Simon M. Smith

A product for permutation groups and topological groups

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.

Infinite primitive directed graphs

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.

Orbital digraphs of infinite primitive permutation groups

Abstract 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) G is called an orbital digraph of G. Each orbit of the stabilizer G α 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 connectivityone orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterized in a previous paper by the author.

Subdegree growth rates of infinite primitive permutation groups

 If G is a group acting on a set Ω, and α , β ∈ Ω, the directed graph whose vertex set is Ω and whose edge set is the orbit ( α , β ) G is called an orbital graph of G . These graphs have many uses in permutation group theory. A graph Γ is said to be primitive if its automorphism group acts primitively on its vertex set, and is said to have connectivity one if there is a vertex α such that the graph Γ\{ α } is not connected. A half-line in Γ is a one-way infinite path in Γ. The ends of a locally finite graph Γ are equivalence classes on the set of half-lines: two half-lines lie in the same end if there exist infinitely many disjoint paths between them. A complete characterisation of the primitive undirected graphs with connectivity one is already known. We give a complete characterisation in the directed case. This enables us to show that if G is a primitive permutation group with a locally finite orbital graph with more than one end, then G has a connectivity-one orbital graph Γ, and that this graph is essentially unique. Through the application of this result we are able to determine both the structure of G , and its action on the end space of Γ. If α ∈ Ω, the orbits of the stabiliser G α are called the α-suborbits of G . The size of an α -suborbit is called a subdegree . If all subdegrees of an infinite primitive group G are finite, Adeleke and Neumann claim one may enumerate them in a non-decreasing sequence ( m r ). They conjecture that the growth of the sequence ( m r ) is extremal when G acts distance transitively on a locally finite graph; that is, for all natural numbers m the stabiliser in G of any vertex α permutes the vertices lying at distance m from α transitively. They also conjecture that for any primitive group G possessing a finite self-paired suborbit of size m there might exist a number c which perhaps depends upon G , perhaps only on m , such that m r ≤ c ( m -2) r -1 . We show their questions are poorly posed, as there exist primitive groups possessing at least two distinct subdegrees, each occurring infinitely often. The subdegrees of such groups cannot be enumerated as claimed. We give a revised definition of subdegree enumeration and growth, and show that under these new definitions their conjecture is true for groups exhibiting exponential subdegree growth above a prescribed bound.

Some infinite permutation groups and related finite linear groups

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 non-trivial 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.

On a theorem of Halin

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

Infinite motion and 2-distinguishability of graphs and groups

\aleph _0 \le |{\text {Aut}}(G)| < 2^{\aleph _0}

Bounding the distinguishing number of infinite graphs

ℵ0≤|Aut(G)|<2ℵ0 and subdegree-finite automorphism group, 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.