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A few years ago, a number of examples were given of Fraisse structures without the SAP in answer to the question raised in A Fraïssé class without the strong amalgamation property. It is well known that any closed subgroup of the full permutation group of the integers is the automorphism group of some Fraisse limit, which may or may not have the SAP, but this limit is far from unique.

For example, if one partitions the integers into pairs and considers the group of all permutations that send pairs to pairs and respect the natural ordering on each pair then one gets a group of permutations isomorphic to the full permutation group. The age of the original structure does not have the SAP because it has non-trivial algebraicity (if you know where the bottom element of a pair goes, you know where the top element goes too). However, the full permutation group is the automorphism group of the empty structure which, of course, has the SAP.

My question is: Is every non-locally-compact, closed subgroup of the full permutation group isomorphic to the group of automorphisms of the limit of some Fraisse structure with the SAP?

Since the SAP allows the construction of very many automorphisms, one cannot expect all closed groups to have this property. I suspect that a counterexample can be found in which the group has a locally compact factor, but do not know this either. But if this does turn out to be the case, then the next question is: Is every group that is not the homomorphic preimage of a locally-compact, closed subgroup of the full permutation group isomorphic to the group of automorphisms of the limit of some Fraisse structure with the SAP?

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