Simplicity of the automorphism groups of some Hrushovski constructions

Abstract

Abstract. we show that the automorphism groups of certain countable structures obtained using the Hrushovski amalgamation method are simple groups. The structures we consider are the 'uncollapsed' structures of infinite Morley rank obtained by the ab initio construction and the (unstable) ℵ 0 -categorical pseudoplanes. The simplicity of the automorphism groups of these follows from results which generalize work of Lascar and of Tent and Ziegler.

An Axiomatic Approach to Free Amalgamation

“…The main examples in [29] have trivial algebraic closure and, in such cases, one may show that the two notions of a stationary independence relation are the same. In [11], Evans, Ghadernezhad, and Tent also consider axioms of ternary relations, which have been relativized to the lattice of algebraically closed sets.…”

Group topologies on automorphism groups of homogeneous structures

“…Simple non-modular Hrushovski structures. In Subsection 2.2 we discussed in more details for some cases how the Hrushovski construction is built and especially the ω-categorical variation of it (see section 5.2. in [EGT16] for more details). Here is a brief reminder of the setting:…”

“…Choosing an unbounded convex function f which is "good" enough one can consider C f η , a subclass of C η , with the free-amalgamation property where the limit structure M f η is ωcategorical and which Aut M f η acts transitively on M f η . It is shown in in Lemma 5.7 in [EGT16] that there is an independence relation defined for the class of -closed subsets of M f η that satisfy all the properties of part 1. in Lemma 4.4. Then using Proposition 3.9 we conclude the following.…”

“…If all points are algebraically closed, then any group topology τ on G coarser than τ st is of the form τ X st for some G-invariant X ⊆ M. By an independence relation it is usually meant some ternary relation | ⌣ on (some) collection of sets of parameters of a structure such that A | ⌣C B captures the intuitive idea that B does not any information about A not already contained in C, the paradigmatic example of which is forking independence. The connections between the existence of an independence relations on a homogeneous structure satisfying certain axioms and the properties of the automorphism group goes back to [TZ13] (see also [EGT16]). Of particular relevance to us is the freedom axiom, explored with detail in [Con17].…”

“…Here we briefly discuss a variation on the Hrushovski's pre-dimension construction method which gives rise to ω-categorical structures. The original version of this is used to provide a counterexample to Lachlan's conjecture, where it is used to construct a nonmodular, supersimple ω-categorical structure (see section 5.2. in [EGT16] for some details).…”