Regular Algebraic Surfaces, Ramification Structures and Projective Planes

A new infinite family of non-abelian strongly real Beauville p-groups for every odd prime p

“…Even ignoring the strongly real condition, Beauville p-groups are more difficult to construct in general -a commonly used trick for showing that condition ( †) of Definition 1.1 is satisfied is to find a Beauville structure such that o(x 1 )o(y 1 )o(x 1 y 1 ) is coprime to o(x 2 )o(y 2 )o(x 2 y 2 ) but this clearly cannot be done in a p-group since every non-trivial element has an order that's a power of p. Further motivation comes from the fact that in some sense 'most' finite groups are p-groups [4,5] and thus establishing the general picture in this case a long way to establishing the wider picture in general. Despite the above difficulties, a number of authors have found a variety of ingenious constructions for them [1,2,3,10,11,12,13,17,19]. Our main result is as follows.…”

“…Further motivation comes from the fact that in some sense ‘most’ finite groups are ‐groups and thus establishing the general picture in this case goes a long way to establishing the wider picture in general. Despite the above difficulties, a number of authors have found a variety of ingenious constructions for them . Our main result is as follows.…”

“…Note that x y xx y −1 ((x y xx y −1 ) −1 ) xyx = x y −1 x −y 2 and conjugating this by xyx gives us xx −y 3 . Conjugating this by (xyx) 3 gives us x y 3 x −y 6 and so we have xx −y 3 (x y 3 x −y 6 ) −1 = xx −y 6 . Since 3 is coprime to the order of y we can easily repeat this to obtain xx −y i for any i.…”

Ramification structures for quotients of the Grigorchuk groups

“…By [5, Cor. 2.3], for some n ∈ N we have Rist Gω (n) ′ ≤ H. Using the notation of Proposition 3.4, let Proposition 3.4, where N σ n+1 ω is the normal closure of (ay σ n+1 ω ) 2 . Therefore (az σ n ω ) 2 , ψ −1 (e, (ay σ n+1 ω ay σ n+1 ω ) g ) ∈ N ′ σ n ω = ϕ v (Rist Gω (v) ′ ) for any element g ∈ G σ n+1 ω and nth-level vertex v. In order to deduce that H is a congruence subgroup, by Proposition 3.6 it suffices to show that ψ…”

Coxeter groups as Beauville groups

“…as well as the more difficult question of which Beauville groups are strongly real Beauville groups. Many authors have investigated these questions for several classes of finite groups including abelian groups [7]; symmetric groups [4,17] as well as decorations of simple groups more generally [11,18,13,14,19,20]; characteristically simple groups [8,10,24,25] and nilpotent groups [1,2,3,8,16,27] (this list of references is by no means exhaustive!) Here we extend this list by considering Coxeter groups.…”