Direct products, overlapping actions, and critical regularity
| Autor: | Kim, Sang-hyun, Koberda, Thomas, Rivas, Cristóbal |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if $H$ and $K$ are two non-solvable groups then a faithful $C^{1,\tau}$ action of $H\times K$ on a compact interval $I$ is {\em not overlapping} for all $\tau>0$, which by definition means that there must be non-trivial $h\in H$ and $k\in K$ with disjoint support. As a corollary we prove that the right-angled Artin group $(F_2\times F_2)*\mathbb{Z}$ has critical regularity one, which is to say that it admits a faithful $C^1$ action on $I$, but no faithful $C^{1,\tau}$ action. This is the first explicit example of a group of exponential growth which is without nonabelian subexponential growth subgroups, whose critical regularity is finite, achieved, and known exactly. Another corollary we get is that Thompson's group $F$ does not admit a faithful $C^1$ overlapping action on $I$, so that $F*\mathbb{Z}$ is a new example of a locally indicable group admitting no faithful $C^1$--action on $I$. Comment: 22 pages, to appear in J. Mod. Dyn |
| Databáze: | arXiv |
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