A Criterion of Smoothness at Infinity for an Arithmetic Quotient of the Future Tube

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.