Noncompact surfaces, triangulations and rigidity

TL;DR

This work addresses rigidity for triangulations of noncompact surfaces by proving every such surface admits a $(3,6)$-tight triangulation that is minimally $3$-rigid, using a constructive toolkit of vertex-splitting, Henneberg $0$-extensions, and joins with low-genus bordered surfaces. It introduces model surfaces $S_eta$ built as infinite connected sums along trees, links invariants like the ideal boundary $eta(S)$ to Kerékjártó’s classification, and provides a complementary nonconstructive girth-inequality framework for $(3,6)$-tight triangulations on compact bordered surfaces. A key contribution is the explicit construction scheme ensuring the resulting graphs are countable unions of finite $(3,6)$-tight graphs, with direct implications for the generic rigidity of countable bar-joint frameworks in $oldsymbol{ m R^3}$. The paper also outlines broader directions, including minimally $3$-rigid realizations on punctured compact surfaces and extensions to rigidity under non-Euclidean norms, highlighting the interplay between topological ends, combinatorial sparsity, and geometric rigidity.

Abstract

Every noncompact surface is shown to have a (3,6)-tight triangulation, and applications are given to the generic rigidity of countable bar-joint frameworks in R^3. In particular, every noncompact surface has a (3,6)-tight triangulation that is minimally 3-rigid. A simplification of Richards' proof of Kerékjártó's classification of noncompact surfaces is also given.

Figures (9)

• Figure 1: (i) A subtree $T$ of $T_{\rm bin}$ with vertices $v_s$ with digit symbol $s$. (ii) An associated connected sum surface $S_\gamma$.
• Figure 2: Depictions of the branching surface $B={\mathbb{S}}_0\backslash 3{\mathbb{D}}$.
• Figure 3: (i) A Hennenberg 0-extension move, adding a degree 3 vertex and 3 incident edges, creating cycles of length 5 and 6. (ii) Further 0-extension moves giving a 5-cycle that is disjoint from the 8-cycle and the 6-cycle.
• Figure 4: $(3,6)$-tight triangulations of the bordered surfaces (i) $B= {\mathbb{P}}\backslash {\mathbb{D}}$ with $g_r(B)=1/2$, (ii) $B= {\mathbb{S}}_1\backslash {\mathbb{D}}$ with $g_r(B)=1$.
• Figure 5: The case $B={\mathbb{P}}\backslash 2{\mathbb{D}}$ for which a triangulation $H$ is required, with $|a|=|d|$ and $|b|=|a|+3$ so that the join of $G$ and $H$ is $(3,6)$-tight.

Theorems & Definitions (36)

• Example 2.1
• Theorem 2.2
• proof : Sketch proof
• Corollary 2.3
• proof
• Remark 2.4
• Definition 3.1
• Theorem 3.2
• proof
• Theorem 3.3