Rough ends of infinite primitive permutation groups
-
Simon M. Smith
Abstract.
If 
is a group of permutations of a set 
, then the suborbits of 
are the orbits of point-stabilizers 
acting on 
. The cardinalities of these suborbits are the subdegrees of 
. Every infinite primitive permutation group 
with finite subdegrees acts faithfully as a group of automorphisms of a locally-finite connected vertex-primitive directed graph 
with vertex set 
, and there is consequently a natural action of 
on the ends of 
.
We show that if 
is closed in the permutation topology of pointwise convergence, then the structure of 
is determined by the length of any orbit of 
acting on the ends of 
.
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 
are the rough ends of 
, and so our result is equivalent to saying that the structure of a closed primitive group 
whose subdegrees are all finite is determined by the length of any orbit of 
on its rough ends.
© 2012 by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Masthead
- Lower signalizer lattices in alternating and symmetric groups
- Group rings in which the group of units is hyperbolic
- The normalizer property for integral group rings of finite solvable T-groups
- Finite -groups all of whose proper subgroups have cyclic Frattini subgroups
- A -group with positive rank gradient
- New criteria for equivalence of locally compact abelian groups
- Rough ends of infinite primitive permutation groups
- Automorphism groups of rooted trees have property (FA): A new proof
- About the metric approximation of Higman's group
Articles in the same Issue
- Masthead
- Lower signalizer lattices in alternating and symmetric groups
- Group rings in which the group of units is hyperbolic
- The normalizer property for integral group rings of finite solvable T-groups
- Finite -groups all of whose proper subgroups have cyclic Frattini subgroups
- A -group with positive rank gradient
- New criteria for equivalence of locally compact abelian groups
- Rough ends of infinite primitive permutation groups
- Automorphism groups of rooted trees have property (FA): A new proof
- About the metric approximation of Higman's group
