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Gianluca Faraco

Publications and source records attributed to Gianluca Faraco.

At least 19 recordsLinked to original sources
arXiv2026

Branched real projective structures on surfaces and geometrisation of representations

We introduce and study branched real projective structures on compact surfaces. Our main result shows that every representation of the fundamental group of a compact surface into $PSL(3,\mathbb R)$ arises as the holonomy representation of a branched real projective structure, regardless of whether the surface is orientable. We also consider the realisation problem with prescribed branching data, relating the branching degree to the second Stiefel--Whitney class of the representation. The results obtained in this direction suggest several interesting avenues for future research.

math.GT↗
arXiv2026

The dark side of translation surfaces: It is not all about dynamics

This survey offers a geometric and topological perspective on translation surfaces, with a focus on aspects that are often overshadowed by the dynamical viewpoint. After presenting several equivalent definitions of translation surfaces, particularly those based on polygonal models and fundamental domains, we explore the structure of their moduli spaces in the spirit of Thurston, including natural stratifications. The final part is devoted to the study of the period map, the realisation of representations as period characters and the fibres of the holonomy map. This approach highlights a foundational, yet sometimes underrated, facet of the theory.

math.GT↗
arXiv2026

On the symplectic geometry of branched hyperbolic surfaces in genus two

We construct analogues of Fenchel-Nielsen coordinates on an open and dense subset of the space of holonomies of branched hyperbolic structures on a closed genus-2 surface. We show that these coordinates satisfy an analogue of Wolpert's magic formula, and thus provide Darboux charts for the Goldman symplectic form. To this end, we revisit the parametrization of hyperbolic structures on a one-holed torus and describe a simple polygonal model that makes both length and twist parameters transparent. Gluing two such polygons leads to the notion of bow-tie representations of a genus-2 surface group. We prove that bow-tie representations account for most holonomies of branched hyperbolic structures, though not all: for example, Le Fils' pentagon representations form a real codimension-2 family of holonomies lying outside the bow-tie locus.

math.GT↗
arXiv2026

Mapping class group orbit closures for Deroin-Tholozan representations

We prove that infinite mapping class group orbits are dense in the character variety of Deroin-Tholozan representations. In other words, the action is minimal except for finite orbits. Our arguments rely on the symplectic structure of the character variety, emphasizing this geometric perspective over its algebraic properties.

math.DS↗
arXiv2025

Isoperiodic foliation of the stratum $Ω\mathcal{M}_1(1,1,-2)$

This paper describes the geometry and topology of leaves of the isoperiodic foliation of the stratum $Ω\mathcal{M}_1(1,1,-2)$. We prove that each leaf is a surface of infinite genus homeomorphic to the Loch Ness monster surface and supports a singular Euclidean structure whose singularities correspond to isoperiodic forms in the lower stratum $Ω\mathcal{M}_1(2,-2)$. Along the way we also characterize the possible groups arising as the Veech group of a leaf and give a description of the large-scale conformal geometry of the wall-and-chamber decomposition of the leaves.

math.GT↗
arXiv2023

On the distality and expansivity of certain maps on spheres

Any affine map on the (n+1)-dimensional Euclidean space gives rise to a natural map on the n-dimensional sphere whose dynamical aspects are not so well-studied in the literature. We explore the dynamical aspects of these maps by investigating about their distality and expansivity.

math.DS↗
arXiv2023

Counterexamples to the simple loop conjecture in higher-dimension

For every $g\ge 2$ and $n\ge4$, we provide an $n-$manifold $M$ and a continuous $2-$sided map $f\colon S\longrightarrow M$, where $S$ is a closed genus $g$ surface, such that no simple loop is contained in $\text{ker}(\,f_*\,)$. This provides a counterexample to the the classical simple loop conjecture for surfaces to manifolds of dimensions at least four.

math.GT↗
arXiv2023

Tessellations of surfaces

A tessellation or tiling is a collection of sets, called tiles, that cover a plane without gaps and overlaps. The present note is an invitation to get to know the beauty and majesty of tessellations and triangulation of orientable surfaces.

math.HO↗
arXiv2022

On the automorphism groups of certain branched structures on surfaces

We consider translation surfaces with poles on surfaces. We shall prove that any finite group appears as the automorphism group of some translation surface with poles. As a direct consequence we obtain the existence of structures achieving the maximal possible number of automorphisms allowed by their genus and we finally extend the same results to branched projective structures.

math.GT↗
arXiv2022

Modular orbits on the representation spaces of compact abelian Lie groups

Let $S$ be a closed surface of genus $g$ greater than zero. In the present paper we study the topological-dynamical action of the mapping class group on the $\Bbb T^n$-character variety giving necessary and sufficient conditions for Mod$(S)$-orbits to be dense. As an application, such a characterisation provides a dynamical proof of the Kronecker's Theorem concerning inhomogeneous diophantine approximation.

math.DS↗
arXiv2021

Translation surfaces and periods of meromorphic differentials

Let $S$ be an oriented surface of genus $g$ and $n$ punctures. The periods of any meromorphic differential on $S$, with respect to a choice of complex structure, determine a representation $χ:Γ_{g,n} \to\mathbb C$ where $Γ_{g,n}$ is the first homology group of $S$. We characterize the representations that thus arise, that is, lie in the image of the period map $\textsf{Per}:Ω\mathcal{M}_{g,n}\to \textsf{Hom}(Γ_{g,n},\mathbb{C})$. This generalizes a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on $S$ with the prescribed holonomy $χ$. Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.

math.GT↗
arXiv2021

Monodromy of Schwarzian equations with regular singularities

Let $S$ be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations $ρ:π_1(S) \to \text{PSL}_2(\mathbb{C})$ that arise as the monodromy of the Schwarzian equation on $S$ with regular singularities at the punctures. Equivalently, we determine the holonomy representations of complex projective structures on $S$, whose Schwarzian derivatives (with respect to some uniformizing structure) have poles of order at most two at the punctures. Following earlier work that dealt with the case when there are no apparent singularities, our proof reduces to the case of realizing a degenerate representation with apparent singularities. This mainly involves explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy. As a corollary, we determine the representations that arise as the holonomy of spherical metrics on $S$ with cone-points at the punctures.

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arXiv2019

Complex projective structures with maximal number of Möbius transformations

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Bely\uı curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms.

math.GT↗
arXiv2019

Geometrization of purely hyperbolic representations in $\text{PSL}_2\Bbb R$

Let $S$ be a surface of genus $g$ at least $2$. A representation $ρ:π_1S\longrightarrow \text{PSL}_2\Bbb R$ is said to be purely hyperbolic if its image consists only of hyperbolic elements other than the identity. We may wonder under which conditions such representations arise as holonomy of a hyperbolic cone-structure on $S$. In this work we will characterize them completely, giving necessary and sufficient conditions.

math.GT↗
arXiv2018

Distances on the moduli space of complex projective structures

Let $S$ be a closed and oriented surface of genus $g$ at least $2$. In this (mostly expository) article, the object of study is the space $\mathcal{P}(S)$ of marked isomorphism classes of projective structures on $S$. We show that $\mathcal{P}(S)$, endowed with the canonical complex structure, carries exotic hermitian structures that extend the classical ones on the Teichmüller space $\mathcal{T}(S)$ of $S$. We shall notice also that the Kobayashi and Carathéodory pseudodistances, which can be defined for any complex manifold, can not be upgraded to a distance. We finally show that $\mathcal{P}(S)$ does not carry any Bergman pseudometric.

math.CV↗