Permanence properties of the Baum-Connes conjecture

By J. Chabert, S. Echterhoff. Documenta Mathematica, 2001

93 citations recorded by the source corpus.

In this paper we study the stability of the Baum-Connes conjecture with coe-cients under various natural operations on the groups. We show that the class of groups satisfying this conjecture is stable under taking subgroups, Cartesian products, and more gen- erally, under certain group extensions. In particular, we show that a group satisfles the conjecture if it has an amenable normal subgroup such that the associated quotient group satisfles the conjecture. We also study a natural induction homomorphism between the topolog- ical K-theory of a subgroup H of G and the topological K-theory of G with induced coe-cient algebra, and show that this map is always bijective. Using this, we are also able to present new examples of groups which satisfy the conjecture with trivial coe-cients.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/53aeabec05b4b5e68849abe4144c9b7c3a83630c

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In Applied Mathematics

1. J. Tu
2. R. Nest
3. S. Echterhoff
4. Vincent Lafforgue
5. J. Chabert
6. W. Lück
7. Guoliang Yu
8. G. Skandalis

In Mathematical Physics

1. W. Lück
2. J. Tu
3. S. Echterhoff
4. R. Nest
5. Vincent Lafforgue
6. Guoliang Yu
7. J. Chabert
8. R. Meyer

Citations in the corpus

• Banach KK-theory and the Baum-Connes conjecture (2003, background): “If the Baum-Connes conjecture with coefficients is true for a group, it is true also for all its closed subgroups; the Baum-Connes conjecture with coefficients is also stable under various kinds of extensions (Chabert [7], Chabert-Echterhoff [8], Oyono [44], and Tu [51]).”
• 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”
• The Baum-Connes Conjecture via Localization of Categories (2003, background): “Therefore, the Baum–Connes conjecture holds for all coefficient algebras in 〈CI〉 because it holds for compactly induced coefficient algebras by [11].”
• The structure of crossed products of irrational rotation algebras by finite subgroups of SL2(ℤ) (2006, background): “14 of [13] imply that Z(2) ⋊H does too.”
• The coarse Baum–Connes conjecture and groupoids (2002, background): “If Z is a classifying space for proper actions, then there exists a Γ-equivariant homotopy f : [0, 1]×Z×Z → Z between the two projections Z ×Z → Z.”