Celebratio Mathematica

Topology ’90: Papers from the research semester in low-dimensional topology held at Ohio State University (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematical Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184397 Zbl 0747.00024 book

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W. D. Neumann: “Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic 3-manifolds,” pp. 243–271 in Topology ’90: Papers from the research semester in low-dimensional topology held at Ohio State University (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematics Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184415 Zbl 0768.57006 incollection

Boris Nikolaevich Apanasov	Related
Laurent C. Siebenmann	Related
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@incollection {key1184415m,
	AUTHOR = {Neumann, Walter D.},
	TITLE = {Combinatorics of triangulations and
		 the {C}hern--{S}imons invariant for
		 hyperbolic 3-manifolds},
	BOOKTITLE = {Topology '90: {P}apers from the research
		 semester in low-dimensional topology
		 held at {O}hio {S}tate {U}niversity},
	EDITOR = {Apanasov, Boris and Neumann, Walter
		 D. and Reid, Alan W. and Siebenmann,
		 Laurent},
	SERIES = {Ohio State University Mathematics Research
		 Institute Publications},
	NUMBER = {1},
	PUBLISHER = {de Gruyter},
	ADDRESS = {Berlin},
	YEAR = {1992},
	PAGES = {243--271},
	URL = {https://www.degruyter.com/document/doi/10.1515/9783110857726/pdf#page=255},
	NOTE = {(Columbus, OH, February--June 1990).
		 MR:1184415. Zbl:0768.57006.},
	ISSN = {0942-0363},
	ISBN = {9783110125986},
}

W. D. Neumann and A. W. Reid: “Arithmetic of hyperbolic manifolds,” pp. 273–310 in Topology ’90: Papers from the research semester in low-dimensional topology held at Ohio State University (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematics Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184416 Zbl 0777.57007 incollection

Boris Nikolaevich Apanasov	Related
Laurent C. Siebenmann	Related
Alan William Reid	Related

W. D. Neumann and A. W. Reid: “Notes on Adams’ small volume orbifolds,” pp. 311–314 in Topology ’90: Papers from the research semester in low-dimensional topology held at Ohio State University (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematics Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184417 Zbl 0773.57009 incollection

In [1992] Colin Adams shows there we just six non-compact orientable hyperbolic 3-orbifolds of volume less than \( v_0/4 \), where \( v_0 = 1.01494146\dots \) is the volume of a regular ideal tetrahedron in \( \mathbb{H}^3 \). We give various descriptions of them and their fundamental groups. In particular, we will see that they are all arithmetic, and we give explicit commensurabilities between some of them. Their fundamental groups can be described as symmetry groups of tesselations, so we will introduce terminology for this. Many of them am also “tetrahedral groups,” so we introduce notation for these too.

These orbifolds show that “cusp density” is not a commensurability invariant of one-cusped orbifolds, answering a question that has been asked by several people. As we discuss in Sect. 3, cusp density is a commensurability invariant for non-arithmetic one-cusped orbifolds.

Boris Nikolaevich Apanasov	Related
Laurent C. Siebenmann	Related
Alan William Reid	Related

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Walter D. Neumann

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