Smith, Simon
(2010)
Subdegree growth rates of infinite primitive permutation groups.
Journal of the London Mathematical Society, 82
(2).
pp. 526548.
ISSN 00246107
Full content URL: https://doi.org/10.1112/jlms/jdq046
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Abstract
A transitive group G of permutations of a set Ω is primitive if the only Ginvariant equivalence relations on Ω are the trivial and universal relations. If α ∈ Ω, then the orbits of the stabilizer G_α on Ω are called the αsuborbits of G; when G acts transitively the cardinalities of these αsuborbits 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 nondecreasing sequence 1 = m_0 ≤ m_1 ≤ . . , the subdegree growth rates of infinite primitive groups that act distancetransitively on locally finite distancetransitive graphs are extremal, and conjecture that there might exist a number c that perhaps depends upon G, perhaps only on the size of some suborbit m, such that m_r ≤ 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 αsuborbits. 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.
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