On the automorphism groups of universal submeasures

• Published in: Topology and its applications Vol. 311; p. 107963
• Main Authors: Meza-Alcántara, David, Nuñez-Rosales, Fernando
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15-04-2022
• Subjects: Automorphisms groups; Fraïssé limits; Hrushovski property; Ramsey property; Universal submeasures; Fraïssé limits; 05D10; 54H20; Hrushovski property; 03C15; Universal submeasures; Automorphisms groups; Ramsey property; 03C13
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2021.107963

We investigate dynamical properties of the automorphism groups of general versions of the universal submeasures defined in [3]. First, we show that, a universal submeasure D-valued exists for every countable (finite or infinite) set D of non-negative real numbers, with 0∈D. Moreover, ordered D-valued universal submeasures exist for all such D. By using the Kechris - Pestov - Todorčević theory, we prove that for all the ordered universal submeasures, automorphism groups are extremely amenable, but they do not have ample generics when D satisfies some additional conditions. Finally, we prove that the class of all finite D-valued submeasures has the Hrushovski property, and the automorphism group of the D-valued universal submeasure is amenable and has ample generics.