Core Surfaces

TL;DR

This paper introduces 'core surfaces', a combinatorial 2D complex structure representing subgroups of surface groups, extending the analogy of Stallings core graphs from free groups to surface groups.

Contribution

It develops a new combinatorial framework called core surfaces for analyzing subgroups of surface groups, analogous to Stallings core graphs for free groups.

Findings

Core surfaces provide a compact representation for finitely generated subgroups.

The theory generalizes Stallings core graphs to surface groups.

This framework facilitates subgroup analysis in surface groups.

Abstract

Let $\Gamma_g$ be the fundamental group of a closed connected orientable surface of genus $g\geq2$. We introduce a combinatorial structure of "core surfaces", that represent subgroups of $\Gamma_g$. These structures are (usually) 2-dimensional complexes, made up of vertices, labeled oriented edges, and $4g$-gons. They are compact whenever the corresponding subgroup is finitely generated. The theory of core surfaces that we initiate here is analogous to the influential and fruitful theory of Stallings core graphs for subgroups of free groups.