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Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
The Coarse Baum-Connes Conjecuture for
Relatively Hyperbolic Groups
Tomohiro Fukaya1 , Shin-ichi Oguni2
1 Department
of Mathematics
Kyoto University
2 Department
of Mathematics
Ehime University
The 4th GCOE International Symposium
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
What is Baum-Connes conjecture?
The Baum-Connes conjecture is a part of Connes’s
noncommutative geometry program.
It can be viewed as a conjectural generalization of the
Atiyah-Singer index theorem, to the equivariant setting.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Atiyah-Singer v.s. Baum-Connes
Atiyah-Singer index theorem says that the analytic index
on closed manifold can be computed by topological data.
t − index(DM ) = a − index(DM )
The Baum-Connes conjecture says that a purely
“TOPOLOGICAL” object coincides with a purely
“ANALYTIC” one.
∼
=
RK∗G (EG) −
→ K∗ (Cr∗ (G))
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Baum-Connes⇒??
(Injective part of )The Baum-Connes conjecture implies many
other conjectures.
The Novikov conjecture: the homotopy invariance of the
higher indexes.
The Gromov-Lawson conjecture: non-existence of the
positive scalar curvature on aspherical manifolds.
...
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Coarse version of the Baum-Connes conjecture
Coarse Baum-Connes conjecture is a Coarse Version of
the Baum-Connes conjecture.
Coarse Baum-Connes conjecture for countable group G
which admits finite classifying space BG implies the
injective part of the Baum-Connes conjecture for G.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
What is Coarse Geometry?
Coarse Geometry is a study of metric space from “large
scale point of view”.
Coarse Geometry has (at least) two origins.
Mostov’s rigidity theorem.
Gromov’s geometric group theory.
One of the most important class of metric spaces coming
from geometric group theory.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Coarse maps
Definition
Let X, Y are metric spaces. A map f : X → Y is coarse if the
following two conditions are satisfied:
1
B ⊂ Y is bounded ⇒ f −1 (B) is bounded.
2
∀R
> 0, ∃ S > 0 s.t. d(x, x0 ) < R ⇒ d(f (x), f (x0 )) < S.
Example
f : Z → Z.
f (n) = 1 does not satisfies (1).
f (n) = n2 does not satisfies (2).
∵ |f (n) − f (m)| = |n2 − m2 | = |n − m||n + m|.
f (n) = 2n + 1 is a coarse map.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Coarse equivalence
Definition
Two maps f , g : X → Y are close (f ' g) if
∃ C > 0 s.t. d(f (x), g(x)) ≤ C for all x ∈ X.
Definition
X and Y are coarsely equivalent if ∃ f : X → Y and ∃ g : Y → X s.t.
f and g are coarse maps.
g ◦ f ' idX and f ◦ g ' idY .
Example
Z and R are coarsely equivalent.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Finitely generated groups are metric spaces
Definition (Cayley graph)
Let G be a finitely generated group with generating set
S = {s1 , . . . , sn }. The Cayley graph Cayley(G, S) is a graph (V, E)
defined by
V = G.
E = {{g, g0 } : g−1 g0 ∈ S} ⊂ G × G.
The coarse equivalent class of Cayley(G, S) does not depend
on the choice of S.
-2
Z is generated by 1, −1.
-1
0
1
2
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Coarse K-homology
Category of Coarse spaces consists by
Objects: Coarse equivalence classes of metric spaces.
Morphisms: Hom(X, Y) = {f : X → Ycoarse map}/close.
Definition
The coarse K-homology KX∗ (−) is a coarse version of the
K-homology which is a covariant functor from the category of
coarse spaces to the category of abelian groups.
Let X be a metric space.
KX∗ (X) represent a TOPOLOGICAL property of X.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Roe algebra and its K-theory
For space X, we can associate X with a C∗ -algebra C∗ (X),
called the Roe algebra.
The K-theory of the Roe algebra, K∗ (C∗ (−)), is also a
covariant functor from the category of coarse spaces to the
category of abelian groups.
Let X be a metric space.
K∗ (C∗ (X)) represent a Analytic property of X.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Coarse Baum-Connes conjecture
There is a natural transformation µ from KX∗ (−) to
K∗ (C∗ (−)).
µ is called the coarse assembly map.
Conjecture
If X is an metric space of bounded geometry, the coarse
assembly map
µ : KX∗ (X) → K∗ (C∗ (X))
is an isomorphism.
Higson and Roe proved the conjecture for δ-hyperbolic
space.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
General Method for Coarse Baum-Connes conjecture
Let X be a metric space.
Guoliang Yu proved several sufficient condition of the coarse
BC conjecture for X.
The asymptotic dimension of X is finite.
Example: asdim(Zn ) = asdim(Rn ) = n.
X has the property A.
X can be coarsely embedded into the Hilbert space.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
δ-hyperbolic metric space
Definition
Let δ ≥ 0. A graph Γ is δ-hyperbolic if all geodesic triangle are
δ-thin, i.e., for any a, b, c ∈ Γ, ab is contained in the
δ-neighborhood of bc ∪ ca.
δ-thin triangle
a
b
δ
c
Tree is 0-hyperbolic
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Hyperbolic groups
Definition
Let G be a finitely generated group. G is a hyperbolic group if
the Cayley graph of G is δ-hyperbolic for some δ ≥ 0.
Example
Free group F2 = ha, bi
a b
SL2 (Z) =
: a, b, c, d ∈ Z, ad − bc = 1 .
c d
π1 (Mg ), where Mg is a closed surface of genus g ≥ 2.
π1 (M), where M is a compact Riemannian manifold with
strictly negative sectional curvature.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
“Definition” of Relatively Hyperbolic Groups
Let G be a finitely generated group.
Let P = {P1 , . . . , Pk } be a finite family of infinite subgroups.
(G, P) is called a relatively hyperbolic group if
G is hyperbolic relative to P.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
The combinatorial horoball
Definition
Let (P, d) be a metric space. The combinatorial horoball based
on P, denoted by H(P), is the graph defined as follows:
H(P)(0) = P × (N ∪ {0}).
H(P)(1) contains the following two type of edges:
1
2
For l ∈ N ∪ {0} and p, q ∈ P, if 0 < d(p, q) ≤ 2l then there is
a horizontal edge connecting (p, l) and (q, l).
For l ∈ N ∪ {0} and p ∈ P, there is a vertical edge
connecting (p, l) and (p, l + 1).
Lemma
H(P) is δ-hyperbolic for some δ > 0.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Notations
Let G be a finitely generated group.
Let P = {P1 , . . . , Pk } be a finite family of infinite subgroups.
Choose a sequence g1 , g2 , . . . in G such that for any
r = 1, . . . , k, the map N → G/Pr : a 7→ gak+r Pr is bijective.
For i = ak + r ∈ N, let P(i) denote a subgroup Pr . Thus the
F
set of all cosets kr=1 G/Pr is indexed by the map
N 3 i 7→ gi P(i) .
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Notations
Let S be a finite generating set of G.
Let dS be the word metric of G associated to S.
Each coset gi P(i) has a proper metric di which is the
restriction of dS .
H(gi P(i) ) is the combinatorial horoball based on (gi P(i) , di ).
The zero-th floor of H(gi P(i) ) can be embedded in
Γ = Cayley(G, S).
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
The augmented space
Definition
The augmented space X(G, P, S) is obtained by pasting
H(gi P(i) ) to Γ for all i ∈ N.
X(G, P, S) = Γ ∪
[
H(gi P(i) ).
i∈N
Definition (Groves-Manning)
G is hyperbolic relative to P if the augmented space X(G, P, S)
is δ-hyperbolic for some δ ≥ 0.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Examples of relatively hyperbolic group
Let A, B be finitely generated groups. Then C = A ∗ B is
hyperbolic relative to {A, B}.
Let M be a complete, finite volume Riemannian manifold
with (pinched) negative sectional curvature
−b2 < K(M) < −a2 < 0.
Then π1 (M) is hyperbolic relative to cusp subgroups.
A non-uniform lattice in real R-rank one simple Lie group.
Let K be a hyperbolic knot (i.e. S3 \ K admits hyperbolic
metric). Then π1 (S3 \ K) is hyperbolic relative to Z2 .
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Known results
Theorem
Let (G, P) be a relatively hyperbolic group.
If asdimP < ∞ for all P ∈ P, then asdimG < ∞ (Osin).
If P is exact for all P ∈ P, then so is G (Ozawa).
If P is coarsely embeddable in l2 for all P ∈ P, then so is G
(Dadarlat-Guentner).
Due to Yu’s work, those results imply the coarse Baum-Connes
conjecture for such groups.
Baum-Connes conjecture
Coarse geometry
Relatively hyperbolic groups
Main theorem
Theorem (Oguni-F)
Let (G, P) be a relatively hyperbolic group. If all P ∈ P satisfies
the following two conditions:
P admits a finite P-simplicial complex which is a universal
space for proper actions.
The coarse Baum-Connes conjecture for P holds.
Then the coarse Baum-Connes conjecture for G also holds.