Rationality theorems for curvature invariants of 2-complexes

TL;DR

The paper proves that the curvature invariants $\rho_+(X)$, $\rho_-(X)$ (and the surface analogues $\sigma_+(X)$, $\sigma_-(X)$) for a finite 2-complex $X$ are rational and computable as extrema of explicit rational linear-programming problems. This is achieved by embedding the problem into a linear system on a rational cone $C(\mathbb{R})$ built from blocks encoding essential origami decompositions of branched 2-complexes, and by showing that the total and area curvatures are linear on this cone through the map $\Phi:\mathrm{Orig}_\Pi(X)\to C(\mathbb{Z})$. The framework unifies irreducible and surface curvature bounds via the same linear-programming paradigm and yields realizability: optimal curvature values are attained by actual branched 2-complexes mapping essentially to $X$. The results underpin a program linking curvature bounds to topological and group-theoretic properties (e.g., asphericity, coherence, and largeness) and connect to recent advances in geometric group theory that use rationality theorems to study subgroup structures and immersion properties of 2-complexes.

Abstract

Let $X$ be a finite, 2-dimensional cell complex. The curvature invariants $ρ_\pm(X)$ and $σ_\pm(X)$ were defined in [13], and a programme of conjectures was outlined. Here, we prove the foundational result that the quantities $ρ_\pm(X)$ and $σ_\pm(X)$ are the extrema of explicit rational linear-programming problems. As a result they are rational, realised, and can be computed algorithmically.

Figures (3)

• Figure 1: A Stallings fold $\Delta\to\Delta'$. Note that we do not in general insist that $v_1\neq v_2$.
• Figure 2: Given an essential origami $\Omega'$ on $\Delta'$, an essential fold $\Delta\to\Delta'$ as in Figure \ref{['fig: Fold']} induces an origami $\Omega$ on $\Delta$ with an edge graph $E_\Omega$ obtained by unfolding $E_{\Omega'}$, as illustrated. These unfolds are always essential.
• Figure 3: Under an essential fold as in Figure \ref{['fig: Fold']}, the vertex graph $V_\Omega$ is obtained by unfolding $V_{\Omega'}$, as illustrated. These unfolds are always essential.

Theorems & Definitions (73)

• Theorem A: Rationality theorem for irreducible curvatures
• Corollary B
• Theorem C: Rationality theorem for surface curvatures
• Corollary D
• Remark 1
• Theorem E
• Lemma 1.1: Stallings' folding lemma
• Definition 1.2: Fold
• Remark 1.3
• Definition 1.4: Cores and core graphs