A 1970 article of J. Tits concerning groups acting on trees introduced an independence property $(\mathrm{P})$ as a condition to produce the first examples of nonlinear nondiscrete locally compact simple groups, answering a question of J. P. Serre. This property has become very important in the recent development of the theory of totally disconnected, locally compact (t.d.l.c.) groups, with the majority of new constructions of compactly generated simple t.d.l.c. groups using $(\mathrm{P})$ or related ideas. In this paper we aim to advance the local-to-global theory of groups acting on trees by developing a `local action' complement to classical Bass--Serre theory. We describe, for a closed group $G$ of automorphisms of a (not necessarily locally finite) tree $T$ something called a local action diagram: a graph decorated with the local actions of $G$. A local action diagram plays a role in our theory that is analogous to a graph of groups in Bass--Serre theory. In place of the universal cover of a graph of groups, we define the universal group of a local action diagram. In this context, the groups $\mathbf{U}(F)$ and $\mathbf{U}(F_1, F_2)$ play analogous roles to the HNN extension and amalgamated free product respectively in Bass--Serre theory. We then show how to determine whether the universal group has certain properties, such as geometric density, compact generation and simplicity, directly from the local action diagram. Our theory allows us to completely describe all closed groups of automorphisms of trees with Tits' independence property $(\mathrm{P})$: they are precisely the universal groups of local action diagrams.

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 additional 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.

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.

Let $c$ and $c'$ be edge or vertex colourings of a graph $G$. We say that $c'$ is less symmetric than $c$ if the stabiliser (in $\operatorname{Aut} G$) of $c'$ is contained in the stabiliser of $c$. We show that if $G$ is not a bicentred tree, then for every vertex colouring of $G$ there is a less symmetric edge colouring with the same number of colours. On the other hand, if $T$ is a tree, then for every edge colouring there is a less symmetric vertex colouring with the same number of edges. Our results can be used to characterise those graphs whose distinguishing index is larger than their distinguishing number.

Let $T$ be a tree (not necessarily locally finite). We give a classification up to conjugacy of the closed subgroups $G$ of $\mathrm{Aut}(T)$ that have Tits' independence property (P) in terms of data called a local action diagram, which is the quotient graph $G \backslash T$ decorated with the `local actions' of $G$. We then show how to determine whether the group has certain properties, such as geometric density, compact generation and simplicity, directly from the local action diagram.

If a graph $G$ has distinguishing number 2, then there exists a partition of its vertex set into two parts, such that no nontrivial automorphism of $G$ fixes setwise the two parts. Such a partition is called a 2-distinguishing coloring of $G$, and the parts are called its color classes. If $G$ admits such a coloring, it is often possible to find another in which one of the color classes is sparse in a certain sense. In this case we say that $G$ has 2-distinguishing density zero. An extreme example of this would be an infinite graph admitting a 2-distinguishing coloring in which one of the color classes is finite. If a graph $G$ contains a vertex $v$ such that, for all $n \in \mathbb N$, any two distinct vertices equidistant from $v$ have nonequal $n$-spheres, then we say that $G$ satisfies the Distinct Spheres Condition. In this paper we prove a general result: any countable connected graph that satisfies the Distinct Spheres Condition is 2-distinguishable with density zero. We present two proofs of this, one that uses a deterministic coloring, and another (that applies only to locally finite graphs) using a random coloring. From this result, we deduce that several important families of countably infinite and connected graphs are 2-distinguishable with density zero, including those that are locally finite and primitive. Furthermore, we prove that any connected graph with infinite motion and subquadratic growth is 2-distinguishable with density zero.

This paper is about the structure of infinite primitive permutation groups and totally disconnected locally compact groups ("tdlc groups'"). The permutation groups we investigate are subdegree-finite (i.e. all orbits of point stabilisers are finite). Automorphism groups of connected, locally finite graphs are examples of subdegree-finite permutation groups. The tdlc groups we investigate all have a maximal subgroup that is compact and open. Tdlc groups with few open subgroups (recently studied by Pierre-Emmanuel Caprace and Timoth\'ee Marquis) are examples of such groups. We prove a classification result, and use it to show that every closed, subdegree-finite primitive group is a primitive subgroup of a product H Wr F_1 [X] F_2 Wr ... [X] F_{m-1} Wr S_m, where H is a closed, subdegree-finite and primitive group that is either finite or one-ended and almost topologically simple. The groups F_i are transitive and finite and m is finite. The product [X] here denotes the recently discovered box product, and all wreath products here act via their product action. We apply this permutational result to tdlc groups. If G is a tdlc group then it contains a compact open subgroup V. The permutation group induced by the action of G on the coset space G/V is called the Schlichting completion of the pair (G,V), and is denoted G//V. Knowledge of this action of G on G/V underpins many recent influential results in tdlc theory. We show that if V is a maximal subgroup of G, and G is non-compact, then G//V is subject to a topological interpretation of our result for primitive groups. We use this to show that in this case, if G//V is nondiscrete, preserves no nontrivial homogenous cartesian decomposition on G/V, and does not split nontrivially over a compact open subgroup, then the monolith of G//V is a nondiscrete, one-ended, topologically simple, compactly generated tdlc group.

If G is a group acting on a set Ω and α, β ∈ Ω, the digraph whose vertex set is Ω and whose arc set is the orbit (α, β) is called an orbital digraph of G. Each orbit of the stabiliser Gα acting on Ω is called a suborbit of G. A digraph is locally finite if each vertex is adjacent to at most finitely many other vertices. A locally finite digraph Γ has more than one end if there exists a finite set of vertices X such that the induced digraph Γ\X contains at least two infinite connected components; if there exists such a set containing precisely one element, then Γ has connectivity one. In this paper we show that if G is a primitive permutation group whose suborbits are all finite, possessing an orbital digraph with more than one end, then G has a primitive connectivity-one orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterised in a previous paper by the author. Date: October 21, 2018. 1991 Mathematics Subject Classification. 20 B 15; 05 C 25. Address: 59 Netherwood Road, London W14 0BP. Formally, Mathematical Institute, University of Oxford, Oxford. Email: simon.smith@chch.oxon.org.

This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph G with a subdegree-finite, infinite automorphism group whose cardinality is strictly less than continuum, has a finite set F of vertices that is setwise stabilized only by the identity automorphism. A bound on the size of such sets, which are called distinguishing, is also provided. To put this theorem of Halin and its generalization into perspective, we also discuss several related non-elementary, independent results and their methods of proof.

We introduce a new product for permutation groups. It takes as input two permutation groups, M and N, and produces an infinite group M [X] N which carries many of the permutational properties of M. Under mild conditions on M and N the group M [X] N is simple. As a permutational product, its most significant property is the following: M [X] N is primitive if and only if M is primitive but not regular, and N is transitive. Despite this remarkable similarity with the wreath product in product action, M [X] N and M Wr N are thoroughly dissimilar. The product provides a general way to build exotic examples of non-discrete, simple, totally disconnected, locally compact, compactly generated topological groups from discrete groups. We use this to solve a well-known open problem from topological group theory, by obtaining the first construction of uncountably many pairwise non-isomorphic simple topological groups that are totally disconnected, locally compact, compactly generated and non-discrete. The groups we construct all contain the same compact open subgroup. To build the product, we describe a group U(M,N) that acts on an edge-transitive biregular tree T. This group has a natural universal property and is analogous to the iconic universal group construction of M. Burger and S. Mozes for locally finite regular trees.

This article began as a study of the structure of infinite permutation groups G in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-N, the minimal condition on normal subgroups. The groups G are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup M which is a divisible abelian p-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a p-adic vector space associated with M. This leads to our second variation, which is a study of the finite linear groups that can arise.

The most common method of assessing the quality of medical education is through a selection of qualitative assessments, usually as part of a programme evaluation. Common qualitative assessments include measurements of students' and teachers' participation, outcome measures such as assessment results, and qualitative assessments such as interviews and questionnaires of students and teachers. Programme evaluation can therefore be a process that is both laborious and subject to accusations of a lack of objectivity. As a result, the development of a quantitative tool that could be used alongside a programme evaluation may be both useful and desirable. A pragmatic scoring system, utilizing routinely collected quantitative data, termed as the Quality Assessment Tool, was developed during the 2013 academic year within the setting of an Emergency Medicine training programme in the UK. This tool was tested against the standard assessment currently used for this programme to establish whether the quantitative tool correlated with the programme evaluation. Second, the individual items within the tool were investigated to identify any correlations with the current assessment of quality established by the programme evaluation. The Quality Assessment Tool appears to be correlated to the quality of training delivered at individual training sites in a single specialty. It certainly identifies those centres delivering the highest quality of training and also identifies those centres whose training is consistently of a lower standard. The assessment tool is less accurate at ranking those training centres whose training is merely 'satisfactory'; whether this is a result of the imprecision of the tool itself or a reflection of the subjective nature of the current assessment (i.e. whether the current evaluation system lacks validity) cannot be stated. In summary, it appears to be possible to use a single quantitative tool to reliably, and with validity, measure the quality of training delivered at a postgraduate medical training centre. Although it is not envisaged that this would, or should, replace ongoing quality assurance systems such as programme evaluations, a validated tool can be derived for a given setting to usefully inform and augment current quality management systems in postgraduate medical education.

We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to all primitive permutation groups with finite point stabilizers.

A group A acting faithfully on a set X is 2-distinguishable if there is a 2-coloring of X that is not preserved by any nonidentity element of A, equivalently, if there is a proper subset of X with trivial setwise stabilizer. The motion of an element a in A is the number of points of X that are moved by a, and the motion of the group A is the minimal motion of its nonidentity elements. For finite A, the Motion Lemma says that if the motion of A is large enough (specifically at least 2 log_2 |A|), then the action is 2-distinguishable. For many situations where X has a combinatorial or algebraic structure, the Motion Lemma implies the action of Aut(X) on X is 2-distinguishable in all but finitely many instances. We prove an infinitary version of the Motion Lemma for countably infinite permutation groups, which states that infinite motion is large enough to guarantee 2-distinguishability. From this we deduce a number of results, including the fact that every locally finite, connected graph whose automorphism group is countably infinite is 2-distinguishable. One cannot extend the Motion Lemma to uncountable permutation groups, but nonetheless we prove that 2-distinguishable permutation groups with infinite motion are dense in the class of groups with infinite motion. We conjecture an extension of the Motion Lemma which we expect holds for a restricted class of uncountable permutation groups, and we conclude with a list of open questions. The consequences of our results are drawn for orbit equivalence of infinite permutation groups.

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

The {\em distinguishing number} of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The {\em distinguishing number} of a graph is the distinguishing number of its full automorphism group acting on its vertex set. A connected graph $\Gamma$ is said to have {\em connectivity 1} if there exists a vertex $\alpha \in V\Gamma$ such that $\Gamma \setminus \{\alpha\}$ is not connected. For $\alpha \in V$, an orbit of the point stabilizer $G_\alpha$ is called a {\em suborbit} of $G$. We prove that every connected primitive graph with infinite diameter and countably many vertices has distinguishing number 2. Consequently, any infinite, connected, primitive, locally finite graph is 2-distinguishable; so, too, is any infinite primitive group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number 2. All of our results follow directly from a versatile lemma which we call The Distinct Spheres Lemma.

If G is a group of permutations of a set Omega , then the suborbits of G are the orbits of point-stabilisers G_\alpha acting on Omega. The cardinalities of these suborbits are the subdegrees of G. Every infinite primitive permutation group G with finite subdegrees acts faithfully as a group of automorphisms of a locally-finite connected vertex-primitive directed graph Gamma with vertex set Omega, and there is consequently a natural action of G on the ends of Gamma. We show that if G is closed in the permutation topology of pointwise convergence, then the structure of G is determined by the length of any orbit of G acting on the ends of Gamma. Examining the ends of a Cayley graph of a finitely generated group to determine the structure of the group is often fruitful. B. Kron and R. G. Moller have recently generalised the Cayley graph to what they call a rough Cayley graph, and they call the ends of this graph the rough ends of the group. It transpires that the ends of Gamma are the rough ends of G, and so our result is equivalent to saying that the structure of a closed primitive group G whose subdegrees are all finite is determined by the length of any orbit of G on its rough ends.

If G is a group of permutations of a set Ω and α∈Ω, then the α-suborbits of G are the orbits of the stabilizer G_α on Ω. The cardinality of an α-suborbit is called a subdegree of G. If the only G-invariant equivalence classes on Ω are the trivial and universal relations, then G is said to be a primitive group of permutations of Ω. In this paper we determine the structure of all primitive permutation groups whose subdegrees are bounded above by a finite cardinal number.

A group G of permutations of a set Ω is primitive if it acts transitively on Ω, and the only G-invariant equivalence relations on Ω are the trivial and universal relations. A digraph Γ is primitive if its automorphism group acts primitively on its vertex set, and is infinite if its vertex set is infinite. It has connectivity one if it is connected and there exists a vertex α of Γ, such that the induced digraph Γ∖{α} is not connected. If Γ has connectivity one, a lobe of Γ is a connected subgraph that is maximal subject to the condition that it does not have connectivity one. Primitive graphs (and thus digraphs) with connectivity one are necessarily infinite. The primitive graphs with connectivity one have been fully classified by Jung and Watkins: the lobes of such graphs are primitive, pairwise-isomorphic and have at least three vertices. When one considers the general case of a primitive digraph with connectivity one, however, this result no longer holds. In this paper we investigate the structure of these digraphs, and obtain a complete characterisation.