### Abstract

In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric extension of this result to higher dimensions. We show that in terms of the number of facets, there are only exponentially many geometric triangulations of space forms with bounded geometry in the sense of Cheeger (curvature and volume bounded below, and diameter bounded above). This establishes a combinatorial version of Cheeger’s finiteness theorem. Further consequences of our work are: (1) there are exponentially many geometric triangulations of S^{d}; (2) there are exponentially many convex triangulations of the d-ball.

Original language | English (US) |
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Pages (from-to) | 233-247 |

Number of pages | 15 |

Journal | Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - 2020 |

Externally published | Yes |

### Keywords

- Bounded geometry
- Collapsibility
- Discrete finiteness Cheeger theorem
- Discrete quantum gravity
- Geometric manifolds
- Simple homotopy theory
- Triangulations

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Algebra and Number Theory
- Statistics and Probability
- Geometry and Topology
- Discrete Mathematics and Combinatorics