Subdegree growth rates of infinite primitive permutation groups

A transitive group $G$ of permutations of a set $\Omega$ is primitive if the only $G$-invariant equivalence relations on $\Omega$ are the trivial and universal relations. If $\alpha \in \Omega$, then the orbits of the stabiliser $G_\alpha$ on $\Omega$ are called the $\alpha$-suborbits of $G$; when $G$ acts transitively the cardinalities of these $\alpha$-suborbits are the subdegrees of $G$. If $G$ acts primitively on an infinite set $\Omega$, and all the suborbits of $G$ are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of $G$ as a non-decreasing sequence $1 = m_0 \leq m_1 \leq ...$, the subdegree growth rates of infinite primitive groups that act distance-transitively on locally finite distance-transitive graphs are extremal, and conjecture there might exist a number $c$ which perhaps depends upon $G$, perhaps only on $m$, such that $m_r \leq c(m-2)^{r-1}$. In this paper it is shown that such an enumeration is not desirable, as there exist infinite primitive permutation groups possessing no infinite subdegree, in which two distinct subdegrees are each equal to the cardinality of infinitely many suborbits. The examples used to show this provide several novel methods for constructing infinite primitive graphs. A revised enumeration method is then proposed, and it is shown that, under this, Adeleke and Neumann's question may be answered, at least for groups exhibiting suitable rates of growth.Comment: 41 page

Rough ends of infinite primitive permutation groups

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. Krön and R. G. Möller 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

Bounding the size of a vertex-stabiliser in a finite vertex-transitive graph

In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a $G$-vertex-transitive graph $\Gamma$. In the main result the group $G$ is quasiprimitive or biquasiprimitive on the vertices of $\Gamma$, and we obtain a genuine reduction to the case where $G$ is a nonabelian simple group. Using normal quotient techniques developed by the first author, the main theorem applies to general $G$-vertex-transitive graphs which are $G$-locally primitive (respectively, $G$-locally quasiprimitive), that is, the stabiliser $G_\alpha$ of a vertex $\alpha$ acts primitively (respectively quasiprimitively) on the set of vertices adjacent to $\alpha$. We discuss how our results may be used to investigate conjectures by Richard Weiss (in 1978) and the first author (in 1998) that the order of $G_\alpha$ is bounded above by some function depending only on the valency of $\Gamma$, when $\Gamma$ is $G$-locally primitive or $G$-locally quasiprimitive, respectively

Graphs, permutations and topological groups

Various connections between the theory of permutation groups and the theory of topological groups are described. These connections are applied in permutation group theory and in the structure theory of topological groups. The first draft of these notes was written for lectures at the conference Totally disconnected groups, graphs and geometry in Blaubeuren, Germany, 2007.Comment: 39 pages (The statement of Krophollers conjecture (item 4.30) has been corrected