Chewing gums, snakes and candle cakes

TL;DR

This paper explores higher Teichmuller spaces of Riemann surfaces with boundaries, illustrating key ideas through explicit examples, and discusses the geometric and combinatorial structures involved.

Contribution

It provides a concrete, computational perspective on the construction of bordered cusped Teichmuller spaces and their relation to classical and Fock-Goncharov frameworks.

Findings

The bordered cusped Teichmuller space arises as a limit of classical spaces via the chewing-gum move.

The chewing-gum move is the inverse of amalgamation in the Fock-Goncharov setting.

Explicit examples illustrate the underlying geometric and combinatorial structures.

Abstract

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.