Smith, Simon M. (2012) Rough ends of infinite primitive permutation groups. Journal of Group Theory, 55 (2). ISSN 14335883
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Item Type:  Article 

Item Status:  Live Archive 
Abstract
If G is a group of permutations of a set Omega , then the suborbits of G are the orbits of pointstabilisers 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 locallyfinite connected vertexprimitive 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.
Keywords:  Infinite permutation groups, CayleyAbels Graph 

Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  College of Science > School of Mathematics and Physics 
ID Code:  27492 
Deposited On:  12 May 2017 09:35 
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