On the commensurability of hyperbolic Coxeter groups

Dotti, Edoardo

doi:10.48550/arxiv.2101.10024

Cited by 1 publication

(7 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The coefficients of G off the diagonal have the following geometric meaning. (3,4), (2,5), (3,5), (4,5) k l ∞ ∞ Fig. 2 The four non-arithmetic Coxeter pyramids with exactly one ideal vertex in H 3…”

Section: Hyperbolic Coxeter Groups and Coxeter Orbifoldsmentioning

confidence: 99%

“…For n ≥ 3, the field K ( ) is the smallest field of definition for , and it is moreover an algebraic number field coinciding with the adjoint trace field of . As a consequence, the Vinberg field is a commensurability invariant for (see [4,Sect. 3]).…”

Section: Hyperbolic Coxeter Groups and Coxeter Orbifoldsmentioning

confidence: 99%

“…To do this we follow two different paths. The first one is algebraic and based on the study of Vinberg spaces and the relevant results of the first author [3,4]. We shall see that this approach has limitations.…”

Section: Example 22mentioning

confidence: 99%

“…4. The subsequent machinery is due to Vinberg, and the new results about commensurability based on it are due to the first author (see [3,4] and the references therein).…”

Section: The Vinberg Space and Commensurabilitymentioning

confidence: 99%

“…1. These necessary conditions rely upon the Vinberg space and the Vinberg form related to an arbitrary hyperbolic Coxeter group, and they were recently established by the first author [3,4]. This investigation leads to first yet incomplete conclusions.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Cusp Density and Commensurability of Non-arithmetic Hyperbolic Coxeter Orbifolds

Dotti

Drewitz

Kellerhals

2022

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Hyperbolic Coxeter Groups and Coxeter Orbifoldsmentioning

confidence: 99%

Section: Hyperbolic Coxeter Groups and Coxeter Orbifoldsmentioning

confidence: 99%

Section: Example 22mentioning

confidence: 99%

“…4. The subsequent machinery is due to Vinberg, and the new results about commensurability based on it are due to the first author (see [3,4] and the references therein).…”

Section: The Vinberg Space and Commensurabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations