GGS-groups: Order of congruence quotients and Hausdorff dimension
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- by Gustavo A. Fernández-Alcober and Amaia Zugadi-Reizabal PDF
- Trans. Amer. Math. Soc. 366 (2014), 1993-2017 Request permission
Abstract:
If $G$ is a GGS-group defined over a $p$-adic tree, where $p$ is an odd prime, we calculate the order of the congruence quotients $G_n=G/\mathrm {Stab}_G(n)$ for every $n$. If $G$ is defined by the vector $\mathbf {e}=(e_1,\ldots ,e_{p-1})\in \mathbb {F}_p^{p-1}$, the determination of the order of $G_n$ is split into three cases, according to whether $\mathbf {e}$ is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on $p$, $n$, and the rank of the circulant matrix whose first row is $\mathbf {e}$. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the $p$-adic tree.References
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Additional Information
- Gustavo A. Fernández-Alcober
- Affiliation: Matematikaren eta Zientzia Esperimentalen Didaktika Saila, Euskal Herriko Unibertsitatea UPV/EHU, 48080 Bilbao, Spain
- MR Author ID: 307028
- Email: gustavo.fernandez@ehu.es
- Amaia Zugadi-Reizabal
- Affiliation: Matematikaren eta Zientzia Esperimentalen Didaktika Saila, Euskal Herriko Unibertsitatea UPV/EHU, 48080 Bilbao, Spain
- Email: amaia.zugadi@ehu.es
- Received by editor(s): August 17, 2011
- Received by editor(s) in revised form: June 29, 2012
- Published electronically: October 16, 2013
- Additional Notes: The authors were supported by the Spanish Government, grant MTM2008-06680-C02-02, partly with FEDER funds, and by the Basque Government, grant IT-460-10
The second author was also supported by grant BFI07.95 of the Basque Government - © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 1993-2017
- MSC (2010): Primary 20E08
- DOI: https://doi.org/10.1090/S0002-9947-2013-05908-9
- MathSciNet review: 3152720