Benedetta Facciotti | Searcharxiv

arXiv2026

Chewing gums, snakes and candle cakes

The aim of these lecture notes, based on lectures given by the second author at the CIME school in Cetraro, is to illustrate a range of ideas surrounding higher Teichmuller spaces of Riemann surfaces with marked boundaries through explicit and computationally tractable examples. After reviewing the classical Teichmuller space of hyperbolic Riemann surfaces with boundary and its combinatorial description in terms of Thurston shear coordinates on a fat-graph, we explain how the bordered cusped Teichmuller space arises as a confluent limit when two boundary components in the Riemann surface collide via the so-called chewing-gum move giving rise to a candle cake. We then revisit these constructions from the Fock-Goncharov perspective, explaining snake calculus for transport matrices in PSL_n(R) and explain how the chewing gum move is the inverse of amalgamation. Rather than focusing on formal proofs, our goal is to illustrate the underlying theorems and constructions in a concrete and intuitive way.

math.GT↗

arXiv2026

Decorated Local Systems and Character Varieties

The focus of this paper is the study of the moduli space of representations of fundamental groupoids of surfaces $\Sigma$ with boundaries with values in $G:=GL_n(\mathbb C)$. In absence of marked points on the boundary, this moduli space is realized in many equivalent ways: as the moduli space of linear local systems on $\Sigma$, as the moduli space of representations of the fundamental groupoid $\Pi_1 (\Sigma)$, as the space of monodromy data and as character variety. By adding marked points to the boundary of $\Sigma$ in order to capture irregular singularities, the Betti moduli space has been generalized in several ways by many authors. Although it is clear that these different approaches describe essentially the same spaces of mathematical objects, exactly how they fit together has not yet been established. Motivated by the broader programme of establishing an explicit and conceptually coherent relationship between the existing approaches to the study of the decorated Betti moduli space, in this paper, we develop a categorical framework that allows for a systematic definition of the \dfn{decorated Betti moduli spaces} space, in the presence of higher order poles, designed to specialize to the different points of view encountered in the literature.

math.AG↗