By Armand Borel

Introduces uniform buildings of lots of the identified compactifications of symmetric and in the neighborhood symmetric areas, with emphasis on their geometric and topological constructions fairly self-contained reference aimed toward graduate scholars and examine mathematicians drawn to the functions of Lie concept and illustration idea to research, quantity thought, algebraic geometry and algebraic topology

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**Compactifications of Symmetric and Locally Symmetric Spaces (Mathematics: Theory & Applications)**

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**Extra info for Compactifications of Symmetric and Locally Symmetric Spaces (Mathematics: Theory & Applications)**

**Example text**

T→+∞ It is clearly an equivalence relation. Denote the set of equivalence classes of geodesics by X (∞), X (∞) = {all geodesics in X }/ ∼ . For any geodesic γ , its equivalence class is denoted by [γ ]. 3 The set X (∞) can be canonically identiﬁed with the unit sphere in the tangent space Tx X at any basepoint x, in particular, the unit sphere in p = Tx0 X . Proof. Clearly, the set of geodesics passing through the basepoint x is parametrized by the unit sphere in Tx X . Since X is simply connected and nonpositively curved, comparison with the Euclidean space implies that any two such different geodesics are not equivalent; and it remains to show that every equivalence class contains a geodesic passing through x.

3 also holds. 2. 4 The sphere X (∞) can be attached at the inﬁnity of X to deﬁne the geodesic compactiﬁcation X ∪ X (∞). 8 below for details on how to deﬁne a topology in terms of convergent sequences). For any [γ ] ∈ X (∞), an unbounded sequence y j in X converges to [γ ] if the geodesic from a basepoint x to y j converges to a geodesic in [γ ]. Comparison with the Euclidean space shows that this topology is independent of the choice of the basepoint x. For convenience, we will use the basepoint x0 = K .

For thorough discussions of parabolic subgroups, see [Wal1–2] [War] [Va]. This section is organized as follows. 14), which plays a basic role in relating the geometry of symmetric spaces to parabolic subgroups. 23). Such relative decompositions are important for the purpose of understanding the topology of the boundary components of compactiﬁcations of symmetric spaces, for example, relations between the boundary components of a pair of parabolic subgroups P, Q satisfying P ⊂ Q. 1 To motivate parabolic subgroups and the induced horospherical decompositions, we consider the example of G = SL(2, R).