgroup of $G$. Then there is a totally disconnected, locally compact (t.d.l.c.) group $\hat{G}_K$ that contains the profinite completion of $K$ as an open compact subgroup and also contains $G$ (modulo the finite residual of $K$) as a dense subgroup. Moreover, given an arbitrary group $G$, then every t.d.l.c. group containing an image of $G$ as a dense subgroup can be realised as a quotient of $\hat{G}_K$ for some commensurated subgroup $K$.
The Tits core G^+ of a totally disconnected locally compact group G is defined as the abstract subgroup generated by the closures of the contraction groups of all its elements. We show that a dense subgroup is normalised by the Tits core if and only if it contains it. It follows that every dense subnormal subgroup contains the Tits core. In particular, if G is topologically simple, then the Tits core is abstractly simple, and if G^+ is non-trivial then it is the unique minimal dense normal subgroup. The proofs are based on the fact, of independent interest, that the map which associates to an element the closure of its contraction group is continuous.
We announce various results concerning the structure of compactly generated simple locally compact groups. We introduce a local invariant, called the structure lattice, which consists of commensurability classes of compact subgroups with open normaliser, and show that its properties reflect the global structure of the ambient group.
Let $S$ be a finitely generated pro-$p$ group. Let $\Emb(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects non-trivially with every non-trivial normal subgroup of $G$. In this paper, we investigate the question of whether or not $\Emb(S)$ has finitely many isomorphism classes. For instance, we give an example where $\Emb(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $\Emb(S)$ contains only finitely many isomorphism classes in the case that $S$ is just infinite.
We obtain some general restrictions on the continuous endomorphisms of a profinite group G under the assumption that G has only finitely many open subgroups of each index (an assumption which automatically holds, for instance, if G is finitely generated). In particular, given such a group G and a continuous endomorphism phi we obtain a semidirect decomposition of G into a 'contracting' normal subgroup and a complement on which phi induces an automorphism; both the normal subgroup and the complement are closed. If G is isomorphic to a proper open subgroup of itself, we show that G has an infinite abelian normal pro-p subgroup.