Expanders, exactness, and exotic completions
Rufus Willett
University of Hawai‘i
Texas A&M University, April 2014
1/1
2/1
The coarse Baum-Connes conjecture
X : metric space.
Assume bounded geometry: for all r ą 0, there is a uniform bound on the
cardinality of all r -balls.
Examples: Cayley graph of group, net in Riemannian manifold...
3/1
The coarse Baum-Connes conjecture
X : bounded geometry metric space.
T “ pTxy qx,y PX : X -by-X indexed complex matrix.
T is finite propagation: there exists r ą 0 such that
dpx, y q ą r
ñ Txy “ 0.
Cu rX s: ˚-algebra of all finite propagation matrices with bounded entries.
Complete in natural representation on l 2 pX q
(uniform) Roe algebra Cu˚ pX q.
4/1
The coarse Baum-Connes conjecture
Pr pX q: simplicial complex with vertex set X , and
px0 , ..., xn q a simplex iff diamptx0 , ..., xn uq ď r .
K˚u pPr pX qq: (uniform) K -homology of Pr pX q, built from generalized
elliptic operators on X .
There is a higher index / assembly map
µ : lim K˚u pPr pX qq Ñ K˚ pCu˚ pX qq.
r Ñ8
(Uniform) coarse Baum-Connes conjecture: this is an isomorphism.
5/1
The coarse Baum-Connes conjecture
(Uniform) coarse Baum-Connes assembly map:
µ : lim K˚u pPr pX qq Ñ K˚ pCu˚ pX qq.
r Ñ8
Idea: topological data lives in left hand side ... but has better properties
under image by µ.
Applications to Novikov conjecture, existence of positive scalar
curvature metrics...
6/1
The coarse Baum-Connes conjecture
pXn q: sequence of (vertex sets of finite) graphs such that:
each Xn is connected;
|Xn | tends to infinity;
there is a uniform bound on all vertex degrees.
X :“ \nPN Xn .
Metric on X : restricts to edge metric on each Xn , and satisfies
dpXn , X zXn q Ñ 8 as n Ñ 8.
Study coarse Baum-Connes for such X as:
general case reduces to this;
understand counterexamples...
7/1
Expanders and ghosts
pXn q: sequence of finite graphs, bounded vertex degree, getting bigger.
∆n : l 2 pXn q Ñ l 2 pXn q graph Laplacian:
ÿ
∆n : δx ÞÑ
δx ´ δy .
dpx,y q“1
Define
∆ :“
ź
∆n : l 2 pX q Ñ l 2 pX q.
n
pXn q is an expander: there exists c ą 0 such that
spectrump∆q Ď t0u \ rc, 8q.
8/1
Expanders and ghosts
Note: ∆ is finite propagation, so in Cu rX s
e ´t∆ P Cu˚ pX q for all t ą 0
if pXn q an expander,
p :“ lim e ´t∆
tÑ8
exists in
Cu˚ pX q.
p is a ghost projection: pxy Ñ 0 as x, y Ñ 8.
Theorem (Higson)
µ : lim K˚u pPr pX qq Ñ K˚ pCu˚ pX qq
r Ñ8
coarse assembly map.
For some expanders, rps R Imagepµq.
(Expect: never in Imagepµq).
Expanders are generic among such sequences...
9/1
Expanders and ghosts
What can we say about coarse Baum-Connes for generic sequences?
(Uniform) coarse Baum-Connes assembly map
µ : lim K˚u pPr pX qq Ñ K˚ pCu˚ pX qq.
r Ñ8
Theorem
For generic sequences of graphs as above (ignoring some technicalities):
1
µ is not surjective;
2
µ is injective;
3
µmax is an isomorphism.
µmax : change norm on Cu rX s to
}T } :“ supt}πpT q} | π a ˚-representation of Cu rX su.
maximal (uniform) coarse Baum-Connes assembly map
˚
µmax : lim K˚u pPr pX qq Ñ K˚ pCu,max
pX qq.
r Ñ8
10 / 1
Exactness
Cu rX s: ˚-algebra of finite propagation matrices with bounded entries.
Cf rX s :“ tT P Cu rX s | tpx, y q | Txy ‰ 0u is finite.u,
Short exact sequence
0 Ñ Cf rX s Ñ Cu rX s Ñ C8 rX s :“
Cu rX s
Ñ 0.
Cf rX s
Exactness issues: does this stay exact on completion?
Want to complete ... need representations of C8 rX s.
11 / 1
Exactness
pY , y0 q: pointed bounded geometry metric space.
pY , y0 q is an ultralimit of X if there is a sequence pxn q in X , tending to
infinity, such that
BX pxn ; r q is isometric to BY py0 ; r q
for all r ą 0, and all but finitely many n.
Example: girthpXn q Ñ 8
all ultralimits of X “ \Xn are trees.
Each gives rise to ˚-homomorphism
C8 rX s Cu rY s ãÑ Bpl 2 pY qq.
12 / 1
Exactness
Consider again
0 Ñ Cf rX s Ñ Cu rX s Ñ C8 rX s :“
Completing
ι
Cu rX s
Ñ 0.
Cf rX s
π
˚
0 Ñ Kpl 2 pX qq Ñ Cu˚ pX q Ñ C8
pX q Ñ 0.
Ghost projection p in kernel of π, but not image of ι.
For generic graphs:
/ K u pptq
˚
/ limr Ñ8 K u pPr pX qq
˚
µ
– µ
/ K˚ pKq
Ău pPr pX qq
/ limr Ñ8 K
˚
ι
/ K˚ pC ˚ pX qq
u
/ .
– µ
π
/ K˚ pC ˚ pX qq
8
/
If we use maximal completions, corresponding bottom row is exact.
13 / 1
Exactness
The maximal assembly map
˚
µ : lim K˚u pPr pX qq Ñ K˚ pCu,max
pX qq
r Ñ8
is ‘generically’ an isomorphism. Is it always an isomorphism?
Note: ‘yes’ ñ Novikov conjecture, Gromov-Lawson-Rosenberg
conjecture...
Theorem (W.-Yu)
‘No.’
Obstruction: geometric property (T).
14 / 1
Groups - warm-up for Paul Baum
G : locally compact group.
Baum-Connes conjecture with coefficients: the higher index map
µ : K˚top pG ; Aq Ñ K˚ pA ¸r G q
is an isomorphism.
Gromov: can ‘embed’ some expanders into groups
counterexamples based on failures of exactness
(Higson-Lafforgue-Skandalis).
15 / 1
Groups - warm-up for Paul Baum
There are counterexamples to isomorphism of the Baum-Connes assembly
map
µ : K˚top pG ; Aq Ñ K˚ pA ¸r G q
based on exactness.
Theorem (W.-Yu)
Maximal completion fixes these.
Maximal completion is not right in general: other property (T)
obstructions exist.
16 / 1
Groups - warm-up for Paul Baum
Theorem (Baum-Guentner-W.)
There exists a crossed product functor ¸E which is minimal subject to:
it is exact;
it takes Morita equivalences to Morita equivalences.
There are no known counterexamples to the BC conjecture for ¸E , and
some counterexamples become confirming examples.
More in Paul Baum’s talk...
17 / 1