The large scale topology and algebraic \(K\)-theory of arithmetic groups. (English) Zbl 1092.22008
A symmetric space is a simply connected Riemannian manifold \(X\)of nonpositive sectional curvature such that for each point \(x\in X\) the geodesic symmetry \(s_{x}:X\rightarrow X\) given by \(\exp _{x}(v)\mapsto \exp _{x}(-v)\) for all \(v\in T_{x}(X) \) is an isometry of \(X\). If \(X\) is noncompact and it has no Euclidean space as a Riemannian, then one says that \(X\) is of noncompact type. If \(G\) is a semisimple Lie group with finite center and no compact factors, and if \(H\) is a maximal compact subgroup, then the homogeneous space \(G/H\) is a symmetric space of noncompact type. This paper is devoted to provide an outline of the proof of the following result: Given the symmetric space \(X\) associated with a split rank algebraic group \(G\), there is an embedding of \(X\) in a space \(X^{\ast }\) such that:
1. \(X^{\ast }\) is compact and Hausdorff,
2. \(X\) is an open dense subset of \(X^{\ast }\),
3. \(X^{\ast }\) is acyclic,
4. the isometries of \(X\) extend to continuous maps of \(X^{\ast }\),
5. there are continuous equivariant maps from \(X^{\ast }\) to other compactifications of \(X\).
The author uses the \(K\)-theory of the Satake compactification \(X^{S}\) [S. Zucker, Comment. Math. Helv. 58, 312–343 (1983; Zbl 0565.22009)] and the Borel-Serre enlargement \(X_{\mathbb{Q}}^{BS}\) [A. Borel and J.-P. Serre, Comment. Math. Helv. 48, 436–491 (1973; Zbl 0274.22011)] to obtain a proof of the integral Novikov conjecture [S. C. Ferry, A. Ranicki and J. Rosenberg (eds.), Novikov conjectures, index theorems and rigidity, (Cambridge Univ. Press, London) (1995; Zbl 0829.00027; Zbl 0829.00028)] for arithmetic groups.
1. \(X^{\ast }\) is compact and Hausdorff,
2. \(X\) is an open dense subset of \(X^{\ast }\),
3. \(X^{\ast }\) is acyclic,
4. the isometries of \(X\) extend to continuous maps of \(X^{\ast }\),
5. there are continuous equivariant maps from \(X^{\ast }\) to other compactifications of \(X\).
The author uses the \(K\)-theory of the Satake compactification \(X^{S}\) [S. Zucker, Comment. Math. Helv. 58, 312–343 (1983; Zbl 0565.22009)] and the Borel-Serre enlargement \(X_{\mathbb{Q}}^{BS}\) [A. Borel and J.-P. Serre, Comment. Math. Helv. 48, 436–491 (1973; Zbl 0274.22011)] to obtain a proof of the integral Novikov conjecture [S. C. Ferry, A. Ranicki and J. Rosenberg (eds.), Novikov conjectures, index theorems and rigidity, (Cambridge Univ. Press, London) (1995; Zbl 0829.00027; Zbl 0829.00028)] for arithmetic groups.
Reviewer: Juan Antonio Pérez (Zacatecas)
MSC:
| 22E40 | Discrete subgroups of Lie groups |
| 20F65 | Geometric group theory |
| 53C35 | Differential geometry of symmetric spaces |
| 11F06 | Structure of modular groups and generalizations; arithmetic groups |
| 18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |
| 54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |
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