Coarse spaces, ultrafilters and dynamical systems

For a coarse space $(X, \mathcal{E})$, $X^\sharp$ denotes the set of all unbounded ultrafilters on $X$ endowed with the parallelity relation: $p||q$ if there exists $E \in \mathcal{E}$ such that $E[P]\in q$ for each $P\in p$. If $(X, \mathcal{E})$ is finitary then there exists a group $G$ of permutations of $X$ such that the coarse structure $\mathcal{E}$ has the base $\{\{ (x,gx): x\in X$, $g\in F\}: F\in [G]^{<ω}, \ id \in F \}.$ We survey and analyze interplays between $(X, \mathcal{E})$, $X^\sharp$ and the dynamical system $(G, X^\sharp)$.

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