  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 $\{\{ (x,gx):
x\in X$, $g\in F\}: F\in [G]^{<\omega}, \ id \in F \}.$ We survey and analyze
interplays between $(X, \mathcal{E})$, $X^\sharp$ and the dynamical system $(G,
X^\sharp)$.
