Rearrangement groups of fractals

Belk, James; Forrest, Bradley

doi:10.1090/tran/7386

Rearrangement groups of fractals
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by James Belk and Bradley Forrest PDF

Trans. Amer. Math. Soc. 372 (2019), 4509-4552 Request permission

Abstract:

We construct rearrangement groups for edge replacement systems, an infinite class of groups that generalize Richard Thompson’s groups $F$, $T$, and $V$. Rearrangement groups act by piecewise-defined homeomorphisms on many self-similar topological spaces, among them the Vicsek fractal and many Julia sets. We show that every rearrangement group acts properly on a locally finite $\mathrm {CAT}(0)$ cubical complex, and we use this action to prove that certain rearrangement groups are of type $F_{\infty }$.

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