# Subdegree growth rates of infinite primitive permutation groups

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Reference: Smith, Simon (Simon Mark), (2005). Subdegree growth rates of infinite primitive permutation groups. DPhil. University of Oxford.Citable link to this page:

Subdegree growth rates of infinite primitive permutation groups

Abstract: If G is a group acting on a set Ω, and α, β ∈ Ω, the directed graph whosevertex set is Ω and whose edge set is the orbit (α, β)G is called an orbitalgraph of G. These graphs have many uses in permutation group theory. Agraph Γ is said to be primitive if its automorphism group acts primitivelyon its vertex set, and is said to have connectivity one if there is a vertex αsuch that the graph Γ\{α} is not connected. A half-line in Γ is a one-wayinfinite path in Γ. The ends of a locally finite graph Γ are equivalenceclasses on the set of half-lines: two half-lines lie in the same end if thereexist infinitely many disjoint paths between them.A complete characterisation of the primitive undirected graphs with connectivityone is already known. We give a complete characterisation in thedirected case. This enables us to show that if G is a primitive permutationgroup with a locally finite orbital graph with more than one end, then Ghas a connectivity-one orbital graph Γ, and that this graph is essentiallyunique. Through the application of this result we are able to determineboth the structure of G, and its action on the end space of Γ.If α ∈ Ω, the orbits of the stabiliser Gα are called the α-suborbits of G. Thesize of an α-suborbit is called a subdegree. If all subdegrees of an infiniteprimitive group G are finite, Adeleke and Neumann claim one may enumeratethem in a non-decreasing sequence (mr). They conjecture that thegrowth of the sequence (mr) is extremal when G acts distance transitivelyon a locally finite graph; that is, for all natural numbers m the stabiliser inG of any vertex α permutes the vertices lying at distance m from α transitively.They also conjecture that for any primitive group G possessinga finite self-paired suborbit of size m there might exist a number c whichperhaps depends upon G, perhaps only on m, such that mr ≤ c(m-2)r-1.We show their questions are poorly posed, as there exist primitive groupspossessing at least two distinct subdegrees, each occurring infinitely often.The subdegrees of such groups cannot be enumerated as claimed. We givea revised definition of subdegree enumeration and growth, and show thatunder these new definitions their conjecture is true for groups exhibitingexponential subdegree growth above a prescribed bound.

Type of Award:DPhil Level of Award:Doctoral Awarding Institution: University of Oxford Notes:This thesis was digitised thanks to the generosity of Dr Leonard Polonsky.

Contributors

Neumann, P. M.More by this contributor

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Peter NeumannMore by this contributor

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Bibliographic Details

Issue Date: 2005Identifiers

Urn: uuid:1baa0e15-363a-4163-b21b-59fcd62d210b

Source identifier: 603828444 Item Description

Type: Thesis;

Language: eng Subjects: Group theory Graph theory Tiny URL: td:603828444

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Author: ** Smith, Simon Simon Mark - institutionUniversity of Oxford facultyMathematical and Physical Sciences Division - - - - Contributors**

Source: https://ora.ox.ac.uk/objects/uuid:1baa0e15-363a-4163-b21b-59fcd62d210b