Direct products, overlapping actions, and critical regularity
| Autor: | Sang-hyun Kim, Thomas Koberda, Cristóbal Rivas |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Algebra and Number Theory
Group (mathematics) Applied Mathematics 05 social sciences Disjoint sets Group Theory (math.GR) Dynamical Systems (math.DS) Combinatorics 03 medical and health sciences Lamplighter group 0302 clinical medicine Free product 0502 economics and business FOS: Mathematics Interval (graph theory) Artin group Mathematics - Dynamical Systems Mathematics - Group Theory 050203 business & management 030217 neurology & neurosurgery Analysis Mathematics |
| 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: | OpenAIRE |
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