Automorphism groups of generic structures: extreme amenability and amenability

Abstract

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.

Automorphism groups and Ramsey properties of sparse graphs

“…We shall say what it means for (K; ≤) to be a Ramsey class and in the next subsection,we state the KPT correspondence and associated results in this context. Similar statements (about special class of maps) can be found in the paper of Zucker [40] and in [15]. In the case where K is closed under substructures and ≤ is just the usual notion of substructure, this is just the usual notion of Ramsey class and the KPT correspondence.…”

“…In Section 3, we describe and use it to prove Theorem (in the more general form of Theorem ). As an additional benefit, we also use it (in Section 3.3) to give a simple proof of a general result (Theorem ) about non‐amenability of which generalises results in . The argument we use shows that in Theorem we may take also having the property that there is no ‐categorical expansion of with amenable automorphism group (Corollary ).…”

Fixed points in compactifications and combinatorial counterparts

“…As indicated in the introduction, the practical use of Theorem 1.13 is so far limited. There are promising exceptions, as the papers by Gadhernezhad, Khalilian and Pourmahdian [GKP18], and by Etesami and Gadhernezhad [EG17], do make use of it to prove that certain automorphism groups of the form Aut(F), where F is a so-called Hrushovski structure, are not amenable. Nevertheless, there is presently no significant instance where Theorem 1.13 can be used to prove that some group is amenable.…”

Convex Ramsey matrices and non-amenability of automophism groups of generic structures

“…A generic structure, similar to a Fraïssé-limit structure, is constructed from a countable class of finite structures, called a smooth class, with an amalgamation property. In [6] following ideas of [8,11] it is shown that the amenability and extreme amenability of the automorphism group of a generic structure corresponds to the structural Ramsey type properties of its smooth class.…”

“…It is shown in Theorem 32 in [6] that the automorphism group of a generic structure is amenable if and only if the automorphism group has the convex Ramsey property with respect to the smooth class (see Definition 4). Then the correspondence is used to show that if the smooth class of a generic structure contains a certain pair of finite closed substructures, called a tree-pair (Definition 39 in [6]), then the automorphism group of the generic structure is not amenable (see Theorem 40 in [6]). Existence of a tree-pair implies that there is an open subgroup of the automorphism group of the generic structure that acts on a tree which is a substructure of the generic structure.…”