A combinatorial model for the moduli of bordered Riemann surfaces and a compactification

TL;DR

This paper introduces a combinatorial model for the moduli space of bordered Riemann surfaces, providing a new compactification method using symmetric metric ribbon graphs and sequences of subgraphs, with detailed results for genus zero.

Contribution

It presents a novel combinatorial construction of a moduli space and its compactification related to bordered Riemann surfaces, extending existing models with explicit graph-based methods.

Findings

Constructed a combinatorial moduli space using symmetric metric ribbon graphs.

Developed a compactification via sequences of non-contractible subgraphs.

Provided an orbi-cell decomposition for genus zero moduli space.

Abstract

We construct a combinatorial moduli space closely related to the KSV-compactification of the moduli space of bordered marked Riemann surfaces. The open part arises from symmetric metric ribbon graphs. The compactification is obtained by considering sequences of non contractible subgraphs. This leads to a partial real blow-up of rational cells that together form a compact orbi-cell space. For genus zero the constructed space gives an orbi-cell decomposition of the corresponding analytic moduli space decorated by real numbers and a compactification of this space. In higher genus the relation is more involved, as we briefly explain. The spaces we construct are of interest in their own right as they are constructed directly from an interesting class of graphs.