Geometric surgeries of three-dimensional flag structures and non-uniformizable examples

TL;DR

The paper develops a higher-rank geometric surgery for three-dimensional flag structures modeled on the flag space $\mathbf{X}$ under $\mathrm{PGL}_{3}(\mathbb{R})$, enabling construction of new examples by gluing along genus-two handlebodies tied to $\alpha$-$\beta$ bouquets. It proves a main existence theorem for flag surgeries, and shows that surgeries on Kleinian and Schottky flag manifolds can yield both Kleinian and non-Kleinian outcomes, with the latter illustrated through deformations and genus-three coverings that force the developing map to be surjective onto $\mathbf{X}$. The work thereby extends Ehresmann–Thurston-type deformation theory to higher rank geometries, providing explicit non-Kleinian closed flag structures and a framework for further exploration of higher-dimensional generalizations. It also clarifies the role of anti-flag involutions in the gluing process and situates flag surgeries within the broader Klein–Maskit paradigm for geometric structure combinations.

Abstract

In this paper, we introduce a notion of geometric surgery for flag structures, which are geometric structures locally modelled on the three-dimensional flag space under the action of ${\mathrm{PGL}}_3(\mathbb{R})$. Using such surgeries we provide examples of flag structures, of both uniformizable and non-uniformizable type.

Figures (2)

• Figure 3.1: The surgery $S$ along $B_M$ and $B_N$. We glue the two manifolds $M$ and $N$ through embeddings $j_M: M\setminus K_M\to S$ and $j_N: N\setminus K_N\to S$. The regions $K_N$ and $K_M$ are deleted and the green region $j_M(U_M\setminus K_M)\cap j_N(U_N\setminus K_N)$ contains the identifications defined by the surgery on a tubular neighborhood of a genus two surface.
• Figure 3.2: The green region is defined as the intersection of the handlebody $H_2$ and the complement of the handlebody $H_1$. It is invariant under the map $\varphi=g^{-1}\circ\kappa\circ g$. The $\alpha-\beta$ bouquet is contained in the handlebody $H_1$.

Theorems & Definitions (37)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof : Proof of Proposition \ref{['proposition-Liouville']}
• Definition 2.4
• Example 2.5