Groups acting properly on “bolic” spaces and the Novikov conjecture

By G. Kasparov, G. Skandalis. 2003

156 citations recorded by the source corpus.

We introduce a class of metric spaces which we call “bolic”. They include hyperbolic spaces, simply connected complete manifolds of nonpositive curvature, euclidean buildings, etc. We prove the Novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a “bolic”, weakly geodesic metric space of bounded geometry.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0e5133561cd22fdbf4e405e2893c75be58af98a3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Guoliang Yu
2. R. Nest
3. Vincent Lafforgue
4. Qin Wang
5. J. Tu
6. S. Echterhoff
7. J. Chabert
8. G. Gong

In Geometry and Topology

1. S. Echterhoff
2. Vincent Lafforgue
3. Qin Wang
4. J. Tu
5. R. Nest
6. H. Oyono-Oyono
7. G. Gong
8. Guoliang Yu

In Mathematical Physics

1. Guoliang Yu
2. S. Echterhoff
3. J. Chabert
4. J. Tu
5. Vincent Lafforgue
6. R. Nest
7. Qin Wang
8. Rufus Willett

Citations in the corpus

• Non-Hausdorff groupoids, proper actions and $K$-theory (2004, methodology): “However, since L2(G) is a C0(X0)-module and cuto functions are continuous on X0 and not on X, the method of [ 13, Theorem 5.4 ]”
• The Baum-Connes Conjecture via Localization of Categories (2003, unknown): “7 occurs in [30].”
• The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups (2001, methodology): “Relations to the K-theory of the maximal compact subgroup 25 References 28”
• Geometrization of the Strong Novikov Conjecture for residually finite groups (2008, background): “The Strong Novikov Conjecture holds for many infinite groups with property T [5, 6, 9, 10, 14, 15, 24, 26, 27].”
• Permanence properties of the Baum-Connes conjecture (2001, background): “The existence of universal proper spaces is shown in [17].”
• Going-down functors, the Künneth formula, and the Baum-Connes conjecture (2004, background): “…F n H (A) −→ F n H (J) ∂n −→ F n+1 H (A/J) −→ · · · If F is a Going-Down functor for G, we define F n (G) := lim Z⊆E(G) F n (C 0 (Z)), where E(G) denotes a locally compact universal example for the proper actions of G, as introduced in [25], and where Z runs through the G-compact subsets of E(G).”
• Banach KK-theory and the Baum-Connes conjecture (2003, background): “…groups acting properly isometrically on a discrete space with good properties at infinity (weakly geodesic, uniformly locally finite, and "bolic" [30]), and in particular hyperbolic groups (i.e., word-hyperbolic in the sense of Gromov), d) groups acting amenably on a compact space ([17]).”
• The Novikov conjecture and geometry of Banach spaces (2012, unknown): “5 in [15], we can show that there exists a Γ-Hilbert space H∆ satisfying the conditions of Definition 3.”