On Minimal Covolume Hyperbolic Lattices

Kellerhals, Ruth

doi:10.3390/math5030043

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2022

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Cited by 2 publications

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“…We assume that P as convex hull of finitely many ordinary or ideal points has at least one vertex on the ideal boundary ∂H n . These orbifolds form a very natural and important family of cusped hyperbolic space forms that include orbifolds of small volume in various dimensions up to n = 18 (see [12,13]).…”

Section: Introductionmentioning

confidence: 99%

Cusp Density and Commensurability of Non-arithmetic Hyperbolic Coxeter Orbifolds

Dotti

Drewitz

Kellerhals

2022

Discrete Comput Geom

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For three distinct infinite families $$(R_m)$$ ( R m ) , $$(S_m)$$ ( S m ) , and $$(T_m)$$ ( T m ) of non-arithmetic 1-cusped hyperbolic Coxeter 3-orbifolds, we prove incommensurability for a pair of elements $$X_k$$ X k and $$Y_l$$ Y l belonging to the same sequence and for most pairs belonging two different ones. We investigate this problem first by means of the Vinberg space and the Vinberg form, a quadratic space associated to each of the corresponding fundamental Coxeter prism groups, which allows us to deduce some partial results. The complete proof is based on the analytic behavior of another commensurability invariant. It is given by the cusp density, and we prove and exploit its strict monotonicity.

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Section: Introductionmentioning

confidence: 99%