Mapping class group dynamics and the holonomy of branched affine structures

Abstract

We classify, up to few exceptions, the orbit closures of the Mod( )-action on the affine character variety χ(Aff(C)). We obtain from this classification that the only obstruction for a non-abelian representation ρ : π 1 −→ Aff(C) to be the holonomy of a branched affine structure on is to be Euclidean and not to have positive volume, where is a closed oriented surface of genus g ≥ 2.

Holonomy of complex projective structures on surfaces with prescribed branch data

“…This answers a question raised in [CFG21, Question 1.3]. In fact Proposition 1.5 is due to a large extent to Ghazouani [Gha18], where he studies the action of the mapping class group on Hom(Γ, Aff(C))/Aff(C) and characterizes the representations of Hom(Γ, Aff(C)) that are the holonomy of an affine structure. Ghazouani does not address the problem of characterizing the representations that are holonomies of affine structures with prescribed branch points.…”

“…In this subsection we geometrize affine representations. An example of a branched projective structure with affine holonomy is an affine structure, as studied for example in [Gha18]. Definition 6.14.…”

Isoperiodic meromorphic forms: two simple poles

“…Proof of Theorem Proof of Theorem 1.10: Apply the Transfer Principle ( [11]) to transfer the properties of the action of the mapping class group on the image of Per to properties of the isoperiodic foliation. The analysis of the action of the mapping class group of Σ g on H 1 (Σ g , C/Z) can be found in [19,Proposition 4.2] where Ratner's Theory is used to provide the described classification of the closures of the orbits. Moore's Theorem allows to deduce the ergodicity property.…”

“…As an application, we compute, using Ratner's theory ( specifically [19]), the closed subsets of moduli spaces that are saturated/invariant by the isoperiodic foliation, and find that they are real analytic submanifolds. A qualitative difference with the case of holomorphic differentials is that some transcendental leaves are closed in these quasiprojective manifolds.…”

Translation surfaces and periods of meromorphic differentials

“…In particular, they showed that any non-elementary representation of the fundamental group of the surface to PSL(2, C) appears as the holonomy of some projective structure on a closed surface, if one allows one branch-point of degree two. The holonomy representations of branched affine structures have been studied in [Gha18]; note that such a representation has image in the affine group Aff(C) and is necessarily elementary. However, that paper is not concerned about the number and order of branch-points.…”