Fra\"iss\'e structures and a conjecture of Furstenberg

TL;DR

This paper investigates the Samuel compactification of automorphism groups of countable structures, addressing a problem posed by Furstenberg and exploring the difference between Samuel compactification and enveloping semigroup, with new ultrafilter concepts.

Contribution

It resolves Furstenberg's problem for multiple automorphism groups and introduces new ultrafilter types for the case of the infinite symmetric group.

Findings

Resolved Furstenberg's problem for several automorphism groups

Identified differences between Samuel compactification and enveloping semigroup in specific cases

Developed new ultrafilter concepts on countable sets

Abstract

We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between $S(G)$, the Samuel compactification, and $E(M(G))$, the enveloping semigroup of the universal minimal flow. We resolve Furstenberg's problem for several automorphism groups and give a detailed study in the case of $G = S_\infty$, leading us to define and investigate several new types of ultrafilter on a countable set.