Topology and geometry of flagness and beltness of simple handlebodies

Lü, Zhi; Wu, Lisu

doi:10.2140/agt.2025.25.55

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Topology and geometry of flagness and beltness of simple handlebodies

Zhi Lü and Lisu Wu

Algebraic & Geometric Topology 25 (2025) 55–106

DOI: 10.2140/agt.2025.25.55

Abstract

We consider a class of right-angled Coxeter orbifolds, called simple handlebodies, which are a generalization of right-angled Coxeter simple polytopes. We generalize the notions of flag and belt in the setting of simple polytopes into the setting of simple handlebodies, and prove the following two topological properties characterized in terms of combinatorics: a simple handlebody is orbifold-aspherical if and only if it is flag; and the orbifold fundamental group of a simple handlebody contains a rank-two free abelian subgroup if and only if this simple handlebody contains an $□$ -belt. Furthermore, together with some results of geometry, it is shown that the existence of some curvatures on manifold double over a simple handlebody can be also characterized in terms of combinatorics.

Keywords

simple handlebody, flagness, belt, orbifold fundamental group, asphericity, hyperbolic structure, nonpositive curvature

Mathematical Subject Classification

Primary: 57R18

References

Publication

Received: 6 March 2022

Revised: 2 July 2023

Accepted: 22 September 2023

Published: 5 March 2025

Authors

Zhi Lü
School of Mathematical Sciences Fudan University Shanghai China

Lisu Wu
College of Mathematics and Sciences Shandong University of Science and Technology Qingdao China

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Volume 25, issue 1 (2025)