Beauville \(p\)-groups of wild type and groups of maximal class (Q6633621)

Beauville groups are finite groups that arise in the construction of an interesting class of complex surfaces, the so-called Beauville surfaces, introduced by \textit{F. Catanese} [Am. J. Math. 122, No. 1, 1--44 (2000; Zbl 0983.14013)] following an example of \textit{A. Beauville} [Astérisque 54, 1--172 (1978; Zbl 0394.14014)].\N\NLet \(G\) be a group, \(x,y \in G\), \(S=\{x,y \}\), \(T=\{x,y,xy\}\) and \(\Sigma(T)=\bigcup_{t \in T, g \in G} \langle t \rangle^{g}\). Then \(G\) is a Beauville group if and only if it is a 2-generator group and \(\Sigma(S_{1}) \cap \Sigma(S_{2})=1\) for some \(2\)-element sets of generators \(S_{1}\) and \(S_{2}\) of \(G\).\N\NBelow the reviewer reports the abstract of the paper. ``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 \(G/\Phi(G)\) 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.''