Beauville p-Groups: A Survey

Abstract

Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G.In this survey we discuss the p-groups that may be used in this way. En route we discuss several open problems, questions and conjectures.

The Business of Armaments

“…According to the firm's historian, patriotism dominated Armstrong's motives: He "had not patented his new inventions and furthermore he had declined the £20,000 reward which was offered him, so the Government took out patents in his name for the guns and their complete outfit, and by an Act of Parliament, withheld publication of the patents." 22 Marshall Bastable also sees Armstrong's actions as an astute political move because "[i]n return he demanded a guarantee that the capital investment in the new company would not be lost if the government decided to build all the guns at Woolwich or contract them out to others." 23 In 1859 Armstrong was given a ten-year contract (backdated to April 1, 1856 so as to cover the expenses that he incurred in developing his gun) as Engineer for Rifled Ordnance to the War Office, a knighthood and an annual salary of £2,000, plus payment for the "draughtsmen employed by him" and his travel expenses.…”

Strongly Real Beauville Groups III

“…The earliest explicit infinite family of Beauville 2-groups were constructed by Barker, Boston, Peyerimhoff and Vdovina in [3,4,5] where, again, more general constructions are also considered. The most comprehensive surveys on Beauville p-groups in general are given by Boston in [16] and more recently by the author in [24].…”

“…Imposing a restriction on the Beauville structures clearly makes this set smaller and thus the problem of determining such a spectrum is more tractable. The following natural definition was first made by the author in [24].…”

Beauville structures for quotients of generalised GGS-groups

“…Beauville groups have been intensely studied in recent times; see surveys [1,7,8,16]. For example, the abelian Beauville groups were classified by Catanese [4]: a finite abelian group G is a Beauville group if and only if G ∼ = C n ×C n for n > 1 with gcd(n, 6) = 1.…”