Amenability of automorphism groups of generic structures
Wednesday, 14.6.17, 16:30-17:30, Raum 404, Eckerstr. 1
In a paper by J. Moore following the seminal work of Kechris-Pestov-Todorsevic a correspondence between a certain combinatorial property of a Fraisse class, called convex Ramsey property, and amenability of the automorphism group of the Fraisse limit has been found. In this paper we review similar results for the automorphism groups of generic structures and especially show that automorphism groups of certain generic structures are not amenable by showing that a certain point-line geometries are realized in the generic structure.
Higher Amalgamation and Finite Covers (of first order structures)
Wednesday, 21.6.17, 16:30-17:30, Raum 404, Eckerstr. 1
The talk will be about the fine structure of (very) well-behaved complete first order theories.\nTotally categorical structures of disintegrated type (i.e. the underlying strongly minimal set is trivial) can analysed by a chain of finite covers. A finite cover of some structure is an extension by a new sort and new relations such that the old structure is stably embedded (i.e. every automorphism of the old structure extends to the cover) and there is some definable finite-to-one function from the new sort to the\nold sorts. \nNow we have that non-trivial phenomena in this chain of finite covers are connected to something called higher amalgamation, that is the ability to amalgamate certain systems of types. We will investigate higher amalgamation over parameters in a more general setting, i.e. in theories with a good notion of independence (e.g. strongly minimal, stable, simple). We give a general finite cover construction to force failure of higher amalgamation and\napply it to the totally categorical structure (Z/4Z)^\bomega such that higher amalgamation over some parameter fails while it holds over the empty set. \nThis tells us that the analysis of general totally categorical structure via covers has another complication. But on the other hand as we can, after adding a sequence of finite covers, force every omega-categorical theory to have higher amalgamation over any parameter set, we could potentially have a starting point for some sort of classification of general totally categorical theories via covers.\n
Free homogeneous structures
Wednesday, 28.6.17, 16:30-17:30, Raum 404, Eckerstr. 1
Free homogeneous structures
Wednesday, 28.6.17, 16:30-17:30, Raum 404, Eckerstr. 1
A countably infinite first order structure is\nhomogeneous if every isomorphism between finitely generated\nsubstructures extends to a total automorphism. By Fraisse\nTheorem, homogeneous structures arise as the Fraisse limits\nof amalgamation classes. Moreover, a free homogeneous\nstructure is a homogeneous relational structure whose age\nhas the free amalgamation property. In a joint work with\nSolecki, we show that free amalgamation classes has a\n'coherent' form of the extension property for partial\nautomorphisms (EPPA). We further discuss some\ngroup-theoretic consequences of this result on the\nautomorphism group of any free homogeneous structure such\nas the existence of ample generics and a dense locally\nfinite subgroup.