On the asymptotic behaviour of the number of Beauville and non‐Beauville p‐groups

Fernández-Alcober, Gustavo A.; Gül, Şükran; Vannacci, Matteo

doi:10.1112/plms.12295

Proc. London Math. Soc.

2019

DOI: 10.1112/plms.12295

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On the asymptotic behaviour of the number of Beauville and non‐Beauville p‐groups

Gustavo A. Fernández-Alcober

Şükran Gül

Matteo Vannacci

Abstract: We find asymptotic lower bounds for the numbers of both Beauville and non‐Beauville 2‐generator finite p‐groups of a fixed order, which turn out to coincide with the best known asymptotic lower bound for the total number of 2‐generator finite p‐groups of the same order. This shows that both Beauville and non‐Beauville groups are abundant within the family of finite p‐groups.

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Cited by 3 publications

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Beauville p-groups of wild type and groups of maximal class

Fernández-Alcober,

Gavioli,

Gül

et al. 2024

Monatsh Math

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Let G be a Beauville p-group. If G exhibits a ‘good behaviour’ with respect to taking powers, then every lift of a Beauville structure of $$G/\Phi (G)$$ G / Φ ( G ) is a Beauville structure of G. We say that G is a Beauville p-group of wild type if this lifting property fails to hold. Our goal in this paper is twofold: firstly, we fully determine the Beauville groups within two large families of p-groups of maximal class, namely metabelian groups and groups with a maximal subgroup of class at most 2; secondly, as a consequence of the previous result, we obtain infinitely many Beauville p-groups of wild type.

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Beauville p-groups of wild type and groups of maximal class

Fernández-Alcober,

Gavioli,

Gül

et al. 2024

Monatsh Math

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Ramification structures for quotients of the Grigorchuk groups

Noce

Thillaisundaram

2021

J. Algebra Appl.

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Groups associated to surfaces isogenous to a higher product of curves can be characterized by a purely group-theoretic condition, which is the existence of the so-called ramification structure. In this paper, we prove that infinitely many quotients of the Grigorchuk groups admit ramification structures. This gives the first explicit infinite family of 3-generated finite 2-groups with ramification structures.

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Generalised Beauville groups

Carta

Fairbairn

2021

Journal of Group Theory

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A Beauville surface is defined by the action of a finite group (Beauville group) on the product of two compact Riemann surfaces. In this paper, we consider higher products and the possibility of a similar action by a finite group, which we call a generalised Beauville group; we prove several results regarding the existence and construction of infinite families of generalised Beauville groups and provide a complete classification of the abelian ones; we list all generalised Beauville groups of orders from 1 to 1023.

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On the asymptotic behaviour of the number of Beauville and non‐Beauville p‐groups

Cited by 3 publications

References 18 publications

Beauville p-groups of wild type and groups of maximal class

Beauville p-groups of wild type and groups of maximal class

Ramification structures for quotients of the Grigorchuk groups

Generalised Beauville groups

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