# Translation-like Actions and Aperiodic Subshifts on Groups

* Corresponding author
1 CARTE - Theoretical adverse computations, and safety
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : It is well known that if $G$ admits a f.g. subgroup $H$ with a weakly aperiodic SFT (resp. an undecidable domino problem), then $G$ itself has a weakly aperiodic SFT (resp. an undecidable domino problem). We prove that we can replace the property "$H$ is a subgroup of $G$" by "$H$ acts translation-like on $G$", provided $H$ is finitely presented. In particular: * If $G_1$ and $G_2$ are f.g. infinite groups, then $G_1 \times G_2$ has a weakly aperiodic SFT (and actually a undecidable domino problem). In particular the Grigorchuk group has an undecidable domino problem. * Every infinite f.g. $p$-group admits a weakly aperiodic SFT.
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Preprints, Working Papers, ...
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https://hal.inria.fr/hal-01187069
Contributor : Emmanuel Jeandel Connect in order to contact the contributor
Submitted on : Wednesday, August 26, 2015 - 9:05:55 AM
Last modification on : Saturday, June 25, 2022 - 7:46:49 PM
Long-term archiving on: : Friday, November 27, 2015 - 10:56:59 AM

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wgroups2.pdf
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### Identifiers

• HAL Id : hal-01187069, version 1
• ARXIV : 1508.06419

### Citation

Emmanuel Jeandel. Translation-like Actions and Aperiodic Subshifts on Groups. 2015. ⟨hal-01187069⟩

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