Effectively closed subgroups of the infinite symmetric group

Abstract

We apply methods of computable structure theory to study effectively closed subgroups of S ∞ S_\infty . The main result of the paper says that there exists an effectively closed presentation of Z 2 \mathbb {Z}_2 which is not the automorphism group of any computable structure M M . In contrast, we show that every effectively closed discrete group is topologically isomorphic to A u t ( M ) \rm {Aut}(M) for some computable structure M M . We also prove that there exists an effectively closed compact (thus, profinite) subgroup of S ∞ S_\infty that has no computable Polish presentation. In contrast, every profinite computable Polish group is topologically isomorphic to an effectively closed subgroup of S ∞ S_\infty . We also look at oligomorphic subgroups of S ∞ S_\infty ; we construct a Σ 1 1 \Sigma ^1_1 closed oligomorphic group in which the orbit equivalence relation is not uniformly HYP. Our proofs rely on methods of computable analysis, techniques of computable structure theory, elements of higher recursion theory, and the priority method.