The Baum–Connes conjecture: an extended survey

Aparicio, Maria Paula Gomez; Julg, Pierre; Valette, Alain

doi:10.1007/978-3-030-29597-4_3

Cited by 20 publications

(14 citation statements)

References 114 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…). The injectivity of µ red is the usual strong Novikov conjecture, which has been verified for a large class of groups [2,5,11,13,16,19,25,26,27].…”

Section: 3mentioning

confidence: 79%

$K$-theory of relative group $C^*$-algebras and the relative Novikov conjecture

Deng¹,

Tian²,

Xie³

et al. 2021

Preprint

View full text Add to dashboard Cite

The relative Novikov conjecture states that the relative higher signatures of manifolds with boundary are invariant under orientation-preserving homotopy equivalences of pairs. In this paper, we study the relative Baum-Connes assembly map for any pair of groups and apply it to solve the relative Novikov conjecture when the groups satisfy certain geometric conditions. Contents 1. Introduction 2. The relative Novikov conjecture 2.1. Roe algebras and localization algebras 2.2. Rips complex 2.3. Relative Roe algebras and relative localization algebras 3. Relative Baum-Connes conjecture 3.1. Relative reduced assembly map 3.2. Relative Baum-Connes conjecture for hyperbolic groups 4. A relative Bott Periodicity 4.1. C * -algebras associated with Hilbert spaces 4.2. C * -algebra with group actions 5. The proofs of the main results 5.1. The maximal strong relative Novikov conjecture 5.2. The reduced strong relative Novikov conjecture 6. Applications to the relative Novikov conjecture References

show abstract

“…). The injectivity of µ red is the usual strong Novikov conjecture, which has been verified for a large class of groups [2,5,11,13,16,19,25,26,27].…”

Section: 3mentioning

confidence: 79%

$K$-theory of relative group $C^*$-algebras and the relative Novikov conjecture

Deng¹,

Tian²,

Xie³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The picture of the unitary dual is rather different between the case of SO 0 (n, 1) and SU(n, 1), which have the Haagerup property (or Gromov's a-T-menability), and the case of Sp(n, 1) and F 4(−20) which have Kazhdan's property T (see [6,Chapter 5]). The complementary series (i.e.…”

Section: Introductionmentioning

confidence: 99%

Slow exponential growth representations of Sp(n, 1) at the edge of Cowling's strip

Julg¹,

Nishikawa²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We obtain a slow exponential growth estimate for the spherical principal series representation ρs of Lie group Sp(n, 1) at the edge (Re(s) = 1) of Cowling's strip (|Re(s)| < 1) on the Sobolev space H α (G/P) when α is the critical value Q/2 = 2n+1. As a corollary, we obtain a slow exponential growth estimate for the homotopy ρs (s ∈ [0, 1]) of the spherical principal series which is required for the first author's program for proving the Baum-Connes conjecture with coefficients for Sp(n, 1).

show abstract

“…For discrete groups, as no such classification exists, other approaches were needed and led to the development of very powerful techniques. For an account of the history of the conjecture and the recent developments, we refer to the survey [GAJV19] and the references therein, as well as to the books [Val02,MV03].…”

Section: Introductionmentioning

confidence: 99%

K-homology and K-theory of pure Braid groups

Azzali¹,

Browne²,

Aparicio³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We produce an explicit description of the K-theory and K-homology of the pure braid group on n strands. We describe the Baum-Connes correspondence between the generators of the left-and right-hand sides for n = 4. Using functoriality of the assembly map and direct computations, we recover Oyono-Oyono's result on the Baum-Connes conjecture for pure braid groups [OO01]. We also discuss the case of the full braid group B3.

show abstract

The Baum–Connes conjecture: an extended survey

Cited by 20 publications

References 114 publications

$K$-theory of relative group $C^*$-algebras and the relative Novikov conjecture

$K$-theory of relative group $C^*$-algebras and the relative Novikov conjecture

Slow exponential growth representations of Sp(n, 1) at the edge of Cowling's strip

K-homology and K-theory of pure Braid groups

Contact Info

Product

Resources

About