Geodesic
A Criterion of Smoothness at Infinity for an Arithmetic Quotient of the Future Tube
Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 1, pp. 40-59

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Let $\Gamma$ be an arithmetic group of affine automorphisms of the $n$-dimensional future tube $\mathcal{T}$. It is proved that the quotient space $\mathcal{T}\!/\Gamma$ is smooth at infinity if and only if the group $\Gamma$ is generated by reflections and the fundamental polyhedral cone (“Weyl chamber”) of the group $d\Gamma$ in the future cone is a simplicial cone (which is possible only for $n\le 10$). As a consequence of this result, a smoothness criterion for the Satake–Baily–Borel compactification of an arithmetic quotient of a symmetric domain of type IV is obtained.
Keywords: symmetric domain, future tube, boundary component, arithmetic quotient, reflection group, automorphic form. 
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È. B. Vinberg; O. V. Schwarzman. A Criterion of Smoothness at Infinity for an Arithmetic Quotient of the Future Tube. Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 1, pp. 40-59. http://geodesic.mathdoc.fr/item/FAA_2017_51_1_a3/