Weyl groups, lattices and geometric manifolds

Everitt, Brent; Howlett, Robert B.

doi:10.1007/s10711-009-9354-5

Cited by 1 publication

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References 27 publications

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“…The torsion in a Coxeter group is well understood (see for instance [11,12] and the references there), so as a corollary we will describe the torsion of Γ n 2 for 2 ≤ n ≤ 7. We will need a precise description of the torsion of Γ 6 2 in §7.…”

Section: Two Families Of Hyperbolic Coxeter Polytopesmentioning

confidence: 99%

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Right-angled Coxeter polytopes, hyperbolic six-manifolds, and a problem of Siegel

2011

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By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic −1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.The n-dimensional manifold Siegel problem. Determine the minimum possible volume obtained by an orientable hyperbolic n-manifold.All hyperbolic manifolds in this paper will be complete Riemannian manifolds of constant sectional curvature −1. The "full" Siegel problem refers to the problem above for orbifolds rather than manifolds. Nevertheless, the manifold Siegel problem is one with a long and venerable history. This paper describes our solution when n = 6. But first, an overview and some background. The Euler characteristic χ creates a big difference between even and odd dimensions. When n is even, the Gauss-Bonnet theorem gives vol(M ) = κ n χ(M ), with κ n = (−2π) n 2 /(n − 1)!! for the volume of an n-dimensional hyperbolic manifold M . As χ(M ) ∈ Z, the most obvious place to look for solutions to the problem is when |χ| = 1. A compact orientable M satisfies |χ(M )| ∈ 2Z, so the minimum volume is most likely achieved by a non-compact manifold. When n is odd, χ(M ) = 0, and a different approach must be found. For these reasons progress in even dimensions has been more rapid.

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Section: Two Families Of Hyperbolic Coxeter Polytopesmentioning

confidence: 99%

“…It is worth noting that the torsion-free subgroup of Γ 6 , with χ = −2, constructed in [12,Theorem 10 (ii)] also maps onto a Z/8 subgroup of Σ 6 .…”

Section: The Algebraic Interpretation Of Our Constructionmentioning

confidence: 99%