Topology and geometry of flagness and beltness of simple orbifolds
Zhi Lü, Lisu Wu
TL;DR
This work extends the framework of flagness and belt theory from simple polytopes to the broader class of simple orbifolds, focusing on simple handlebodies. It proves that orbifold-asphericity for a simple handlebody $Q$ is equivalent to being flag, and that a rank-two free abelian subgroup in the orbifold fundamental group $ ext{π}^{ ext{orb}}_1(Q)$ exists precisely when $Q$ contains a $oxed{ ext{square}}$-belt, establishing a Combinatorial Sphere Theorem and a Combinatorial Flat Torus Theorem in this setting. Using the Davis–D2 basic construction, nonpositive curvature arguments, and HNN-extension calculus, the authors derive curvature characterizations for manifold doubles $M_Q$ and connect 3D cases to right-angled hyperbolic structures via a Pogorelov-style combinatorial description. These results yield combinatorial criteria for various geometric structures (hyperbolic, CAT(0), scalar curvature) on covers and doubles of simple handlebodies, and provide a unified approach to hyperbolization questions in the right-angled Coxeter orbifold framework. The work thus links orbifold topology, cubical geometry, and curvature theory through explicit combinatorial data of belts and nerves.
Abstract
We consider a class of right-angled Coxeter orbifolds, named as simple orbifolds, which are a generalization of simple polytopes. Similarly to manifolds over simple polytopes, the topology and geometry of manifolds over simple orbifolds are closely related to the combinatorics and orbifold structure of simple orbifolds. We generalize the notions of flag and belt in the setting of simple polytopes into the setting of simple orbifolds. To describe the topology and geometry of a simple orbifold in terms of its combinatorics, we focus on {\em simple handlebodies} (that is, simple orbifolds which can be obtained from simple polytopes by gluing some disjoint specific codimension-one faces). We prove the following two main results in terms of combinatorics, which can be understood as "Combinatorial Sphere Theorem" and "Combinatorial Flat Torus Theorem" on simple handlebodies: (A) A simple handlebody is orbifold-aspherical if and only if it is flag. (B) There exists a rank-two free abelian subgroup in $π_1^{orb}(Q)$ of an orbifold-aspherical simple handlebody $Q$ if and only if it contains an $\square$-belt. Furthermore, based on such two results and some results of geometry, it is shown that the existence of some curvatures on a certain manifold cover (manifold double) over a simple handlebody $Q$ can be characterized in terms of the combinatorics of $Q$. In 3-dimensional case, together with the theory of hyperbolic 3-manifolds, we can induce a pure combinatorial equivalent description for a simple $3$-handlebody to admit a right-angled hyperbolic structure, which is a natural generalization of Pogorelov Theorem.