Abstract:

We say that a group G is inner ultrahomogeneous (IUH) if every isomorphism between finitely generated subgroups of G is given by conjugation by an element of G. By classical group theory, it is relatively easy to show that every existentially closed group is IUH, and similarly for e.c. locally finite and e.c. torsion-free groups. In particular, every infinite group is contained in an IUH group of the same cardinality, and there are natural Fraisse classes of groups with IUH limits (finite groups, finitely presentable groups*, all groups in a given countable model of set theory). It turns out that the finite exponent IUH groups have exponent bounded by (2^100)! (although there are no examples of finite exponent IUH groups with more than 6 elements, so the true bound may be much lower). On the other hand, the infinite exponent groups all have very large abelian subgroups (they contain either all torsion or all torsion-free countable abelian groups), and are model-theoretically very wild (e.g. they are never aleph_0-saturated, have no q.e., they have TP2, IP_n for all n, the strict order property...), while simultaneously, they are often dynamically very tame. I will give an outline of these results, with some more precise statements and ideas of proofs. I'm cheating a little bit here, since finitely presentable groups are not a hereditary class. The actual Fraisse class consists of all finitely generated groups in a given countable model of set theory.

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