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GGS-groups: Order of congruence quotients and Hausdorff dimension


Authors: Gustavo A. Fernández-Alcober and Amaia Zugadi-Reizabal
Journal: Trans. Amer. Math. Soc. 366 (2014), 1993-2017
MSC (2010): Primary 20E08
DOI: https://doi.org/10.1090/S0002-9947-2013-05908-9
Published electronically: October 16, 2013
MathSciNet review: 3152720
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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.


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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
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

DOI: https://doi.org/10.1090/S0002-9947-2013-05908-9
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
Article copyright: © Copyright 2013 American Mathematical Society

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