Subdegree growth rates of infinite primitive permutation groups

Abstract

A transitive group $G$ of permutations of a set $Ω$ is primitive if the only $G$-invariant equivalence relations on $Ω$ are the trivial and universal relations. If $α\in Ω$, then the orbits of the stabiliser $G_α$ on $Ω$ are called the $α$-suborbits of $G$; when $G$ acts transitively the cardinalities of these $α$-suborbits are the subdegrees of $G$. If $G$ acts primitively on an infinite set $Ω$, 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.