Ordered structures and large conjugacy classes

Abstract

This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraïssé limit) that is extremely amenable, and has ample generics. As Fraïssé limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraïssé limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. We also provide general conditions implying that there is no comeager conjugacy class, comeager 2-dimensional diagonal conjugacy class or non-meager 2-dimensional topological similarity class in the automorphism group of an ordered Fraïssé limit. We provide a number of applications of these results.2010 Mathematics Subject Classification. 03E15, 54H11.

On the Automorphism Group of the Universal Homogeneous Meet-Tree

“…Our main results are similar to the ones in [10,18] regarding (Q, <) and the universal partial order (although it is not known if the universal partial order has a generic pair); more recently, a new preprint [12] appeared giving similar results on two different structures (the universal ordered boron tree-roughly speaking, a graph theoretic binary tree with a lexicographical order-and the universal ordered poset). The latter's motivation came from a different yet related question of finding an ultrahomogeneous ordered structure whose automorphism group has ample generics and is extremely amenable (in other words, by [8], its age has the Ramsey property).…”

“…Another approach to trees is by graph theory: we can identify every finite tree with an acyclic directed graph, but the Fraïssé limits of classes of finite trees in this language will be quite different (for instance, the "order" on each branch will not be dense). In [12], the authors study the existence of generic automorphisms in this context. ♦…”

Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups

“…In particular, we can recover Theorems 3.12 and 4.4 from [9]. Recall that a boron tree structure B is formed from leaves of a connected, acyclic graph G all of whose vertices have order 1 or 3, together with a quaternary relation R defined by the following condition: R(a, b, c, d) iff the unique paths connecting a with b, and c with d, are disjoint.…”

“…Finally, we study groups of ball-preserving bijections of ordered ultrametric spaces, objects that seem not to have been explicitly considered so far, although they have implicitly appeared in the literature devoted to structural Ramsey theory. For example, the Ramsey expansion of the class of boron trees studied by Jasinski [6], and Kwiatkowska and Malicki [9], or Ramsey expansions of structures that can be naturally identified with Ważewski dendrites, studied by Kwiatkowska [8], can be naturally viewed as ordered ultrametric spaces with ball-preserving mappings as morphisms. In Theorems 5.10 and 5.11, we prove that groups of ball-preserving bijections of ordered ultrametric Urysohn spaces with rational distances have a comeager conjugacy class but they do not have a comeager 2-diagonal conjugacy class.…”

“…In Theorems 5.10 and 5.11, we prove that groups of ball-preserving bijections of ordered ultrametric Urysohn spaces with rational distances have a comeager conjugacy class but they do not have a comeager 2-diagonal conjugacy class. This, in particular, gives alternative, and much shorter proofs of Theorems 3.12 and 4.4 from [9].…”