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By Robin Hartshorne, C. Musili
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Such a metric is referred to as an invariant metric and in this metric, the manifold ???? is complete. J. Getz and M. 1007/978-3-0348-0351-9_4, © Springer Basel 2012 41 42 Chapter 4. Review of Arithmetic Quotients A locally symmetric space or arithmetic quotient is the quotient Γ∖???? by an arithmetic subgroup Γ ⊂ ????(ℚ). It is Hermitian if the symmetric space ???? is Hermitian, that is, if it carries a ????(ℝ)-invariant complex structure. If Γ is torsionfree then Γ∖???? is a smooth manifold, otherwise it is an orbifold.
Commutative diagrams were typeset using Paul Taylor’s diagrams package. The body of the book was typeset with LATEX2????. , [Hud]) that a closed convex linear cell is the convex hull of ﬁnitely many points in Euclidean space. A convex linear cell complex ???? is a ﬁnite collection of closed convex linear cells in some ℝ???? such that if ???? ∈ ???? then every face of ???? is in ????, and if ????, ???? ∈ ???? then the intersection ???? ∩ ???? is in ????. The underlying closed subset of Euclidean space is denoted ∣????∣. Such a complex is a regular cell complex, meaning that each (closed) cell is homeomorphic to a closed ball: no identiﬁcations occur on its boundary.
Let ???? = ∣????∣ and let ???? = ∣????∣ − ∣????∣. Although ???? is not a union of cells, it is a union of interiors of cells. We refer to this decomposition of ???? as a pseudo cell decomposition (or a pseudo-triangulation if ???? is a simplicial complex). Every cell ???? ∈ ???? has two orientations. A choice of orientation for ???? determines a unique orientation for each codimension 1 face ???? < ???? such that the orientation of ???? followed by the inward pointing vector − ????→ ???? agrees with the orientation of ????. The complex ???? is purely ????-dimensional if every cell is the face of some ????-dimensional cell and there are no cells of dimension greater than ????.