The Baum-Connes conjecture for hyperbolic groups

By Igor Mineyev, Guoliang Yu. 2001

134 citations recorded by the source corpus.

Abstract.We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/db0ac2eafda6c9ab262b97a6fbcf05c2a0dcfc50

How researchers see this paper

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

In Geometry and Topology

1. W. Lück
2. Arthur Bartels
3. H. Reich
4. D. Groves
5. J. Manning
6. Thomas Schick
7. Igor Mineyev
8. P. Piazza

In Mathematical Physics

1. W. Lück
2. H. Reich
3. Arthur Bartels
4. D. Groves
5. J. Manning
6. Thomas Schick
7. Igor Mineyev
8. Vincent Lafforgue

Citations in the corpus

• Some geometric groups with Rapid Decay (2003, background): “Word hyperbolic groups were shown to satifsy Baum-Conne in full generality in [23].”
• Introduction To The Baum-Connes Conjecture (2002, background): “Yu in [65] that every subgroup of Gromov hyperbolic group admits a proper, cocompact, isometric action on a strongly bolic metric space, hence satisfies Conjecture 1.”
• The Kadison-Kaplansky conjecture for word-hyperbolic groups (2002, background): “Recently Mineyev-Yu [ MY ] and Lafforgue (unpublished) proved the Baum-Connes conjecture for word-hyperbolic groups.”
• The Borel Conjecture for hyperbolic and CAT(0)-groups (2009, background): “The latter depends ultimately on work of Mineyev and Yu [43, 44].”
• Non-Hausdorff groupoids, proper actions and $K$-theory (2004, background): “o [25, 27]). When G is a group, the map µis known to be an isomorphism in many cases, such as almost connected groups, linear p-adic groups [7], a-T-menable groups [10] and hyperbolic discrete groups [22]. For groupoids, the situation is, unexpectedly, somewhat different: Higson, Lafforgue and Skandal”
• Banach KK-theory and the Baum-Connes conjecture (2003, background): “They are morally the same, and in particular they respectively contain all closed subgroups of reductive Lie groups, and all hyperbolic groups (for general hyperbolic groups see [42], and [37] for a slightly different approach).”
• Dehn filling in relatively hyperbolic groups (2006, background): “Applications of Mineyev’s construction are myriad (see, for example, [22, 23, 24, 25, 26, 33]).”
• Flows and joins of metric spaces (2005, methodology): “We will moreover extend the construction in [31] in two ways to provide a metric d̂ = d̂X which is more general in the following sense: • d̂ is defined on all of X , rather than just on a discrete group Γ.”
• Bordism, rho-invariants and the Baum-Connes conjecture (2004, background): “all Gromov hyperbolic groups [50] or cocompact discrete subgroups of Sl(3,C) [29].”
• On the Farrell–Jones Conjecture and its applications (2007, background): “We mention that subgroups of word-hyperbolic groups satisfy the Baum-Connes Conjecture by a result of Mineyev-Yu [ 48, Theorem 20 ] based on the work of Laorgue [37] (see also [61]).”