The reduced `p -cohomology in degree 1 and
harmonic functions
Antoine Gournay
University of Warwick, May 21st 2015
A.Gournay (TU Dresden)
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`p H & harmonic functions
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Why `p -cohomology?
A ball packing in a manifold M is a [countable] set of closed balls in M so
that any two balls intersect at most in a point. The incidence graph of a ball
packing is a graph whose vertices are the balls and there is an edge if the ball
touches.
Theorem (Koebe 1936)
A finite graph can be packed in
R2 if and only if it is planar.
Quasi-round packing: replace balls by generic domains, require there is a
K so that the ration “outer radius / inner radius” is K .
There is no obstruction for quasi-round packings of finite graphs in
A.Gournay (TU Dresden)
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R3.
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Why `p -cohomology?
Theorem (Koebe 1936)
A finite graph can be packed in
R2 if and only if it is planar.
There is no obstruction for quasi-round packings of finite graphs in
Theorem (Benjamini & Schramm 2009)
R3.
R
If an infinite graph can be quasi-roundly packed in d either it is d-parabolic or
it has non-trivial reduced `d -cohomology in degree 1.
() fk k j
g
d-parabolic
inf ∇f `p f has finite support and f (x0 ) = 1 = 0.
“Easy” to understand, e.g. 2-parabolicity is recurrence. A Cayley graph is
d-parabolic if and only if it has polynomial growth of degree d.
f j
g
In general, dpar = inf d d-parabolic belongs to [disop ; dgr ] where disop is
the isoperimetric dimension (see later) and dgr the minimal polynomial degree
growth of balls.
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What is `p -cohomology?
In degree 1, the `p -cohomology of a graph G = (V ; E ) is defined via
incidence operators between vertices and edges. Take E V V symmetric,
and let
∇: V
E
f
∇f (x ; y ) = f (y ) f (x )
f ! Rg ! f ! Rg
7!
In graphs of bounded valency, ∇ : `p (V )
!
p
`
(E ) is a bounded operator.
f
! R j ∇f 2 p (E )g.
It is endowed with a semi-norm kf kD = k∇f k . (“semi-” ! constant
The space of p-Dirichlet functions is Dp (G) = f : V
p
`
p
`
functions).
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What is `p -cohomology?
∇:
fV ! Rg ! fE ! Rg
f
7! ∇f (x y ) = f (y )
f (x )
;
kf kD
p
= k∇f k p
`
Definition
The reduced `p -cohomology in degree 1 of a graph is
p
`
H (G ) =
1
Theorem (Élek 1998, Pansu
Im ∇ \ `p (E )
p
`
∇`p (V )
Dp (G)
=
p
`
(V ) + cst
Dp
?)
Fix a bound on the geometry (valency, curvature and injectivity radius). Then
the [reduced] `p -cohomology [in degree 1] is an invariant of quasi-isometry.
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A simple (yet important) example.
p
`
H p (G) = Dp (G)=`p (V ) +
RD
p
(G)
Example:
g:
0
0
1
1
v
1
1
1
1
gn :
0
0
1
n 1
n
2
n
1
n
0
0
v
v
v
v
v
v
gn finitely supported so
∇(g
v
v
2
v
p
`
v
v
v
v
v
(V ) for any p.
gn ) takes n times the value 1=n
A.Gournay (TU Dresden)
v
=) kg
1
`p H & harmonic functions
gn
kD
p
= (n=np )1 p = n
=
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1=p0
.
6 / 30
A simple (yet important) example.
p
`
H p (G) = Dp (G)=`p (V ) +
RD
p
(G)
Example:
g:
0
0
1
1
v
1
1
1
1
gn :
0
0
1
n 1
n
2
n
1
n
0
v
0
p
H 1 (G).
v
v
v
v
v
=) kg
gn
kD
v
p
D
!
g if p
A.Gournay (TU Dresden)
v
v
= (n=np )1 p = n
=
p
Then gn
v
>
1=p0
v
v
v
v
.
8
1. Thus 1 < p < ∞; [g ] = 0
1
v
`p H & harmonic functions
2
`
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A simple (yet important) example.
p
`
H p (G) = Dp (G)=`p (V ) +
RD
p
(G)
Example:
g:
0
0
1
1
v
1
1
1
1
gn :
0
0
1
n 1
n
2
n
1
n
0
0
v
v
v
v
v
v
v
v
v
fg
A.Gournay (TU Dresden)
v
v
v
v
v
2]1 ∞[ and 1H 1(G) ' R.
Id
q, the map p H 1 (G) ! q H 1 (G) is not always injective...
In fact `p H 1 (G) = 0 if p
Remark: If p <
v
;
`
`
`
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Ends
1
`
H (G) is intimately related to the ends of a graph.
1
Definition (Freudenthal, 193?)
An end of a graph = (V ; E ) is a function from finite subsets of V to infinite
ones, such that
ξ(F ) is an infinite connected component of F c ;
F ; F 0 V (finite), ξ(F ) ξ(F 0 ) = .
8
\
6 ?
Examples:
A finite graph has 0 ends.
Z2) has 1 end.
The infinite line (a Cayley graph of Z) has 2 ends.
Regular trees of even valency 3 (Cayley graphs of free groups) have ∞
The infinite grid (a Cayley graph of
many ends.
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Ends
Lemma
The number of ends is a quasi-isometry invariant.
Theorem (Hopf, 1944)
The number of ends of a Cayley graph is 0, 1, 2 or ∞.
Idea: 3 ends
=) ∞ ends
Theorem (Stallings, 1971)
Z
[The Cayley graph of] a group has 2 ends iff it contains
as a finite index
subgroup. It has ∞ many ends iff it is a “non-trivial” amalgamated product or
HNN extension.
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Ends and `1 H
1
1
`
H (G) = D1 (G)=`1 (V ) +
1
Lemma (“well-known”)
Rends G
If G has finitely many ends, `1 H (G) =
1
Preliminary claim: D1 (G)
A.Gournay (TU Dresden)
RD
( ) 1
1
(G)
.
∞ (V )...?
`
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Ends and `1 H
1
1
`
H (G) = D1 (G)=`1 (V ) +
1
Lemma (“well-known”)
Rends G
If G has finitely many ends, `1 H (G) =
1
Preliminary claim: D1 (G)
Hint: f (y )
RD
( ) 1
1
(G)
.
∞ (V )...?
`
f (x ) = ∑e2P ∇f (e) for P a path from x to y .
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Ends and `1 H
1
1
`
H (G) = D1 (G)=`1 (V ) +
1
Lemma (“well-known”)
Rends G
If G has finitely many ends, `1 H (G) =
1
jf (y )
f (x )
RD
( ) 1
1
(G)
.
j = ∑ ∇f (e) k∇f k
1 (E ) :
`
2
e P
Shows more: the `1 norm of ∇f tends to 0 outside larger and large balls.
On the [infinite] connected components of Bnc (Bn
vertex), f becomes uniformly constant as n
∞.
!
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= balls centred at some
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Ends and `1 H
1
1
`
H (G) = D1 (G)=`1 (V ) +
1
Lemma (“well-known”)
Rends G
If G has finitely many ends, `1 H (G) =
1
jf (y )
f (x )
RD
( ) 1
1
(G)
.
j = ∑ ∇f (e) k∇f k
1 (E ) :
`
2
e P
Shows more: the `1 norm of ∇f tends to 0 outside larger and large balls.
On the [infinite] connected components of Bnc (Bn
vertex), f becomes uniformly constant as n
∞.
!
= balls centred at some
2 D1(G) on each end:
βf (ξ) := nlim
!∞ f (xn ) where xn 2 ξ(Bn )
Thence, one defines a value of f
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Ends and `1 H
1
1
`
H (G) = D1 (G)=`1 (V ) +
1
Lemma (“well-known”)
Rends G
If G has finitely many ends, `1 H (G) =
1
jf (y )
f (x )
( ) 1
1
(G)
.
j = ∑ ∇f (e) k∇f k
2
1 (E ) :
`
e P
2 D1(G):
Boundary value of f
RD
βf (ξ) := nlim
!∞ f (xn ) where xn 2 ξ(Bn )
β is linear and continuous on D1 (G).
Also:
1
`
(V ) Ker β
=) β sends `1 (V ) + cst
D1 (G)
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to constant functions.
`p H & harmonic functions
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Ends and `1 H
1
1
`
H (G) = D1 (G)=`1 (V ) +
1
Lemma (“well-known”)
Rends G
If G has finitely many ends, `1 H (G) =
1
RD
( ) 1
1
(G)
.
We have a [linear & continuous] map β which associates to g
function on the ends.
1
`
(V ) + R
D1 (G)
is sent to constant functions.
Remains to show that if βg is constant, then g
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2 D1(G) a
1
`p H & harmonic functions
2
1
`
(V ) + R
D1 (G)
.
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Truncation Lemma 1
Definition
Say g : V
! R takes only one value at infinity if there is a K 2 R so that for any
ε > 0 one can find a Fε V finite so that
g (Fεc )
]K
ε; K + ε[:
Lemma (“maximum principle revisited”)
If g
2 Dp (G) takes only one value at infinity then [g ] = 0 2
Proof: WLOG K
gε ( x ) =
k
g (x )
εg (x )= g (x )
j
gε is finitely supported, so
k
H 1 (G).
= 0. Define gε as
fε = g
p
`
k
g fε Dp = ∇gε
0; so tends to 0.
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k
p (E )
`
2
p
`
j
if g (x ) < ε
if g (x ) ε
(V ) for any p.
is pointwise bouded by ∇g and tends pointwise to
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`p H & harmonic functions
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Aim ...
... to describe `p H 1 as an ideal boundary.
1
`
1
H is related to the ends [“well-known”].
[Bourdon & Pajot 2003]: In the hyperbolic case, there a strong link between
-cohomology in degree 1 and some [Besov] space of functions on the visual
boundary; also pc = inf p `p H 1 = 0 is a lower bound on the conformal
dimension.
p
`
f j
6 g
A natural idea, is to try to look at the “values” of this function on the
“Poisson boundary”.
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Questions and answers
Question (Gromov 1992)
If G is the Cayley graph of an amenable group is
]1; ∞[? [In fact, in all degrees]
p
`
fg
H 1 (G) = 0 for any p
2
Theorem (Gromov 1992, ...)
If G is the Cayley graph of a virtually nilpotent group then
any p ]1; ∞[.
2
p
`
fg
H 1 (G) = 0 for
Holds for any group with [virtually] infinite center. [Tessera 2009] show this
also holds for polycyclic groups and some more.
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Isoperimetric profiles
Definition
2Z
9
Let d
1 . A graph G satisfies a d-dimensional isoperimetric profiles (noted
ISd ) if K > 0 such that, F V finite,
8 jF j1 d K j∂F j
1
9
It has a strong isoperimetric profile (noted ISω ) if K
finite, F
K ∂F
j j j j
Examples: Cayley graphs of
>
0 such that,
8F V
Zd satisfy ISd .
A group is amenable iff its Cayley graph does not satisfy ISω . (Restatement
of Følner)
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Isoperimetric profiles
Satisfying ISα (for α
Hyperbolic
2 Z1 [fωg) is invariant under quasi-isometries.
=) ISω =) ISd ; 8d.
8 6 ) ISω. For example, Cayley graphs of Z2 oα Z where
But ISd ; d =
α(1) = 21 11 .
Theorem (Varopoulos 1985 + Gromov 1981 + Wolf 1968)
has polynomial growth of degree
d ()
Cay( ; S ) does not have ISd +1 .
=) groups which are not virtually nilpotent have ISd for all d.
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Harmonic functions
(n )
(n )
Let Px be the measure defined by Px (y ) = the probability that a simple
random walker starting at x ends up in y after n steps.
This gives a kernel: P (n) g (x ) :=
A function g : V
R
(n )
g (y )dPx
(y ).
! R is harmonic if P 1 g = g (mean value property).
( )
H (G) :=space of harmonic functions.
Hb (G) := H (G) \ `∞ (V ) =space of bounded harmonic functions.
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Boundary values
Theorem (G., 2013)
Assume G satisfies ISd . Let 1
Dp (G)
H (G) such that
[
1p d 2
<
=
!
p
<
d =2.
There is a linear map π
:
2 Dp (G), then π(g ) 2 Dq (G) for all q d dp2p .
if g 2 Dp (G) \ ∞ (V ), then π(g ) 2 ∞ (V ).
[g ] = 0 2 p H 1 (G) () π(g ) is a constant function
() g takes only one value at ∞.
if g
>
`
`
`
This extends a result of Lohoué (1990) which requires ISω .
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Truncation Lemma 2
Lemma (Holopainen & Soardi 1994)
2
6
2
Assume g Dp (G) is such that [g ] = 0 `p H 1 (G). Let gK be the function with
values truncated in [ K ; K ]. Then for some K0 and any K > K0 , [gK ] = 0
p 1
` H (G ).
6
2
Proof goes along the same lines as Truncation Lemma 1.
Corollary
fg
To show `p H 1 (G) = 0 , it suffices to show [g ] = 0 for any g
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`p H & harmonic functions
2 Dp (G) \
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∞ (V ).
`
17 / 30
Corollaries
Theorem (G., 2013)
Assume G satisfies ISd . Let 1
Dp (G)
H (G) such that
[
1p d 2
<
=
!
p
<
d =2.
There is a linear map π
:
2 Dp (G), then π(g ) 2 Dq (G) for all q d dp2p .
if g 2 Dp (G) \ ∞ (V ), then π(g ) 2 ∞ (V ).
[g ] = 0 2 p H 1 (G) () π(g ) is a constant function
() g takes only one value at ∞.
if g
>
`
`
`
Corollary 1
fg
Indeed, π(g ) = 0, 8g 2 Dp (G) \
fg
If G satisfies ISd and Hb (G) = 0 , then `p H 1 (G) = 0 for all p
∞ (V ),
`
2 [1
;
d
2
[.
so Theorem gives [g ] = 0.
Truncation Lemma 2 allows to conclude.
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Corollaries
Theorem (G., 2013)
Assume G satisfies ISd . Let 1
Dp (G)
H (G) such that
[
1p d 2
<
=
!
p
<
d =2.
There is a linear map π
:
2 Dp (G), then π(g ) 2 Dq (G) for all q d dp2p .
if g 2 Dp (G) \ ∞ (V ), then π(g ) 2 ∞ (V ).
[g ] = 0 2 p H 1 (G) () π(g ) is a constant function
() g takes only one value at ∞.
if g
>
`
`
`
Corollary 2
If G is the Cayley graph of a group
Then the identity map `p H 1 (G)
!
Z
which is not virtually- and 1
1
` H (G ) is injective.
q
fg
fg
p
<
q < ∞.
[Cheeger-Gromov 1992] show `2 H (G) = 0 for amenable groups
= for amenable groups, `p H 1 (G) = 0 for all 1 < p 2.
1
)
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Corollaries
Theorem (G., 2013)
Assume G satisfies ISd . Let 1
Dp (G)
H (G) such that
[
1p d 2
<
=
!
p
<
d =2.
There is a linear map π
:
2 Dp (G), then π(g ) 2 Dq (G) for all q d dp2p .
if g 2 Dp (G) \ ∞ (V ), then π(g ) 2 ∞ (V ).
[g ] = 0 2 p H 1 (G) () π(g ) is a constant function
() g takes only one value at ∞.
if g
>
`
`
`
Corollary 3
\
f
g ) 8q d dp2p p H 1(G) = f0g
p H 1(G) = f0g =) H (G) \ Dp (G) = fconstantsg
If G has ISd and 1 p < d =2, then
Hb (G) Dp (G) = constants =
<
+
;`
:
`
This uses that π is the identity on harmonic functions.
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Corollaries
Theorem (G., 2013)
Assume G satisfies ISd . Let 1
Dp (G)
H (G) such that
[
1 p<d =2
!
p
<
d =2.
There is a linear map π
:
2 Dp (G), then π(g ) 2 Dq (G) for all q d dp2p .
if g 2 Dp (G) \ ∞ (V ), then π(g ) 2 ∞ (V ).
[g ] = 0 2 p H 1 (G) () π(g ) is a constant function
() g takes only one value at ∞.
if g
>
`
`
`
Corollary 4
fg
Assume G0 G is a connected spanning subgraph of G so that G0 has ISd and
p 1
0
` H (G ) = 0 for 1
p < d =2. `p H 1 (G) = 0 .
fg
If g 2 Dq (G) then g 2 Dq (G0 ). This implies g takes only one value at ∞ as
seen on G0 . But this is also the case on G.
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Corollaries
Corollary 4
fg
Assume G0 G is a connected spanning subgraph of G so that G0 has ISd and
0
p 1
p < d =2. `p H 1 (G) = 0 .
` H (G ) = 0 for 1
fg
Corollary 5
6 fg
Let L =
and has at most two ends and H is infinite and has most two ends.
Then the lamplighter L H has no harmonic function with gradient in `p (p < ∞).
o
[Thomassen 1978] shows that up to a quasi-isometry (actually bi-Lipschitz
map of constant 6) both L and H contain a connected spanning subgraph
which is a line, half-line or cycle. Hence they contain a of G0 = H 0 L0 where
H 0 = Cn ; or and L0 = or . This G0 has Hb (G0 ) = 0 and ISd for all d.
N Z
N Z
fg
o
) gradient in
! These graphs have lots of bounded harmonic functions ( =
∞ ).
`
Any Cayley graph has harmonic functions with gradient in `∞ (= Lipschitz).
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Corollaries
Corollary 4
fg
Assume G0 G is a connected spanning subgraph of G so that G0 has ISd and
p 1
0
` H (G ) = 0 for 1
p < d =2. `p H 1 (G) = 0 .
fg
This can also be interpreted as a “forbidden subgraph” result: if
H 1 (G) = 0 then there are no spanning connected subgraphs G0 with
Hb (G0 ) = constants and ISd (for some d > 2p).
p
`
6 fg
f
g
Zd for d
e.g. no spanning
is a spanning line!
Corollary 6
>
2p... Recall: [Thomassen 1978] very often there
6 fg
Assume G has ISd and `p H 1 (G) = 0 for 1 p
the Poisson boundary is quasi-isometry invariant
<
d =2. Then some part of
Both ISd and `p H 1 are QI-invariant, so always get a non-trivial bounded
harmonic function by looking at π(g ) where [g ] = 0 `p H 1 and g bounded.
6
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The proof
Theorem (G., 2013)
Assume G satisfies ISd . Let 1
Dp (G)
H (G) such that
[
1 p<d =2
!
p
<
d =2.
There is a linear map π
:
2 Dp (G), then π(g ) 2 Dq (G) for all q d dp2p .
if g 2 Dp (G) \ ∞ (V ), then π(g ) 2 ∞ (V ).
[g ] = 0 2 p H 1 (G) () π(g ) is a constant function
() g takes only one value at ∞.
if g
>
`
`
`
n
π is naively defined: π(g ) = nlim
!∞ P g.
The “difficult” parts are:
π is [well-]defined.
π(g ) is a constant implies g takes only one value at ∞.
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π defined, take one
P (n) g (x ) Cauchy?
! P n g (x )
( )
P (m) g (x ) =
R
(n)
gdPx
R
(m)
gdPx
.
Problem/Idea ?
R
gdξ
R
gdφ
= hg j ξ φi
=???
= h∇g j??i
k∇g k p k??k p
`
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`
0
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Duality
∇ the adjoint of ∇:
∇:
Note:
fE ! Rg ! fV ! Rg
f
7! ∇f (x ) = ∑y x f (y
() (I
= ∇ ∇ then g = 0
P )g
;
x)
∑y x f (x ; y )
= 0.
Definition
ξ; φ finitely supported probability measures. A transport pattern from ξ to φ is a
finitely supported function τξ;φ : E ! R so that ∇ τξ;φ = φ ξ.
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π defined, take two
P (n) g (x ) Cauchy?
! P n g (x )
( )
P (m) g (x ) =
R
(n)
gdPx
R
(m)
gdPx
.
Problem/Idea ?
R
gdξ
R
gdφ
= hg j ξ φi
= hg j ∇ τφ ξ i
= h∇g j τφ ξ i
;
jj k∇g k kτφ ξk
;
p
`
A.Gournay (TU Dresden)
1
`p H & harmonic functions
;
p0
`
2015/05/21
22 / 30
π defined, take two
P (n) g (x ) Cauchy?
! P n g (x )
( )
P (m) g (x ) =
R
(n)
gdPx
R
(m)
gdPx
.
Problem/Idea ?
P (m+k ) g (x )
P (n ) g ( x )
= hg j Px(m+k ) Px(m) i
= hg j ∇ τP (m) P (m+k ) i
x
x
= h∇g j τP (m) P (m+k ) i
;
jj k∇g k kτP
x
p
`
How to define τ
(m)
Px
A.Gournay (TU Dresden)
(m+k )
Px
;
;
x
(m)
x
(m+k )
Px
;
k
p0
`
?
1
`p H & harmonic functions
2015/05/21
22 / 30
π defined, take two
P (n) g (x ) Cauchy?
! P n g (x )
( )
P (m) g (x ) =
R
(n)
gdPx
R
(m)
gdPx
.
Problem/Idea ?
P (m+k ) g (x )
P (n ) g ( x )
= hg j Px(m+k ) Px(m) i
= hg j ∇ τP (m) P (m+k ) i
x
x
= h∇g j τP (m) P (m+k ) i
;
jj k∇g k kτP
x
p
`
How to define τ
(m)
Px
(m+k )
Px
;
;
x
(m)
x
(m+k )
Px
;
k
p0
`
?
∇ is linear so (a possible choice is):
τP (m) ;P (m+k ) =
x
A.Gournay (TU Dresden)
x
1
k 1
∑ τP
i =0
(m+i )
x
(m++i +1)
Px
;
`p H & harmonic functions
2015/05/21
22 / 30
Simple but inefficient...
How to define τ
(m)
Px
(m+1)
Px
;
?
There is a very natural transport pattern:
take a random step!
kP
Then τ
(m)
x
(m+1)
Px
;
A.Gournay (TU Dresden)
k K kPxm k
p0
( )
`
p0
`
1
(where K depends on the valency).
`p H & harmonic functions
2015/05/21
23 / 30
Transport: to infinity and beyond!
jP m
( +k )
g (x )
P (n) g (x )
j k∇g k kτP
p
`
m +k 1
(m)
x
(n)
So it suffices to check that ∑n0 Px
(m+k )
Px
;
k k∇g k ∑ kPxi k
p
`
p
`
0
i =m
()
p0
`
is in `p (V ).
0
Theorem (Varopoulos 1985)
If G has ISd , then for some K
>
k
(n)
0, Px
k
∞ (V )
`
Kn
d =2
.
Thus
kPxn kq
( )
q (V )
`
kPxn kq 1V kPxn k
( )
( )
∞( )
`
1 (V )
`
K 0n
d (q 1)=2
and ∑ P (n) converges in `q if q 0 < d =2.
A.Gournay (TU Dresden)
1
`p H & harmonic functions
2015/05/21
24 / 30
π(g ) =cst
) constant at ∞
=
Want to prove:
π(g ) is a constant function =
)
g is constant at infinity.
WLOG the constant π(g ) is 0.
Will prove:
8ε
>
9
0, nε such that x
2 B3n =) jg (x )j
=
ε
<
K ε.
Make a well-chosen splitting of the scalar product:
h j
P (n) g (x )
A.Gournay (TU Dresden)
g (x ) = ∇g τ
1
`p H & harmonic functions
(n)
δx ;Px
i
2015/05/21
25 / 30
π(g ) =cst
from π(g )
) constant at ∞
=
0, to prove: for some K 0
8ε 0, 9nε such that x 2 B3n =) jg (x )j
>
>
=
ε
<
K ε.
Given ε > 0, define nε to be so that
∇g Bncε < ε
k k
n
sup
∑i n kPx k
x 2V
( )
ε
P (n ) g ( x )
p0 <
`
ε
g (x )
= h∇g j τδ P (n) i
x x
= h∇g j τδ P (n) ijBnε + h∇g j τδ
;
x; x
(n)
x ;Px
ijB
c
nε
k k
2nd term: ∇g is small, and there is an absolute bound on τ: ∑i 0 P (i )
1st term: ∇g is not small, but the τ will be small:
can replace τ
(n) by τ (n )
(n) because it takes at least nε steps to go
δx ;Px
P ε ;Px
from x to Bnε .
A.Gournay (TU Dresden)
1
`p H & harmonic functions
2015/05/21
26 / 30
π(g ) =cst
from π(g )
) constant at ∞
=
0, to prove: for some K 0
8ε 0, 9nε such that x 2 B3n =) jg (x )j
>
>
=
ε
<
K ε.
Given ε > 0, define nε to be so that
∇g Bncε < ε
k k
n
sup
∑i n kPx k
x 2V
( )
p0 <
`
ε
ε
Massage 2nd term:
jP n g (x )
( )
j jh∇g j τδ
jh∇g j τδ
jh∇g j τδ
jh∇g j τδ
g (x )
(n)
x ;Px
(n)
x ;Px
(n)
x ;Px
(n)
x ;Px
A.Gournay (TU Dresden)
1
ijB
ijB
ijB
ijB
nε
nε
nε
nε
j + jh∇g j τδ P ijB j
j + k∇g k B kτδ P k B
i
j+
ε
K1 ∑i 0 kPx k
j+
ε
K2
`p H & harmonic functions
(n)
x; x
p( c )
nε
`
c
nε
(n)
x; x
p0 ( c )
nε
`
()
p0
`
2015/05/21
27 / 30
) constant at ∞
π(g ) =cst
from π(g )
=
0, to prove: for some K 0
8ε 0, 9nε such that x 2 B3n =) jg (x )j
>
>
=
ε
<
K ε.
Given ε > 0, define nε to be so that
∇g Bncε < ε
k k
n
sup
∑i n kPx k
x 2V
( )
p0 <
`
ε
ε
Massage 1st term:
jP n g (x )
( )
j jh∇g j τδ P ijB j
jh∇g j τP P ijB j
k∇g k kτP P k
k∇g k ∑i n kPxi k
k∇g k ε
g (x )
+εK2
+εK2
(nε )
(n)
nε
x
x
(n)
(nε )
p (B ) +εK2
nε
(n)
nε
x; x
;
p
`
;
x
p
`
p
`
A.Gournay (TU Dresden)
1
`p H & harmonic functions
ε
x
`
()
0
p0
`
+εK2
+εK2
2015/05/21
28 / 30
π(g ) =cst
from π(g )
) constant at ∞
=
0, to prove: for some K 0
8ε 0, 9nε such that x 2 B3n =) jg (x )j
>
>
=
ε
<
K ε.
Given ε > 0, define nε to be so that
∇g Bncε < ε
k k
n
sup
∑i n kPx k
x 2V
Up to now: 8x 2 B3n ,
( )
p0 <
`
ε
=
ε
ε
jP n g (x )
( )
Letting n
! ∞:
j K3 ε
g (x )
jg (x )j K3ε
as claimed.
A.Gournay (TU Dresden)
1
`p H & harmonic functions
2015/05/21
29 / 30
Some questions
Q1: Is there an amenable group with a non-constant (bounded or not)
harmonic functions whose gradient is in c0 ?
Q2: If G is the Cayley graph of an amenable group (with
Hb (G) = constants , what is the maximal d so that G has a connected
spanning subgraph G0 with ISd and Hb (G0 ) = constants ?
6 f
g
f
g
(n)
Q3.a: Is there an explicit and more efficient transport pattern from δx to Px ?
Q3.b: In an amenable Cayley graph, is there a Følner sequence and an
explicit (not too unefficient) transport pattern from δe to 1lFn ?
Q.4: For two infinite connected sets A and B let
nA;B (`) = number of mutually disjoint paths of length
` from A to B :
Estimates for the growth (in `) of nA;B (`) in an amenable Cayley graph (of
exponential growth)
A.Gournay (TU Dresden)
1
`p H & harmonic functions
2015/05/21
30 / 30