## 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 | American English |
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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 |

### Bibliographical note

Funding Information:Acknowledgments. Karim Adiprasito acknowledges support by ISF Grant 1050/16, ERC StG 716424 - CASe and the Knut och Alice Wallenberg Foundation. Bruno Benedetti acknowledges support by NSF Grants 1600741 and 1855165, the DFG Collaborative Research Center TRR109, and the Swedish Research Council VR 2011-980. Part of this work was supported by NSF under grant No. DMS-1440140, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, in Fall 2017. Both authors are thankful to the anonymous referees and to Günter Ziegler for corrections, suggestions, and improvements.

Publisher Copyright:

© European Mathematical Society.

## Keywords

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