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Contents
1. Introduction
2. Language, notation, and groups
3. More on $G_{n,r,\mathrm {d}}$, $G_{n,r,\mathrm {p}}$, $G_{n,r}$, and $H_{n,r,{\sim }}$
4. Local actions
5. On the rational group $\mathcal {R}_n$ and the related groups $\mathcal {R}_{n,r}$
6. Automorphisms admit few local actions
7. Finding our place in the rational group $\mathcal {R}_{n,r}$
8. Combinatorial properties of strongly synchronizing transducers
9. The natural quotient $\mathcal {B}_{n,r}\twoheadrightarrow \mathcal {O}_{n,r}$
10. Root functions and detecting flavours of synchronicity
11. Open problems
A. The inversion construction



