Small profinite structures (Q2759048)

The author develops some of his geometric-stability-theoretic results in a more abstract setting of profinite structures. A topological space \(U\) is said to be profinite if there is a countable directed system of closed equivalence relations on \(U\) with finite classes such that all the classes form a basis of \(U\). The group of homeomorphisms of \(U\) respecting all these equivalence relations is the inverse image of a direct system of finite groups and so is a profinite group acting continuously on \(U\). A profinite structure is defined to be a profinite topological space \(U\) together with a distinguished closed subgroup \(G\) of the above group of homeomorphisms. NEWLINENEWLINENEWLINEThe notion can be put into a model-theoretic framework as follows. Let \(M\) be a multi-sorted structure with countably many finite sorts, and \(\Phi\) a parameter-free type in variables \((x_i: i\in I)\), where \(x_i\) is of sort \(M_i\). The set \(U\) of realizations of \(\Phi\) in \(M\) is a closed subset of the topological product of all \(M_i\), where the sets \(M_i\) are equipped with the discrete topology. Then \(U\), together with the group \(G\) of all transformations of \(U\) induced by automorphisms of \(M\), forms a profinite structure in which the \(G\)-invariant relations on \(U\) are exactly the relations on \(U\) definable in \(M\) by types without parameters. Moreover, any profinite structure is shown to be obtained in that way. For this reason, for any profinite structure \((U,G)\), the author calls \(G\)-invariant relations on \(U\) definable. NEWLINENEWLINENEWLINEA profinite structure \((U,G)\) is said to be small if for any \(n<\omega\) the number of \(G\)-orbits in \(U^n\) is at most countable. The notion is motivated by the known notion of a small theory: a countable complete first order theory \(T\) is called small if \(T\) has at most countably many \(n\)-types over the empty set. Small theories give rise to small profinite structures as follows. Suppose \(T=T^{\text{eq}}\), and \(M\) is a monster model of \(T\). Let \(X\) be a \(\emptyset\)-type-definable subset of \(M\), and \(f_i:X\to \text{ acl}(\emptyset)\) be \(\emptyset\)-definable functions, \(i<\omega\). Then \(U=\{(f_i(a):i<\omega): a\in X\}\) is a closed subset of \(\text{ acl}(\emptyset)^\omega\) with the product topology, where \(\text{ acl}(\emptyset)\) is equipped with the discrete topology. Moreover, if \(G\) is the group of transformations of \(U\) induced by automorphisms of \(M\) then \((U,G)\) is a profinite structure, which is small if \(T\) is small. Such a profinite structure is said to be interpreted in the theory \(T\). NEWLINENEWLINENEWLINEIn his studies of small superstable theories inspired by Vaught's conjecture, the author introduced the notions of meager forking, \(m\)-independence, \(m\)-normality, and \(\mathcal M\)-rank [``Meager forking and \(m\)-independence'', Doc. Math., J. DMV, Extra Vol. ICM Berlin, 1998, Vol. II, 33-42 (1998; Zbl 0907.03015)]. In the paper under review he defines and studies analogs of these notions (as well as of some other known model-theoretic notions like local modularity) for small profinite structures, basing on the known idea of correspondence between types and orbits. The main result: If \((U,G)\) is a small profinite structure, and \(\mathcal O\) is a nontrivial locally modular \(G\)-orbit of \(\mathcal M\)-rank 1 in \(U\) then some open subset \(\mathcal O'\) of \(\mathcal O\) is a definable group. The result has earlier analogs, first in the totally categorical context (B.~Zilber), and then in the stable context (E.~Hrushovski) and in the o-minimal context (Y.~Peterzil). The author [``Geometry of *-finite types'', J. Symb. Log. 64, 1375-1395 (1999; Zbl 0957.03045)] proved the result for profinite structures interpreted in small theories. It is not clear whether there are interesting applications of the main result beyond the special case of interpreted profinite structures.