Non-genericity phenomena in ordered Fraïssé classes. (English) Zbl 1257.03058

Slutsky, Konstantin

J. Symb. Log. 77, No. 3, 987-1010 (2012).

This paper studies two equivalence relations on \(n\)-tuples in topological groups. Suppose that \(G\) is a topological group and \(f=(f_1,\dots,f_n)\) and \(g=(g_1,\dots,g_n)\) are \(n\)-tuples from \(G\). We say that \(fE^N_{TS}g\) (\(f\) and \(g\) are topologically similar) if the map \(f_i\mapsto g_i\) extends to an isomorphism between the groups generated by these tuples. We say that \(fE^N_Gg\) (\(f\) and \(g\) are diagonally conjugate) if there is \(\alpha\in G\) such that \(\alpha f_i\alpha^{-1}=g_i\) for each \(i\). Clearly \(fE^n_Gg\) implies \(fE_{TS}^ng\) so that the relation \(E^n_G\) is finer than \(E^n_{TS}\).
Hodkinson showed that if \(G\) is the group of order-preserving automorphisms of the rational numbers, then all \(E^2_G\) classes are meager. In this paper, the author extends this result by showing that all \(E^2_{TS}\) classes are meager for this group. Also, an analogous theorem is proven if \(G\) is the group of order-preserving isometries of the ordered rational Urysohn space.

Reviewer: Isaac Goldbring (Chicago)

MSC:

03C13	Model theory of finite structures
22F50	Groups as automorphisms of other structures

Keywords:

automorphisms of rationals; Urysohn space; topological similarity; diagonal conjugacy classes

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