Closed groups induced by finitary permutations and their actions on trees (Q2759040)

The symmetric group \(\text{Sym}(\omega)\) is a complete metric space with respect to \(d(g, h) = \sum \{ 2^{-n} : n\) natural, \(g(n) \neq h(n)\) or \(g^{-1}(n) \neq h^{-1}(n) \}\), and it is known that a subgroup \(G\) of \(\text{Sym}(\omega)\) is closed if \(G\) is the automorphism group of some structure \(M\) with domain \(\omega\). In this framework the paper under review studies the subgroups \(G\) of \(\text{Sym} (\omega)\) such that \(G\) is the automorphism group of some saturated structure \(M\) on \(\omega\) and at the same time \(G\) is the closure of the subgroup of all elements with finite support (the saturation assumption on \(M\) arises quite naturally from both model theoretic and algebraic reasons). NEWLINENEWLINENEWLINEFirst the author observes that, for such a group \(G\), \(M\) has to be an \(\omega\)-stable weakly minimal structure with a disintegrated locally finite algebraic closure. But what is most relevant is the analysis of the permutation group \(G\), showing that \((\omega, G)\) is a \textsl{2-dimensional cell}. This means that \(G\) is closed (as we know) and moreover \(\omega\) partitions into \(G\)-invariant classes \(\omega = \bigcup_i Y_i\) and every infinite \(Y_i\) decomposes as the product of a countable infinite set \(X_i\) and a finite \(F_i\), \(G\) preserves the equivalence relation \(E_i\) on \(Y_i\) whose classes are the fibres \(\{ x \} \times F_i\) with \(x \in X_i\) and induces \(\text{Sym}(X_i)\) by the action on the quotient set of \(E_i\); moreover the group induced by \(G\) on \(Y_i\) is also induced by the pointwise stabilizer \(\bigcup_{j \neq i} Y_j\). NEWLINENEWLINENEWLINEThe second part of the paper treats the 2-dimensional cells, and their isometric action on real trees; in particular, it studies whether such a permutation group has a fixed point under an action on a real tree.