We investigate correspondences between extreme amenability and amenability of automorphism groups of Fraïssé-Hrushovski generic structures that are obtained from smooth classes, and their Ramsey type properties of their smooth classes, similar to [7,13]. In particular, we focus on some Fraïssé-Hrushovski generic structures that are obtained from pre-dimension functions. Using these correspondences, we prove that automorphism groups of ordered Hrushovski generic graphs are not extremely amenable in both cases of collapsed and uncollapsed. Moreover, we prove that automorphism groups of Fraïssé-Hrushovski generic structures that are obtained from pre-dimension functions with rational coefficients are not amenable.

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.

We show that there are simple groups with a spherical BN-pair of rank 2 which are non-Moufang and hence not of algebraic origin.

We classify all group topologies coarser than the topology of stabilizers of finite sets in the case of automorphism groups of countable free-homogeneous structures, Urysohn space and Urysohn sphere, among other related results.

Introduction 1 2. Setting 2 3. Generic sequences of automorphisms 4 4. Proof of SIP for strong amalgamation classes 8 5. The SIP for the Zilber field, covers and modular invariants 9 References 13