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Colin D. Reid

Publications and source records attributed to Colin D. Reid.

At least 19 recordsLinked to original sources
arXiv2025

On commensurators of free groups and free pro-p groups

We study the commensurators of free groups and free pro-$p$ groups, as well as certain subgroups of these. We prove that the commensurator $Comm(F)$ of a non-abelian free group of finite rank $F$ is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of $Comm(F)$ and show that some groups in this family are simple. For a prime $p$, we also consider the p-commensurator $Comm_p(F)$, which is the commensurator of $F$ viewed as a group with pro-$p$ topology. By contrast with $Comm(F)$, we prove that $Comm_p(F)$ has a simple subgroup of index at most 2. Further, while the isomorphism class of $Comm(F)$ does not depend on the rank of $F$, we prove that the isomorphism class of $Comm_p(F)$ depends on the rank of $F$ and determine the exact dependency. If $\mathbf F$ is the pro-$p$ completion of $F$ (which is a free pro-$p$ group), $Comm(\mathbf F)$ is a totally disconnected locally compact (tdlc) group containing $\mathbf F$ as an open subgroup. We use $Comm_p(F)$ to construct an abstractly simple subgroup of $Comm(\mathbf F)$ containing $\mathbf F$ as well as a family of non-discrete tdlc groups which are compactly generated and simple.

math.GR↗
arXiv2025

Locally compact piecewise full groups of homeomorphisms

We study when a piecewise full group (a.k.a. topological full group) of homeomorphisms of the Cantor space $X$ can be given a non-discrete totally disconnected locally compact (t.d.l.c.) topology and give a criterion for the alternating full group (in the sense of Nekrashevych's group A(G) to be compactly generated. As a result, starting from qualitative criteria, we obtain a large class of t.d.l.c. groups such that the derived group is non-discrete, compactly generated, open and simple, putting previous constructions of Neretin, Roever and Lederle in a more systematic context. We also show some notable properties of Neretin's groups apply to this class in general. General consequences are derived for the theory of simple t.d.l.c. groups, prime among them the universal role that alternating full groups play in the class of simple t.d.l.c. groups that are non-discrete, compactly generated and locally decomposable. Some of the theory is developed in the setting of topological inverse monoids of partial homeomorphisms of $X$. In particular, we obtain a sufficient condition to extend the topology to a monoid equipped with all restrictions with respect to compact open subsets of $X$ and all joins of compatible pairs of elements. The compact generation criterion is also naturally expressed in this context.

math.GR↗
arXiv2024

Compressible subgroups and simplicity

In this article we give sufficient conditions for a group to have simple derived subgroup; the argument is based on generalising properties observed for extremely proximal micro-supported actions on the Cantor space, and generalises previous results of Matui, Le Boudec and others in this direction. We give a sufficient condition for a non-trivial normal subgroup (not assumed closed) of a locally compact group $G$ to be open, also based on the theory of micro-supported actions. This shows in particular that many of the class of robustly monolithic groups introduced by Caprace--Reid--Wesolek are simple-by-discrete.

math.GR↗
arXiv2024

The scale function for locally compact groups acting on non-positively curved spaces

Let $G$ be a totally disconnected, locally compact (t.d.l.c.) group. The scale $s_G(g)$ of $g \in G$ in the sense of Willis is given by the minimum value of the index $|gUg^{-1}:U \cap gUg^{-1}|$ as $U$ ranges over the compact open subgroups; the theory associated to the scale has been very successful in describing general dynamical features of automorphisms of t.d.l.c. groups. We focus on the case where $G$ acts properly and continuously by isometries on a geodesic space $X$, where $X$ is complete CAT(0) or proper and Gromov-hyperbolic, and $g \in G$ is hyperbolic. In this context, we find geometric descriptions of the parabolic and contraction groups, tidy subgroups, and structures in the $G$-action that encode the scale, including criteria for $g$ to have scale $1$.

math.GR↗
arXiv2024

Rigid stabilizers and local prosolubility for boundary-transitive actions on trees

Let $G$ be a group acting $2$-transitively on the boundary of a locally finite tree, and exclude the situation (which is a genuine exception) where $G$ has both $\mathrm{P}Γ\mathrm{L}_3(4)$ and $\mathrm{P}Γ\mathrm{L}_3(5)$ as local actions. We show that for each half-tree $T_a$, the local action of the rigid stabilizer of $T_a$ at the root of $T_a$ contains the soluble residual of the point stabilizer of the local action of $G$. In particular, $G$ is locally prosoluble if and only if its local actions have soluble point stabilizers; if $G$ is not locally prosoluble, then it has micro-supported action on the boundary. We also prove some strong restrictions on the local actions of end stabilizers in $G$. These results are partly inspired by Radu's classification of groups acting boundary-$2$-transitively on trees with local action containing the alternating group, and partly based on the author's recent classification of finite permutation groups that preserve an equivalence relation, act faithfully on blocks and act transitively on pairs of points from different blocks.

math.GR↗
arXiv2024

Multiple transitivity except for a system of imprimitivity

Let $Ω$ be a set equipped with an equivalence relation $\sim$; we refer to the equivalence classes as blocks of $Ω$. A permutation group $G \le \mathrm{Sym}(Ω)$ is $k$-by-block-transitive if $\sim$ is $G$-invariant, with at least $k$ blocks, and $G$ is transitive on the set of $k$-tuples of points such that no two entries lie in the same block. The action is block-faithful if the action on the set of blocks is faithful. In this article we classify the finite block-faithful $2$-by-block-transitive actions. We also show that for $k \ge 3$, there are no finite block-faithful $k$-by-block-transitive actions with nontrivial blocks.

math.GR↗
arXiv2023

An introduction to the local-to-global behaviour of groups acting on trees and the theory of local action diagrams

The primary tool for analysing groups acting on trees is Bass--Serre Theory. It is comprised of two parts: a decomposition result, in which an action is decomposed via a graph of groups, and a construction result, in which graphs of groups are used to build examples of groups acting on trees. The usefulness of the latter for constructing new examples of `large' (e.g. nondiscrete) groups acting on trees is severely limited. There is a pressing need for new examples of such groups as they play an important role in the theory of locally compact groups. An alternative `local-to-global' approach to the study of groups acting on trees has recently emerged, inspired by a paper of Marc Burger and Shahar Mozes, based on groups that are `universal' with respect to some specified `local' action. In recent work, the authors of this survey article have developed a general theory of universal groups of local actions, that behaves, in many respects, like Bass--Serre Theory. We call this the theory of local action diagrams. The theory is powerful enough to completely describe all closed groups of automorphisms of trees that enjoy Tits' Independence Property (P). This article is an introductory survey of the local-to-global behaviour of groups acting on trees and the theory of local action diagrams. The article contains many ideas for future research projects.

math.GR↗
arXiv2023

Growing trees from compact subgroups

We establish a new connection between local and large-scale structure in compactly generated totally disconnected locally compact (t.d.l.c.) groups $G$, finding a sufficient condition for $G$ to have more than one end in terms of its compact subgroups. The condition actually results in an action of a quotient group $G/N$ on a tree with faithful micro-supported action on the boundary, where $N$ is compact, and is closely related to the Boolean algebra formed by the centralisers of the subgroups of $G/N$ with open normaliser. As an application, we find a sufficient condition, given a one-ended t.d.l.c. group $G$, for all direct factors of open subgroups of $G$ to be trivial or open.

math.GR↗
arXiv2022

Locally normal subgroups and ends of locally compact Kac-Moody groups

A locally normal subgroup in a topological group is a subgroup whose normaliser is open. In this paper, we provide a detailed description of the large-scale structure of closed locally normal subgroups of complete Kac-Moody groups over finite fields. Combining that description with the main result from arXiv:2111.07066, we show that under mild assumptions, if the Kac-Moody group is one-ended (a property that is easily determined from the generalised Cartan matrix), then it is locally indecomposable, which means that no open subgroup decomposes as a nontrivial direct product.

math.GR↗
arXiv2021

Orientation of piecewise powers of a minimal homeomorphism

We show that given a compact minimal system $(X,g)$ and an element $h$ of the topological full group $τ[g]$ of $g$, then the infinite orbits of $h$ admit a locally constant orientation with respect to the orbits of $g$. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where $G$ is any virtually polycyclic subgroup of $τ[g]$. We also use the orientation of orbits to give a refinement of the index map and to describe the role in $τ[g]$ of the submonoid generated by the induced transformations of $g$. Finally, we consider the problem, given a homeomorphism $h$ of the Cantor space $X$, of determining whether or not there exists a minimal homeomorphism $g$ of $X$ such that $h \in τ[g]$.

math.DS↗
arXiv2021

A class of well-founded totally disconnected locally compact groups

Motivated by the problem of finding a "well-foundedness principle" for totally disconnected, locally compact (t.d.l.c.) groups, we introduce a class $\mathscr{E}^{\mathscr{S}}$ of t.d.l.c. groups, containing P. Wesolek's class $\mathscr{E}$ of (regionally) elementary groups but also including many groups in the class $\mathscr{S}$ of nondiscrete compactly generated topologically simple t.d.l.c. groups. The class $\mathscr{E}^{\mathscr{S}}$ carries a well-behaved rank function and is closed under taking directed unions, open subgroups, closed normal subgroups, extensions and quotients. The class $\mathscr{E}^{\mathscr{S}}$ also includes other well-studied families of t.d.l.c. groups that are not contained in $\mathscr{E}$, including all locally linear t.d.l.c. groups, all complete geometric Kac--Moody groups over finite fields, the Burger--Mozes groups $U(F)$ where $F$ is primitive, and $2^{\aleph_0}$ more examples of groups in $\mathscr{S}$ that arise as groups acting on trees with Tits' independence property (P). On the other hand, $\mathscr{E}^{\mathscr{S}}$ excludes the Burger--Mozes groups $U(F)$ where $F$ is nilpotent and does not act freely. By contrast, a larger class $\mathscr{E}^{[\mathrm{Sim}]}$ (with similar closure properties to $\mathscr{E}^{\mathscr{S}}$) is closed under forming actions on trees with property (P).

math.GR↗
arXiv2021

Totally disconnected locally compact groups with just infinite locally normal subgroups

We obtain a characterization of totally disconnected, locally compact groups $G$ with the following property: given a locally normal subgroup $K$ of $G$, then there is an open subgroup of $K$ that is a direct factor of an open subgroup of $G$. This property is motivated by J. Wilson's structure theory of just infinite groups, and indeed, when $G$ has trivial quasi-centre, the condition turns out to be equivalent to the condition that $G$ is locally isomorphic to a finite direct product of just infinite profinite groups. In the latter situation we obtain some global structural features of $G$, building on an earlier result of Barnea--Ershov--Weigel and also using tools developed by P.-E. Caprace, G. Willis and the author for studying local structure in totally disconnected locally compact groups.

math.GR↗
arXiv2021

Decomposition of locally compact coset spaces

In a previous article by the author and P. Wesolek, it was shown that a compactly generated locally compact group $G$ admits a finite normal series $(G_i)$ in which the factors are compact, discrete or irreducible in the sense that no closed normal subgroup of $G$ lies properly between $G_{i-1}$ and $G_{i}$. In the present article, we generalize this series to an analogous decomposition of the coset space $G/H$ with respect to closed subgroups, where $G$ is locally compact and $H$ is compactly generated. This time, the irreducible factors are coset spaces $G_{i}/G_{i-1}$ where $G_{i}$ is compactly generated and there is no closed subgroup properly between $G_{i-1}$ and $G_{i}$. Such irreducible coset spaces can be thought of as a generalization of primitive actions of compactly generated locally compact groups; we establish some basic properties and discuss some sources of examples.

math.GR↗
arXiv2021

Chief factors in Polish groups

In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier refinement theorem. We also prove a trichotomy for the structure of topologically characteristically simple Polish groups. The development of the theory of chief factors requires two independently interesting lines of study. First we consider injective, continuous homomorphisms with dense normal image. We show such maps admit a canonical factorization via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalizations of direct products and use these to isolate a notion of semisimplicity for Polish groups.

math.GR↗
arXiv2021

Discrete locally finite full groups of Cantor set homeomorphisms

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological full groups). We determine that any piecewise full group that is locally compact in the compact-open topology on the group of self-homeomorphisms of the Cantor set must be uniformly discrete, in a precise sense that we introduce here. Uniformly discrete groups of self-homeomorphisms of the Cantor set are in particular countable, locally finite, residually finite and discrete in the compact-open topology. The resulting piecewise full groups form a subclass of the ample groups introduced by Krieger. We determine the structure of these groups by means of their Bratteli diagrams and associated dimension ranges ($K_0$ groups). We show through an example that not all uniformly discrete piecewise full groups are subgroups of the ``obvious'' ones, namely, piecewise full groups of finite groups.

math.GR↗
arXiv2020

A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups

We classify the locally compact second-countable (l.c.s.c.) groups $A$ that are abelian and topologically characteristically simple. All such groups $A$ occur as the monolith of some soluble l.c.s.c. group $G$ of derived length at most $3$; with known exceptions (specifically, when $A$ is $\mathbb{Q}^n$ or its dual for some $n \in \mathbb{N}$), we can take $G$ to be compactly generated. This amounts to a classification of the possible isomorphism types of abelian chief factors of l.c.s.c. groups, which is of particular interest for the theory of compactly generated locally compact groups.

math.GR↗
arXiv2020

Groups acting on trees with Tits' independence property (P)

Local actions (actions of a vertex stabiliser on the neighbours of that vertex) have become an important approach to group actions on trees since J. Tits' introduction in 1970 of the independence property (P) and especially since a 2000 paper by M. Burger and Sh. Mozes. This `local-to-global' approach has been critical in the development of the theory of totally disconnected locally compact groups because it allows the construction of nondiscrete group actions on trees while keeping control over the action of a vertex stabiliser, in a way that is not practical under the classical Bass-Serre approach. The majority of constructions of nonlinear nondiscrete locally compact simple groups use (P) and its generalisations. In this article we give a full classification and description of all closed group actions on trees with Tits' independence property (P) using a new coherent theory for local actions that applies to all actions on trees. This theory is a `local action' complement to classical Bass-Serre theory. On the one hand, our theory gives a decomposition of a group acting on a tree into a `local action diagram' (a decorated graph that encodes all `local' information), and on the other hand a construction of a group acting on a tree from a given local action diagram. One can read directly from the local action diagram whether the resulting group has certain properties, like geometric density, compact generation and simplicity.

math.GR↗