Surface braid groups, Finite Heisenberg covers and double Kodaira fibrations

Diagonal double Kodaira structures on finite groups

“…• the construction of diagonal double Kodaira structures on extra-special p-groups of any exponent, see Theorems 2.7 and 2.10. This extends the equivalent statements for extra-special p-groups of exponent p proved in [CaPol19]; • an explicit upper bound for the slope of a diagonal double Kodaira fibration, see Proposition 3.12 and Remark 3.13.…”

“…For more details on the basic definitions and results of this section, we refer the reader to the Introduction and to [CaPol19], especially Sections 1 and 3. Recall that a Kodaira fibration is a smooth, connected holomorphic fibration f 1 : S −→ B 1 , where S is a compact complex surface and B 1 is a compact complex curve, which is not isotrivial.…”

“…In [CaPol19] the author (in collaboration with A. Causin) introduced a new topological method to construct double Kodaira fibrations, based on the so-called Heisenberg covers of Σ b × Σ b , where Σ b denotes a real, closed, connected, orientable surface of genus b (from now on, we will simply write "a real surface of genus b"). These are finite Galois covers S −→ Σ b × Σ b , whose branch locus is the diagonal ∆ ⊂ Σ b × Σ b and whose Galois group is isomorphic to a finite Heisenberg group.…”

“…These are finite Galois covers S −→ Σ b × Σ b , whose branch locus is the diagonal ∆ ⊂ Σ b × Σ b and whose Galois group is isomorphic to a finite Heisenberg group. In this note we rephrase the group-cohomological methods of [CaPol19] in a purely algebraic way, by introducing the so-called diagonal double Kodaira structures on a finite group G, see Definition 1.1. These are special presentations of G, whose existence is equivalent to the fact that G is a good quotient of some higher genus pure braid group on two strands, where "good" means that the natural braid called A 12 in [GG04] has non-trivial image under the quotient map.…”

“…With this new and compact terminology, we give a short account of some of the main results contained in [CaPol19], namely…”