ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M.
GROMOV
LIOR SILBERMAN
A BSTRACT. Written at the request of GAFA’s Editorial Board, with the goal of explicating
some aspects of M. Gromov’s paper [2]. Section 1 recalls the construction of the random
group, while section 2 contains a proof that the random group has property (T) based on
the ideas from the preprint. The appendix defines CAT(0) spaces and works out in detail
some geometric propositions from the preprint used in the proof.
1. R ANDOM G ROUPS
In this application of the probabilistic method, a “random group” will be a quotient of
a given group. We fix the set of generators (and possibly some initial relations) and pick
extra relations “at random” to get the group:
Let Γ be a finitely generated group, specifically generated by the finite symmetric set
~ will denote the
S (of size 2k). Let G = (V, E) be a (locally finite) undirected graph. E
~ = {(u, v), (v, u) | {u, v} ∈ E}. Given a map
(multi)set of oriented edges of G, i.e. E
~
(“S-coloring”) α : E → S and an oriented path p~ = (~e1 , . . . , ~er ) in G, we set α(~
p) =
α(~e1 )·. . .·α(~er ). We will only consider the case of symmetric α (i.e. α(u, v) = α(v, u)−1
~ Then connected components of the labelled graph (G, α) look like
for all (u, v) ∈ E).
pieces of the Cayley graph of a group generated by S. Such a group can result from
“patching together” labelled copies of G, starting from the observation that in a Cayley
graph the cycles correspond precisely to the relations defining the group. Following this
N
idea let Rα = {α(~c) | ~c a cycle in G}, Wα = hRα i (normal closure in Γ) and
Γα = Γ/Wα .
The Γα will be our random groups, with α(~e) chosen independently uniformly at random from S, subject to the symmetry condition. Properties of Γα then become random
variables, which can be studied using the techniques of the probabilistic method (e.g. concentration of measure). We prove here that subject to certain conditions on G (depending on k), the groups Γα furnish examples of Kazhdan groups with high probability as
|V | → ∞.
We remark that Γα is presented by the k generators subject to the relations corresponding to the labelled cycles in the graph, together with the relations already present in Γ
(unless Γ is a free group). In particular if G is finite and Γ is finitely presented then so is
Γα .
Remark 1.1. This can be done in greater generality: Let Γ be any group, let ρ be a symmetric probability measure on Γ (i.e. ρ(A) = ρ(A−1 ) for any measurable A ⊂ Γ), and
~ → Γ.
denote by ρE the (probability) product measure on the space A of symmetric maps E
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
homepage: http://www.math.princeton.edu/~lior.
1
2
LIOR SILBERMAN
This is well defined since ρ is symmetric and justifies the power notation. Now proceed
as before. The case considered in this paper is that of the standard measure, assigning
1
probabilities 2k
to the generators and their reciprocals.
Remark 1.2. Assume that Γ = hS|Ri is a presentation of Γ w.r.t. S. Then Γα =
hS|R ∪ Rα i is a quotient of hS|Rα i. Since a quotient of a (T)-group is also a (T)-group,
it suffices to prove that the latter groups has property (T) with high probability. In other
words, we can assume for the purposes of the next section that Γ = Fk is the free group
on k generators (k ≥ 2).
2. G EOMETRY, R ANDOM WALKS ON T REES AND P ROPERTY (T).
2.1. Overview. In this section we prove that with high probability Γα has property (T).
The idea is to start with a vector in a representation, and consider the average of its translates by the generators. Typically iterating the averaging produces a sequence rapidly converging to a fixed point. The proof of this breaks down in the following parts:
(1) Property (T) can be understood in geometric language by examining random walks
on the group Γα .
(2) A general analysis of random walks on trees gives some technical results.
(3) The spectral gap of G can be expressed as a bound on the long-range variation of
functions on G in terms of their short-range variation.
(4) (“Transfer of the spectral gap”): With high probability (w.r.t the random choice
of α), a similar bound holds on the variation of certain equivariant functions on Γ
(these are Γ-translates of vectors in a representation of Γα ).
(5) By a geometric inequality and an estimate on the random walk on the tree Γ,
averaging such a function over the action of the the generators n times produces a
function whose (short-range) variation is bounded by the long-range variation of
the original function.
(6) Combining (3), (4) and (5) shows that repeated averaging over the action of the
generators converges to a fixed point. The rate of convergence gives a Kazhdan
constant.
2.2. Property (T). Let Γ be a locally compact group generated by the compact subset S
(not to be confused with the Γ of the previous section).
Definition 2.1. Let π : Γ → Isom(Y ) be an isometric action of Γ on the metric space Y .
For y ∈ Y set diS (y) = supγ∈S dY (π(γ)y, y).
Say that y ∈ Y is ε-almost-fixed if diS (y) ≤ ε. Say that {yn }∞
n=1 ⊆ Y represents an
almost-fixed-point (a.f.p.) if limn→∞ diS (yn ) = 0.
Definition 2.2. (Kazhdan, [3]) Say that Γ has property (T) if there exists ε > 0 such
that every unitary representation of Γ which has ε-almost fixed unit vectors is the trivial
representation. Such an ε is called a Kazhdan constant for Γ w.r.t S. The largest such ε is
called the Kazhdan constant of Γ w.r.t. S.
Remark 2.3. It is easy to see that the question of whether Γ has property (T) is independent
of the choice of S. Different choices may result in different constants though.
An alternative definition considers “affine” representations. For the purpose of most
of the discussion, the choice of origin in the representation space is immaterial, as we
consider an action of the group through the entire isometry group of the Hilbert space,
rather than through the unitary subgroup fixing a particular point (informally, we allow
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
3
action by translations in addition to rotations). In this case we will say the Hilbert space is
“affine”. One can always declare an arbitrary point of the space to be the “origin”, letting
the norm denote distances from that point and vector addition and scalar multiplication
work “around” that point. However, the results will always be independent of such a
choice.
Theorem 2.4. (Guichardet-Delorme, see [1, Ch. 4]) Γ has property (T) iff every affine
(=isometric) action of Γ on a Hilbert space Y has a fixed point.
We now return to the group Γ of the previous section and introduce the geometric language used in the remainder of the discussion. As explained above we specialize to the
case of Γ being a free group. Let Y be a metric space, π : Γα → Isom(Y ). Since Γα is a
quotient of Γ we can think of π as a representation of Γ as well. Setting X = Cay(Γ, S)
(a 2k-regular tree) allows us to separate the geometric object X from the group Γ acting
on it (by the usual Cayley left-regular action). We can now identify Y with the space1
B Γ (X, Y ) of Γ-equivariant functions f : X → Y (e.g. by taking the value of f at 1).
We are interested in bounding the distances dY (sy, y) for s ∈ S. More precisely we
will bound
1 X 2
dY (γy, y).
2|S|
γ∈S
Under the identification of Y with B Γ (X, Y ) this is:
X
µX (x → x0 )d2Y (f (x0 ), f (x)),
x0 ∈X
1
where µX is the standard random walk on X (i.e. µX (x → x0 ) = 2k
if x0 = xγ for
0
some generator γ ∈ S and µX (x → x ) = 0 otherwise). We note that since f and µX are
Γ-equivariant, this “energy” is independent of x, and we can set:
1X
µ(x → x0 )d2Y (f (x0 ), f (x))
EµX (f ) =
2 0
x
and call it the µX -energy of f . To conform with the notation of section B.4 in the appendix,
one should formally integrate w.r.t. a measure ν̄ on Γ\X, but that space is trivial. In this
language f is constant (i.e. is a fixed point) iff EµX (f ) = 0 and {fn }∞
n=1 represent an
a.f.p. iff limn→∞ EµX (fn ) = 0.
In much the same way we can also consider longer-range variations in f , using the nstep random walk instead. µnX (x → x0 ) will denote the probability of going from x to x0
in n steps of the standard random walk on X, EµnX (f ) the respective energy. Secondly, we
can apply the same notion of energy to functions on a graph G as well (no equivariance
here: we consider all functions f : V → Y ). Here µnG will denote the usual random walk
1
on the graph, νG will be the standard measure on G (νG (u) = 2|E|
deg u) and EµnG (f ) =
P
1
n
2
u,v νG (u)µG (v)dY (f (u), f (v)). The “spectral gap” property of G can then be written
2
as the inequality (Lemma 2.10, and note that the RHS does not depend on n!)
1
EµnG (f ) ≤
Eµ (f ),
1 − λ1 (G) G
where λr (G) = max{|λi |r | λi is an e.v. of G and λri 6= 1}.
The core of the proof, section 2.4, carries this over to X with a worse constant. There
is one caveat: we prove that with high probability, for every equivariant f coming from a
1For the motivation for this notation see appendix B.
4
LIOR SILBERMAN
10.5
representation of Γα , the inequality Eµ2l
(f ) ≤ 1−λ
2 (G) Eµ2 (f ) holds for some value of l,
X
X
large enough. We no longer claim this holds for every l.
Leveraging this bound to produce an a.f.p. is straightforward: we use the iterated averages Hµl X f (x), which are simply
X
l
(HµX ) f (x) = HµlX f (x) =
µlX (x → x0 )f (x0 ).
x0
Geometric considerations show that if l is large enough then (approximately, see Proposition 2.15 for the accurate result) EµX (HµlX f ) EµlX (f ), and together with the spectral
gap property this shows that continued averaging gives an a.f.p. which converges to a fixed
point. Moreover, if the representation is unitary (i.e. the action of Γ on Y fixed a point
0 ∈ Y ), if f represents a unit vector (i.e. the values of f are unit vectors) and if EµX (f )
is small enough to start with, then this fixed point will be nonzero. This gives an explicit
Kazhdan constant (for details see section 2.6).
One technical problem complicates matters: the tree X is a bipartite graph. It is thus
more convenient to consider the random walk µ2X and its powers instead, since they are all
supported on the same half of the tree. Then the above considerations actually produce a
Γ2 -f.p. where Γ2 is the subgroup of Γ of index 2 consisting of the words of even length.
If Wα (i.e. G) contains a cycle of odd order then Γ2α = Γα and we’re done. Otherwise
[Γα : Γ2α ] = 2 and Γ2α C Γα . Now averaging w.r.t Γα /Γ2α produces a Γα -f.p. out of a
Γ2α -f.p.
2.3. Random walks on trees. Let Td be a d-regular tree, rooted at x0 ∈ Td . Consider
the distance from x0 of the random walk on Td starting at x0 . At each step this distance
1
increases by 1 with probability pd = d−1
d and decreases by 1 with probability qd = d ,
except if the walk happens to be at x0 when it must go away. Except for this small edge
effect, the distance of the walk from x0 looks very much like a binomial random variable.
Formally let Ω1 = {+1, −1} with measure pd on +1, qd on −1 and let Ω = ΩN
1 with
product measure. We define two sequences of random variables on Ω: the usual Bernoulli
walk:
n
X
Xn (ω) =
ωi
i=1
as well as the “distance from x0 ” one by setting Y0 (ω) = 0 and:
(
Yn (ω) + ωn+1
Yn (ω) ≥ 1
Yn+1 (ω) =
.
1
Yn (ω) = 0
Let µnd (r) = P(Yn = r) and bnd (r) = P(Xn = r). If µT is the standard random walk on
the tree then by spherical symmetry µnd (r) = δd (r)µnT (x0 → y0 ) where δd (r) is the size
of the r-sphere in T and dT (x0 , y0 ) = r. In similar fashion bnd (r) is the probability that
the skewed random walk on Z with probabilities pd , qd goes from 0 to r in n steps. We
also add ηd = pd − qd and σd2 = 4pd qd (the expectation and variance of ωn ). Of course
µnd (r) = bnd (r) = 0 if r 6≡ n (2) and otherwise
n−r
n+r
n
n
bd (r) = n+r pd 2 qd 2 .
2
We drop the subscript 0 d0 for the remainder of the section. The following Proposition
collects some facts about the Bernoulli walk:
Proposition 2.5. E(Xn ) = nη, σ 2 (Xn ) = nσ 2 and:
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
5
(1) (Large deviation inequality, e.g. see [4])
2
P (Xn > ηn + εn) , P (Xn < ηn − εn) ≤ e−2nε
so that
2
P (|Xn − ηn| > εn) ≤ 2e−2nε .
(2) (Non-recurrence) Let r ≥ 0 and assume p > q. Then
r
q
P (∀n Xn > −r) = 1 −
.
p
The trivial observation that Xn (ω) ≤ Yn (ω) implies P(Yn ≤ r) ≤ P(Xn ≤ r) leading
to the (one-sided) deviation estimate
r 2
P (Yn ≤ r) ≤ e−2n(η− n ) .
(2.1)
In the other direction the recurrence relation

n
n

 p · µ (r − 1) + q · µ (r + 1)
n+1
n
n
µ
(r) =
µ (0) + q · µ (2)


q · µn (1)
r≥2
r=1
r=0
implies µn (r) ≤ p1 bn (r), with equality for r = n (proof by induction, also carrying the
stronger assertion µn (0) ≤ bn (0)).
√
Corollary 2.6. P |Yn − ηn| > θ n log n ≤ p2 n−2θ .
P
This is a crucial point, since this will allow us to analyze expressions like x0 µnX (x, x0 )f (x0 )
only when dX (x, x0 ) ∼ ηn, making a trivial
√ estimate otherwise. In fact, from now on we
will only consider the range |r − ηn| ≤ 2 n log n.
Lemma 2.7. (Reduction to Bernoulli walks) Let t ≤ n . Then
12 ηt
X
q
−η 2 t/2
P(Yn = r) −
P(Yt = j)P(Xn−t = r − j) ≤ e
+
.
p
1
ηt≤j≤t
2
2
Proof. As has been remarked before, P(Yt < 21 ηt) ≤ P(Xt < 12 ηt) ≤ e−η t/2 . Furthermore, defining X̃n = Yt + Xn − Xt we clearly have:
n
o n
o
ω ∈ Ω | Yt (ω) ≥ j0 ∧ Yn 6= X̃n ⊆ ω ∈ Ω | Yt (ω) ≥ j0 ∧ ∃u > t : X̃u = 0 .
However, by Proposition 2.5(2) P(Yt ≥ j0 ∧ ∃u > t : X̃u = 0) ≤
time translation-invariance of the usual random talk,
j0
q
p
. Also by the
P(Yt = j | X̃n = r) = P(Xn − Xt = r) = P(Xn−t = r − j).
√
Proposition 2.8. Let |r − ηn| ≤ 2 n log n. Then for some constants c1 (d), c2 (d) independent of n,
√
12 η√n !
√
n+2
log n
q
n
n
−η 2 n/2
µ
(r) − µd (r) ≤ √ c1 (d) · µd (r) + c2 (d) e
+
d
p
n
6
LIOR SILBERMAN
√
Proof. Set t = b nc. By the Lemma,
−η 2 t/2
|P(Yn+2 = r) − P(Yn = r)| ≤ 2e
X
+
21 ηt
q
+2
+
p
P(Yt = j) |P(Xn−t+2 = r − j) − P(Xn−t = r − j)| .
1
2 ηt≤j≤t
An explicit computation shows:
|P(Xn−t+2 = r − j) − P(Xn−t
Since
√
log n
= r − j)| ≤ c1 (d) √ P(Xn−t = r − j).
n
p
2
log n/n ≤ 1 the result follows with c2 (d) = (2 + c1 (d))(eη /2 + (p/q)η/2 ).
We also need a result about general trees. Let T be a tree rooted at x0 ∈ T such that the
degree of any vertex in T is at least 3. We would like to prove that the random walk on T
tends to go further away from x0 than the random walk on T3 due to the higher branching
rate.
1
r 2
Proposition 2.9. Let r < 13 n. Then µnT (x0 → BT (x0 , r)) ≤ e−2n( 3 − n ) .
Proof. Label the (directed) edges exiting each vertex x ∈ T with the integers 0, . . . , deg(x)−
1 such that the edge leading toward x0 is labelled 0 (any labelling if x = x0 ). Assume that
the maximal degree occurring in BT (x0 , n) is d, and let Ω = {0, . . . , d! − 1} with uniform
Pi
measure. Again we define two random variables on Ωn . First, let Xi (ω) = j=1 X(ωj )
3
where X(ωj ) = −1 if d!
ωj < 1, X(ωj ) = +1
define Yi : Ωn → T by
j if not. Secondly
k
i−1 )
ωi leaving Yi−1 (ω).
Y0 ≡ x0 and Yi (ω) follows the edge labelled deg(Y
d!
It is easy to see that the Yi give a model for the standard random walk on T , while the Xi
are a Bernoulli walk on Z with probability p3 of going to the right. Moreover, by induction
it is clear that dT (x0 , Ti (ω)) ≥ Xi (ω). The deviation inequality 2.5 now concludes the
proof.
2.4. The spectral gap and its transfer.
Lemma 2.10. Let G = (V, E) be a finite graph, let νG be the standard probability measure
on V (νG (u) = deg(u)
2|E| ), and let µG be the standard random walk on G (µG (u → v) =
1
~ Set λ2 (G) = max λ2 | λ 6= ±1 is an eigenvalue of Hµ
if (u, v) ∈ E).
(note
deg(u)
G
that λ2 (G) < 1). Let Y be an affine Hilbert space, and let f : V → Y . Then Eµ2n
(f ) ≤
G
1
2
1−λ2 (G) EµG (f ) for all n ∈ N.
Proof. Choose an arbitrary origin in Y , making it into a Hilbert space. Then
X q
1X
2
EµqG (f ) =
νG (u)
µG (u → v) kf (u) − f (v)kY
2
u∈V
v∈V
h
i
1 X
2
2
νG (u)µqG (u → v) kf (u)kY + kf (v)kY − 2 hf (u), f (v)iY .
=
2
u,v∈V
µ is ν-symmetric, i.e. ν(x)µq (x → y) = ν(y)µq (y → x) and therefore
(2.2)
*
+
X
X q
EµqG (f ) =
νG (u) f (u), f (/u) −
µG (u → v)f (v) = f, (I − Hµq G )f L2 (ν
G)
u∈V
v
,
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
7
where we used HµqG = (HµG )q . Just like the case Y = C we can now decompose (the
vector-valued) f in terms of the (scalar!) eigenfunctions of HµG (the normalized adjacency
matrix). Assume these are ψ0 , . . . , ψn−1 corresponding to the eigenvalues λ0 = 1 ≥ λ1 ≥
· · · ≥ λ|V |−1 ≥ −1,
P such that λi0 is the first smaller than 1 and λi1 is the last larger than
−1. Writing f = ai ψi (ai ∈ Y ) and specializing to the case q = 2n we get:
Eµ2n
(f ) =
G
i1
X
2
(1 − λ2n
i ) |ai | .
i=i0
For i0 ≤ i ≤ i1 ,
2
1 − λ2t
i = (1 − λi )
t−1
X
2
λ2j
i ≤ (1 − λi )
j=0
∞
X
λ(G)2j =
j=0
1 − λ2i
,
1 − λ2 (G)
and thus
Eµ2n
(f ) =
G
i1
X
2
(1 − λ2n
i ) |ai | ≤
i=i0
i1
X
1
1
2
(1 − λ2i ) |ai | =
E 2 (f ).
2
1 − λ (G) i=i
1 − λ2 (G) µG
0
Now recall the definition of the factor group Γα in Section 1. We would like to transfer
the “boundedness of energy” property to the random walk on the random group Γα . Let
Xα = Cay(Γα ; S) with the standard S-labelling and Γ-action. Given a vertex u ∈ V and
an element x ∈ Γα there exists a unique homomorphism of labelled graphs,
αu→x : Gu → Xα
(Gu is the component of G containing u) which takes u to x and maps a directed edge of
~ to an edge (y, y · α(~e)) of the Cayley graph Xα = Cay(Γα ; S). Given the measure
~e ∈ E
∗
(µG (u →)) on Γα . We can
µG (u →) on Vu we thus have its pushforward measure αu→x
then define a pushforward random walk on Γα by averaging over all choices of u ∈ V :
X
X
∗
−1
µ̄X,α (x → A) =
ν(u)αu→x
(µG (u → A)) =
ν(u)µG (αu→x
(A)).
u
u
It is clear that µ̄X,α is Γ-invariant. The µ̄X,α -odds of going from x to y are the average
over u of the odds of going from a vertex u to a vertex v such that the path from u to v is
α-mapped to x−1 y.
We now fix u0 ∈ V, x0 ∈ Γα (e.g. x0 = 1). Let Γα act by isometries on the metric
space Y . Then we can identify an element of Y with a Γα -equivariant map f : Xα →
Y through the value f (x0 ). Any such function can be clearly pulled back to a function
f ◦ αu0 →x0 : V → Y . Moreover,
1 X
EµqG (f ◦ αu0 →x0 ) =
νG (u)µqG (u → v)d2Y (f (αu0 →x0 (u)), f (αu0 →x0 (v)))
2
u,v∈V
and by the equivariance of f this equals:
X
1 X 2
1
dY (f (e), f (x)) ·
νG (u)µqG (u → v) = |df |2µ̃q (e) = Eµ̃qX,α (f ),
X,α
2
2
x∈X
αu→e (v)=x
pushing forward any walk µqG to a random walk µ̃qX,α on Xα . We can now rewrite the
Lemma as Eµ̃2n
(f ) ≤ 1−λ12 (G) Eµ̃2X,α (f ).
X,α
8
LIOR SILBERMAN
The space Xα , however, is difficult to analyze – in particular it varies with α. We would
rather consider the tree X = Cay(Γ; S), which we also take with the S-labelling and Γaction. Fixing x ∈ X, every path in G has a unique α-pushforward to an path on X starting
at x and preserving the labelling. By averaging over all paths of length q in G we get a
Γ-invariant random walk on X, which we will denote by µ̄qX,α :
X
µ̄qX,α (x → x0 ) =
νG (p0 )µqG (~
p)
|~
p|=q;αp0 →x (~
p)=x0
(this notation is acceptable since the pushforward of this walk by the quotient map X →
Xα will give the walk µ̃qX,α on Xα ). The relevant space of functions is now all the Γequivariant functions f : X → Y such that wf = f for all w ∈ Wα . It is clear that
averaging such a function f w.r.t. this walk on X or on Xα will give the same answer,
allowing us to only consider walks on the regular tree X. The final form of the Lemma is
then
1
E 2 (f ).
(2.3)
Eµ̄2n
(f ) ≤
X,α
1 − λ2 (G) µ̄X,α
We now use the results of section 2.3 to obtain (with high probability) a similar inequality
about the variation of functions w.r.t. the standard walk µqX :
For fixed q, x and x0 , we think of the transition probability µ̄qX,α (x → x0 ) as a function
def
of α, in other words a random variable. It expectation will be denoted by µ̄qX,G (x → x0 ) =
q
0
0
Eµ̄qX,α (x → x0 ). We show that µ̄2n
X,G (x → x ) is a weighted sum of the µX (x → x ),
where small values of q give small contributions. Thus any bound on Eµ̄2n
(f ) can be
X,G
q
used to bound some Eµ2l
(f ). We then show that with high probability µ̄X,α (x → x0 ) is
X
close to its expectation, so that equation (2.3) essentially applies to Eµ̄2n
(f ) as well.
X,G
We recall that the girth of a graph G, denoted g(G), is the length of the shortest nontrivial closed cycle in G. If q < 21 g(G) then any ball of radius q in G is a tree. We also
denote the minimal vertex degree of G by δ(G) = min{deg(v) | v ∈ V }.
Lemma 2.11. Let 2n < 21 g(G). Then there exist nonnegative weights PG2n (2l) such that
Pn
2n
l=0 PG (2l) = 1, and
def
0
2n
0
µ̄2n
X,G (x → x ) = Eµ̄X,α (x → x ) =
n
X
0
PG2n (2l)µ2l
X (x → x ).
l=0
Moreover if δ(G) ≥ 3 then
def
X
Q2n
G =
PG2n (2l) ≤ e−n/9 .
l≤n/6
Proof. Since
def
X
0
µ̄2n
X,α (x → x ) =
νG (p0 )µ2n
p),
G (~
|~
p|=2n;αp0 →x (~
p)=x0
the expectation is
0
µ̄2n
X,G (x → x ) =
X
νG (p0 )µ2n
p)P(αp0 →x (p2n ) = x0 ).
G (~
|~
p|=2n
Where the probability is w.r.t. the choice of α.
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
9
Let p~ be a path of length 2n in G. By the girth assumption, the ball of radius 2n around
p(0) is a tree. Thus there is a unique shortest path p~0 in it from p0 to p2n , and
αp0 →x (~
p) = αp0 →x (~
p0 ) = x · α(~e01 ) · α(~e02 ) · · · · · α(~e0|~p0 | )
(just remove the backtracks by the symmetry of α). Moreover, the α(~e0i ) are independent
random variables since those edges are distinct. We conclude that the probability that
αp0 →x (~
p) = x0 is equal to the probability of the |~
p0 |-step random walk on X getting from
0
x to x . Thus:
n
X
0
0
µ̄2n
(x
→
x
)
=
PG2n (2l)µ2l
X,G
X (x → x )
l=0
where
X
PG2n (2l) =
νG (p0 )µ2n
p).
G (~
|~
p|=2n;|~
p0 |=2l
2n
(2l) is the probability of
For any u ∈ V Let Tu = BG (u, 2n). This is a tree, and if PG,u
the 2n-step P
random walk on Tu starting at u reaching the distance
P 2l from u then clearly
2n
(2l). In particular it is clear that l PG2n (2l) = 1. Moreover
PG2n (2l) = u νG (u)PG,u
P
2n
since we assume that the minimal degree in G is 3, we have l≤n/6 PG,u
≤ e−2n/18 by
2n
Proposition 2.9. Averaging over u gives the bound on QG .
Lemma 2.12. In addition to the assumptions of the previous Lemma, let deg(u) ≤ d for
all u ∈ V , where d is independent of |V |. Then with probability tending exponentially to 1
with |V |,
1 2n
0
0
µ̄2n
X,α (x → x ) ≥ µ̄X,G (x → x )
2
for all x, x0 ∈ X and
µ̄2X,α (x → x0 ) ≤ µ2X (x → x0 )
for all x 6= x0 ∈ X.
Proof. The random variable µ̄qX,α (x → x0 ) is a Lipschitz function on a product measure
space: return to the definition of µ̄qX,α (x → x0 ) as the average of the pushforwards of
random walks centered at the various vertices of G. Changing the value of α on one edge
only affects random walks starting at vertices with distance at most q−1 from the endpoints
of the edge. Since each such contribution can change by at most 1 the average can change
by at most
2(d − 1)q−1
τ≤
,
|V |
which is therefore the Lipschitz constant2. By the concentration of measure inequality (see
[4, Corollary 1.17]) the probability of µ̄qX,α (x → x0 ) deviating from its mean by at least ε
(in one direction) is at most
2
− 2τε2 |E|
e
−
≤e
ε2
4d(d−1)2q−2
|V |
since 2|E| ≤ d|V |.
Fixing q, x we now consider all the µ̄qX,α (x → x0 ) (different x0 ) together. Let Nk (q) be
the number of random variables:
Nk (q) = |{x0 ∈ X | dX (x, x0 ) ≤ q and dX (x, x0 ) ≡ q
2w.r.t. the Hamming metric on the product space.
(mod 2)}| ,
10
LIOR SILBERMAN
and let ε(k, d, q) be the infimum over such x0 (essentially over their distances – the walk
µ̄qX,G is spherically symmetric) and over rooted trees T of depth q and degrees bounded
between 3 and d of the expression3
X
X
µ̄qX,T (x → x0 ) =
µqT (~
p)µlX (x → x0 ),
l≤q |~
p|=q;|~
p0 |=l
the probability of some
µ̄qX,α (x
0
→ x ) being less than half its mean is then at most
−
Nk (2n) · e
ε(k,d,2n)2
16d(d−1)4n−2
|V |
.
To see this note that if the girth of G is larger than 2q the µ̄qX,G is the νG -average of µ̄qX,Tu
where Tu is the ball of radius q around u in G, which is a tree rooted at u.
As to µ̄2X,α (x → x0 ), note that if x 6= x0 but dX (x, x0 ) = 2 then µ̄2X,G (x → x0 ) =
PG2 (2)µ2X (x → x0 ) (to get from x to x0 we can only consider the paths of length 2 in G)
so that the probability of µ̄2X,α (x → x0 ) > µ2X (x → x0 ) for some such x0 is at most:
−
Nk (2) · e
2 (2))2 µ2 (x→x0 ) 2
(1−PG
X
4d(d−1)2
(
)
|V |
.
Combining the deviation estimates with the spectral gap of the graph, we obtain the
main result:
Proposition 2.13. Assume 4 ≤ 2n < 12 g(G) and that for every u ∈ V , 3 ≤ deg(u) ≤ d.
Then with probability at least
−
1 − Nk (2n) · e
ε(k,d,2n)2
16d(d−1)4n−2
|V |
(1−P 2 (2))2
G
− 16k2 (2k−1)
2 d(d−1)2 |V |
− Nk (2) · e
for every Y , π : Γα → Isom(Y ) and every equivariant f : Γα → Y there exists an l
(depending on f ) such that 21 ηd n < l ≤ n and
Eµ2l
(f ) ≤
X
2
1
·
E 2 (f ).
1 − e−2/9 1 − λ2 (G) µX
Proof. Let l0 = 21 ηd n. By Lemma 2.12 we know that with high probability (as in the
statement of this Proposition),
Eµ̄2n
(f ) ≥
X,α
1
E 2n (f ),
2 µ̄X,G
and
Eµ̄2X,α (f ) ≤ Eµ2X (f )
(The second inequality follows from the assumed bound on the random walk µ̄2X,α since
terms with x = x0 don’t contribute on either side). Combining these two inequalities with
the graph spectral gap (equation 2.3) we get:
2
Eµ̄2n
E 2 (f )
(f ) ≤
X,G
1 − λ2 (G) µX
We now use Lemma 2.11 in the form:
X P 2n (2l)
1
1
G
2n (f ) ≥
Eµ̄2n
(f
)
≥
E
Eµ2l
(f )
µ̄
2n
−2/9
X,G
X,G
X
1 − QG
1 − Q2n
1−e
G
l0 <l≤n
3The sum is over paths p
~ starting at the marked root of the tree.
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
where
2n
PG
(2l)
l0 <l≤n 1−Q2n
G
P
l0
11
= 1 by the definition of Q2n
G , to get:
X P 2n (2l)
2
1
G
Eµ2l
(f ) ≤
E 2 (f ).
2n
−2/9
X
1 − QG
1 − λ2 (G) µX
1−e
<l≤n
Finally, the smallest of the Eµ2l (f ) is at most equal to their average, so the desired conclusion holds for that particular l.
Remark 2.14. Modifying the second Lemma and assuming n large enough, the factor
2
could be replaced with a bound arbitrarily close to 1. For brevity we also note
1−Q2n
G
2
1−e−2/9
≤ 10.5.
2.5. Geometry. We begin with some motivation from the case
of a unitary representation.
The derivation of equation (2.2) then shows that EµqX (f ) = f, (I − Hµq X )f (this inner
product is Γ-invariant since the origin is now assumed to be Γ-invariant). Hµ is self-adjoint
w.r.t. this inner product, so
D
E
2n
2
2n f
Eµ2X (Hµ2n
f
)
=
f,
H
I
−
H
H
= Eµ4n
(f ) − Eµ4n+2 (f ).
µ
µ
µ
X
X
X
X
X
X
µ4n
X (x
0
µ4n+2
(x
X
0
Since
→ x ),
→ x ) are in some sense close for large n, this should imply
that averages of f have small energy (see the rigorous discussion below). This is precisely
what we need in order to prove the existence of fixed points.
This analysis is insufficient for our purposes, however. We would like to analyze
affine (isometric) actions when inner products are no longer invariant, and even actions
on CAT(0) metric spaces, where the equation wouldn’t even make sense. Indeed, it turns
out that the the non-positive curvature of Hilbert space is all that is needed here: an analogue of the above formula is proved in the appendix (Proposition B.25, for the random
walk µ = µ2X ) in two parts, reading:
(2.4)
Z
Z
2n+2
2
1
0
2
2n
EµX (HµX f ) ≤
dν̄X (x)
dµX (x → x0 ) − dµ2n
f (x), f (x0 )),
X (x → x ) dY (Hµ2n
X
2 Γ\X
X
and
Z
Z
1
0 2
dν̄X (x)
dµ2n
f (x), f (x0 )) ≤ Eµ2n
(f ).
X (x → x )dY (Hµ2n
X
X
2 Γ\X
X
In the present case Γ\X is a single point, and the outer integrals can be ignored (any x ∈ X
is a “fundamental domain” for Γ\X).
As just indicated, we would like to prove that averaging indeed reduces the variation of
f by producing an inequality of the form4:
Eµ2X (Hµ2n
f ) ≤ o(1) · Eµ2n
(f ).
X
X
2n+2
0
If µ2n+2
(x → x0 ) were all close to the respective µ2n
(x → x0 ) ≤
X (x → x ) (e.g. µX
X
2n
0
(1 + o(1))µX (x → x )) we would be done immediately. Unfortunately, such an inequality does not hold for all x0 . Fortunately, such an inequality does hold for most x0 . The
exceptions lie in the “tails” of the distribution: x0 which are very far or very close to x.
0
Moreover, in these cases both µ2n+2
(x → x0 ) and µ2n
X (x → x ) are extremely small and a
X
2
0
simple estimate for dY (Hµ2n
f (x), f (x )) suffices, leading to an inequality of the form:
X
Eµ2X (Hµ2n
f ) ≤ o(1) · Eµ2n
(f ) + o(1) · Eµ2X (f ).
X
X
4Here lies the main motivation for only looking at walks of even length: one of µn (x → x0 ), µn+1 (x → x0 )
X
X
is always be zero since the tree X is bipartite, meaning such an argument could not work.
12
LIOR SILBERMAN
To be precise let D = max{dY (f (x), f (x0 )) | dX (x, x0 ) = 2} (compare the displacement diS of the introduction). Then D2 ≤ 2(2k)(2k − 1)Eµ2X (f ) ≤ 8k 2 Eµ2X (f ). Also if
dX (x, x0 ) ≤ 2n and has even parity then d2Y (f (x), f (x0 )) ≤ nD2 by the triangle
√ inequality and the inequality of the means. By the convexity of the ball of radius nD around
f (x) we then have d2Y (f (x), Hµ2n
f (x)) ≤ nD2 (Hµ2n
f (x) is an average of such f (x0 )).
X
X
Hence if d(x, x0 ) ≤ 2n and is even:
(2.5)
d2Y (Hµ2n
f (x), f (x0 )) ≤ 2 · nD2 ≤ 16nk 2 Eµ2X (f ).
X
0
We
pcan now split the integration in equation (2.4) into the region where |dX (x, x ) − 2η2k n| ≤
2 (2n) log(2n) and its complement. In the first region the difference
2n+2
0
dµX (x → x0 ) − dµ2n
X (x → x )
is small by Proposition 2.8. The measure of the second region is small by Corollary 2.6,
and we can use there the simple bound (2.5) on the integrand. All-in-all this gives:
p
log(2n)
32k 2 −3
2
2n
Eµ2n
(f
)
+
n Eµ2X (f )+
EµX (HµX f ) ≤ c1 (2k) √
X
p2k
2n


η2k √n/2
√
2
q2k
 n2 Eµ2 (f ).
+16k 2 c2 (2k) e−η2k n/2 +
X
p2k
The first term is the main term for the first region. The second is the bound on the contribution from the second region. The third is the contribution from the error term in Proposition
2.8 (again the first region) using simple bound and multiplying by n to account for the possible (even) values of r. This last term goes to zero quickly as n → ∞ so we can conclude:
Proposition 2.15. There exists constants c3 (k), c4 (k) depending only on k such that for
all n:
√
log n
1
√ Eµ2n
Eµ2X (Hµ2n
(f ) + c3 (k) 3 Eµ2X (f ).
f
)
≤
c
(k)
3
X
X
n
n
2.6. Conclusion. We first prove the main theorem.
Theorem 2.16. If G is an expander, 3 ≤ deg(u) ≤ d for all u ∈ V and the girth of G is
large enough then Γα has property (T) with high probability.
Remark 2.17. Formally we claim: given k ≥ 2, d ≥ 3 and λ0 < 1 there exists an explicit
g0 = g(k, λ0 ) such that if the girth of G is at least g0 , λ2 (G) ≤ λ20 and the degree of every
vertex in G is between 3 and d, then the probability of Γα having property (T) is at least
1 − ae−b|V | where a, b are explicit and only depend on the parameters k, d and λ0 .
√
log l0
10.5
Proof. Choose n large enough such that for some l0 ≤ 61 n we have r = c3 (k) √
+
l0 1−λ2 (G)
1
c3 (k) l3 < 1.
0
By Proposition 2.13 if g(G) ≥ 4n (note that this minimum girth essentially only depends on λ,k and the desired smallness of r) then with high probability (going to 1 at least
as fast as 1 − ae−b|V | for some a, b depending only on n, d, k) for any affine representation
Y of Γα and any equivariant f : X → Y we can find l in the range l0 < l ≤ n such that
10.5
Eµ2l
(f ) ≤
E 2 (f ).
X
1 − λ2 (G) µX
By Proposition 2.15 and the choice of n we thus have
Eµ2X (Hµ2l
(f )) ≤ rEµ2X (f )
X
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
13
for all f ∈ B Γα (Xα , Y ). This means that the sequence of functions defined by fq+1 =
Hµ2l
fq (the choice of l of course differs in each iteration) represents an almost-fixed point
X
for the action of the subgroup Γ2α on Y where Γ2α is the subgroup of Γα generated by all
words of even length. In fact, Eµ2X (fq ) ≤ rq Eµ2X (f0 ). Using d2Y (Hµ2n
f (x), f (x)) ≤
X
2
2
2
q
8nk Eµ2X (f ) (derived in section 2.5), we see that dY (fq+1 , fq ) ≤ 8nk r Eµ2X (f0 ). Since
r < 1 this makes {fq } into a Cauchy sequence, which thus converges to a fixed-point f∞ .
If f∞ is not Γα -fixed, then the midpoint of the interval [f∞ , γf∞ ] is for any γ ∈ Γα \ Γ2α .
Moreover,
1 q
√
dY (f∞ , f0 ) ≤
8nk 2 Eµ2X (f0 ).
1− r
This means that
√
1− r
ε= √
2 16nk 2
is a Kazhdan constant for Γα w.r.t S: Let π : Γα → U (Y ) be a unitary representation, and
let f0 be a unit vector, ε-almost invariant for S. Then dY (γ1 γ2 f0 , f0 ) < ε for all γ1 , γ2 ∈ S
so Eµ2X (f0 ) < 21 ε2 which means dY (f∞ , f0 ) < 12 , in particular f∞ 6= 0. Note that f∞
is Γ2α -invariant, so that the closed subspace Y0 ⊂ Y of Γ2α -invariant vectors is nonempty.
Pick γ ∈ S, y ∈ Y0 and consider y + π(γ)y. This vector is clearly Γα -invariant and we
are thus done unless π(γ)y = −y for all y ∈ Y0 . In that case we have γf∞ = −f∞ , and
since ε ≤ 18 we obtain the contradiction:
2 = dY (f∞ , π(γ)f∞ ) ≤ dY (f∞ , f0 ) + dY (f0 , γf0 ) + dY (γf0 , γf∞ )
1 1 1
+ + .
<
2 8 2
A slightly different version is actually needed for the result in [2]. For a fixed integer j,
let Gj be the graph obtained from G by subdividing each edge of G into a path of length j,
adding j − 1 vertices in the process. On large scales, the new graph resembles the original
one (e.g. every ball of radius < 12 g(G) · j is a tree), but the minimum degree is no longer 3:
most vertices, in fact, now have degree 2. We now indicate how to adapt the proof above
to this case.
We first remark that the spectral gap of Gj can be bounded in terms of the spectral gap
of G. In fact, if f : V (Gj ) 7→ R is an eigenfunction on Gj with eigenvalue cos ϕ = λ,
then f V is an eigenfunction on G with eigenvalue cos jϕ. To see this observe first that
if (u, v) ∈ E then f (u), f (v) determine the values of f along the subdivided edge since
λf (wi ) is the average of the neighbouring values for any internal vertex wi of the subdivided edge. Plugging this in the expression of λf (u) as an average over the neighbours
in Gj gives the desired result. Now λ cannot be too close to ±1since that that would
imply ϕ too close to 0 or π, making jϕ close (but not equal) to a multiple of π, so that
cos jϕ would be too close to ±1, contradicting the spectral gap of G. Write this bound as
1
2
1−λ2 (Gj ) ≤ c(λ (G), j).
We next adjust Proposition 2.13. Define ε(k, d, j, 2n) to be the maximum of µ2n
T (x →
0
x ) over all trees of depth 2n rooted at x, which are obtained by subdividing edges in trees
with degrees in the interval [3, d]. We then have:
14
LIOR SILBERMAN
Proposition. Assume that 200j ≤ 2n < 12 g(Gj ) and that for every u ∈ V , 3 ≤ deg(u) ≤
d. Then with probability at least
−
1 − Nk (2n) · e
ε(k,d,j,2n)2
16d(d−1)4n−2
|V |
2 (2))2
(1−PG
j
− 8kd(2k−1)(d−1)
2 |V |
− Nk (2) · e
for every Y , π : Γα → Isom(Y ) and every equivariant f : Γα → Y there exists an l
n
(depending on f ) such that 12 ηd 8j log
n < l ≤ n and
Eµ2l
(f ) ≤
X
11
E 2 (f ).
1 − λ2 (Gj ) µX
Proof. Also note that PG2 j (2) can be readily bounded since we know all possible balls
of radius 2 in Gj . The only other change needed in the proof is a reevaluation of the
bound on the inverse of the probability that the 2n-step random walk on the graph Gj
n
travels a distance at least 21 ηd 8j log
n from its starting point (the factor 2 used to compare
2n
0
2n
0
µX,α (x → x ) to µX,G (x → x ) remains). The idea of this is to think of the random
walk on Gj as a series of ’macro-steps’, each consisting of a random walk on the ’star’
of radius j centered at a vertex of G until a neighbouring vertex in G is reached. In this
context we will term ’micro-steps’ the steps of this last random walk, i.e. the usual steps
from before. The sequence of ’macro-steps’ is a random walk on G (every neighbour
is clearly reach with equal probability), which we know to travel away from the origin
with high probability, assuming enough ’macro-steps’ are taken. The expected number of
’micro-steps’ until reaching an ’endpoint’ is j 2 , so on first approximation we can think of
the 2n-step walk on Gj as a variant of the 2n
j 2 -step walk on G (up to a correction of length
j at the first and last steps). Of course, there can deviations. It clearly suffices to estimate
the probability that all ’macro-steps’ take less than 8j 2 log n ’micro-steps’ to complete,
2n
since if that happens then we must have made at least j −2 8 log
n ’macro-steps’. Then the
probability of the final distance from the origin being less than 12 ηd j 2 82n
log n · j is at most
−2/9
e
for the same reason as in the original proposition. A bound for an individual ’macrostep’ follows from a large deviation estimate. Noting that there are at most 2n macro-steps
in the process completes the bound.
Corollary 2.18. Given k, d, λ0 and j there exists an explicit g0 = g(k, λ0 , j) such that
if the girth of G is at least g0 , λ2 (G) ≤ λ20 and the degree of every vertex in G is between 3 and d, then the probability of a random group Γα resulting from a labelling of Gj
having property (T) is at least 1 − ae−b|V | where a, b are explicit and only depend on the
parameters k, d, λ0 and j.
Proof. Identical to the main theorem,
except we now choose n large enough so that l0 <
√
log l0
1
n
√
η
satisfies
r
=
c
(k)
11
· c(λ2 (G), j) + c3 (k) l13 < 1.
d
3
2
8j log n
l
0
0
A PPENDIX A. CAT(0) SPACES AND CONVEXITY
Let (Y, d) be a metric space.
Definition A.1. A geodesic path in Y is a rectifiable path γ : [0, l] → Y such that
d(γ(a), γ(b)) = |a − b| for all a, b ∈ [0, l]. The space Y is called geodesic if every
two points of Y are connected by a geodesic path. We say that Y is uniquely geodesic if
that path is unique.
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
15
Definition A.2. Let Y be a geodesic space, λ > 0. Say that f : Y → R is (λ-)convex if
f ◦ γ is (λ-)convex for every geodesic path γ. We say f : [a, b] → R is λ-convex if it is
convex, and furthermore (D+ f )(y) − (D− f )(x) ≥ λ(y − x) for every a < x < y < b.
(informally, the second derivative of the function is bounded away from zero).
We need the following property of λ-convex functions on intervals:
Lemma A.3. Let f : [a, b] → R be λ-convex. Then max f ([a, b]) − min f ([a, b]) ≥
1
2
8 λ(b − a) .
Proof. Assume first that f is monotone nondecreasing. By convexity, the convergence of
f (x+h)−f (x)
to (D+ f )(x) as h → 0+ is monotone. The limit is nonnegative. By the
h
monotone convergence Theorem,
Z b−δ
Z
Z
1 b−δ+h
1 a+h
(D+ f )(x)dx = lim+
f (x)dx − lim+
f (x)dx
h→0 h b−δ
h→0 h a
a
and by continuity of f in the interior of the interval we get:
Z b−δ
(D+ f )(x)dx = f (b − δ) − f (a)
a
letting δ → 0 and using the monotone convergence Theorem again (D+ f is nonnegative
by assumption) we obtain:
Z b
(D+ f )(x)dx ≤ f (b) − f (a)
a
now by the λ-convexity assumption, (D+ f )(x) ≥ λ(x−a) (since (D+ f )(x) ≥ (D− f )(x) ≥
(D+ f )(a) + λ(x − a) and (D+ f )(a) ≥ 0). Thus
Z b
(b − a)2
(A.1)
f (b) − f (a) ≥
λ(x − a)dx = λ
2
a
In the general case, let f obtain its minimum on [a, b] at c ∈ (a, b). Then f is monotone
non-increasing on [a, c] and nondecreasing on [c, b]. It follows that f (a)−f (c) ≥ λ2 (a−c)2
and f (b) − f (c) ≥ λ2 (b − c)2 . Since either c − a ≥ 21 (b − a) or b − c ≥ 12 (b − a) we are
done.
Lemma A.4. Let f be a λ-convex function, bounded below, on a complete geodesic metric
space Y . Then f has a unique global minimum.
Proof. Let m = inf{f (y) | y ∈ Y }, and for ε > 0 consider the closed set
Yε = {y ∈ Y | f (y) ≤ m + ε}.
We claim that limε→0 diam(Yε ) = 0 and therefore that their intersection is nonempty. Let
x, y ∈ Yε , and consider a geodesic γ : [0, d(x, y)] → Y connecting x, y. g = f ◦ γ is a
2-convex function on [0, d(x, y)] and since x, y ∈ Yε we have g(0), g(d(x, y)) ≤ m + ε,
and thus m ≤ g(t) ≤ m + ε for all t ∈
q[0, d(x, y)]. By the previous Lemma we get
d(x, y)2 ≤
8ε
λ
and therefore diam(Yε ) ≤
8ε
λ
as promised.
Definition A.5. A geodesic space (Y, d) will be called a CAT(0) space if for every three
points p, q, r ∈ Y , and every point s on a geodesic connecting p, q, it is true that d(s, r) ≤
|SR| where P, Q, R ∈ E2 (Euclidean 2-space) form a triangle with sides |P Q| = d(p, q), |QR| =
d(q, r), |RP | = d(r, p) and S ∈ P Q satisfies |P S| = d(p, s).
16
LIOR SILBERMAN
Remark A.6. This clearly implies the usual CAT(0) property: if s ∈ [p, q], t ∈ [p, r] and
S ∈ P Q, T ∈ P R such that |P S| = d(p, s), |P T | = d(p, t) then d(s, t) ≤ |ST |.
From now on let Y be a complete CAT(0) space.
Lemma A.7. (Explicit CAT(0) inequality) Let P, A, B be a triangle in E2 with sides
|P A| = a, |P B| = b, |AB| = c + d, and let Q ∈ AB satisfy |Q − A| = c, |Q − B| = d.
Let l = |P Q|. Then:
d 2
c 2
l2 =
a +
b − cd
c+d
c+d
Moreover, let p, q, r, s ∈ Y where s lies on the geodesic containing p, q. Then
d2 (r, s) ≤
Proof. cos(∠BAP ) =
d(s, q) 2
d(p, s) 2
d (q, r) +
d (p, r) − d(p, s)d(s, q)
d(p, q)
d(p, q)
c2 +a2 −l2
2ac
=
(c+d)2 +a2 −b2
2(c+d)a
l2 = c2 + a2 − c(c + d) −
and therefore
c
c 2
a2 +
b
c+d
c+d
as desired. The second part is a restatement of the definition of a CAT(0) space.
Corollary A.8. Fix y0 ∈ Y . Then the function f (y) = d2 (y, y0 ) is strictly convex along
geodesics. In fact, it is 2-convex.
Proof. Let y1 , y2 , y3 ∈ Y be distinct and lie along a geodesic in that order. By Lemma A.7
f (y2 ) ≤
d(y2 , y3 )
d(y1 , y2 )
f (y1 ) +
f (y3 ) − d(y1 , y3 )d(y2 , y3 ),
d(y1 , y3 )
d(y1 , y3 )
and since d(y1 , y3 )d(y2 , y3 ) > 0 we have strict convexity. In particular, we obtain the two
inequalities
f (y3 ) − f (y1 )
f (y2 ) − f (y1 )
≤
− d(y2 , y3 )
d(y2 , y1 )
d(y3 , y1 )
and
f (y3 ) − f (y2 )
f (y3 ) − f (y1 )
≥
+ d(y1 , y2 )
d(y3 , y2 )
d(y3 , y1 )
which together imply:
f (y3 ) − f (y2 ) f (y2 ) − f (y1 )
−
≥ d(y1 , y2 ) + d(y2 , y3 ) = d(y1 , y3 )
d(y3 , y2 )
d(y2 , y1 )
As to the second part, let y1 < y4 lie along a geodesic path γ. Let y1 < y2 < y3 < y4 .
Then applying the last inequality twice, for the triplets (y1 , y2 , y3 ) and (y2 , y3 , y4 ) we
obtain:
f (y4 ) − f (y3 ) f (y2 ) − f (y1 )
−
≥ d(y1 , y3 ) + d(y2 , y4 )
d(y4 , y3 )
d(y2 , y1 )
letting y2 → y1 and y3 → y4 the LHS converge to the difference of the right- and leftderivatives of f ◦ γ at y1 and y4 respectively, while the RHS converges to 2d(y1 , y4 ) as
desired.
Lemma A.9. The metric d : Y × Y → R is convex.
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
17
Proof. Let a1 , a2 , b1 , b2 ∈ Y and let γi : [0, 1] → Y be uniformly parametrized geodesics
from ai to bi . We need to prove:
d(γ1 (t), γ2 (t)) ≤ (1 − t)d(a1 , a2 ) + td(b1 , b2 ).
Consider first the case γ1 (0) = γ2 (0) (i.e. a1 = a2 ). Then γi (t) are two points along two
edges of the geodesic triangle a1 , b1 , b2 . By similarity of triangles in E2 and the strong
CAT(0) condition we are done.
In the general case, let pi = γi (t). Let γ0 : [0, 1] → Y be the uniformly parametrized
geodesic from a1 to b2 , and let r = γ 0 (t). Then by the special case for γ1 , γ0 which begin at
a1 we have d(p1 , r) ≤ td(b1 , b2 ). Similarly by the special case for γ2−1 , γ0−1 which begin
at b2 are have d(p2 , r) ≤ (1 − t)d(a2 , a1 ). By the triangle inequality we are done.
A PPENDIX B. R ANDOM WALKS ON METRIC SPACES
This appendix follows quite closely section 3 of [2] supplying proofs of the results. The
target is Proposition B.25, the geometric result needed for the property (T) proof.
B.1. Random Walks and the Center of Mass. Let X be a topological space, and let
MX be the set of regular Borel probability measures on X (if X is countable & discrete
this is the set of non-negative norm 1 elements of l1 (X)). Topologize MX as a subset of
the space of finite regular Borel measures on X (with the total variation norm). This makes
MX into a closed convex subset of a Banach space.
Definition B.1. A random walk (or a diffusion) on X is a continuous map µ : X → MX ,
whose value at x we write as µ(x →). The set of random walks will be denoted by WX .
(1) The composition of a measure ν ∈ MX with the random walk µ ∈ WX is the
measure:
Z
(ν · µ)(A) =
dν(x)µ(x → A).
X
This is well defined since µ is continuous and bounded. It is clearly a probability
measure. It seems natural to also think of this definition in terms of a vector-valued
integral.
(2) The composition (or convolution) of two random walks µ, µ0 is the random walk:
Z
(µ ∗ µ0 )(x →) =
dµ(x → x0 )µ0 (x0 →).
X
The integral is continuous so this is, indeed, a random walk.
(3) We will also write
def
µn = µ ∗ · · · ∗ µ,
| {z }
n
and also dµn (x → y) for the probability (density) of going from x to y in n
independent steps.
If X is discrete, we think of µ(x → y) = µ(x →)(y) as the transition probability for a
Markov process on X. In this case the stationary Markov process
(
1
x∈A
δ(x → A) = δx (A) =
0
x∈
/A
is a random walk under the above definition. It isn’t in the non-discrete case since the map
x → δx is weakly continuous but not strongly continuous. It might be possible to define
random walks as weakly continuous maps X → MX but this would require more analysis.
18
LIOR SILBERMAN
Definition B.2. Let ν be a (possibly infinite) regular Borel measure on X, µ ∈ WX .
Say that µ is ν-symmetric, or that ν is a stationary measure for µ if for all measurable
A, B ⊆ X
Z
Z
µ(x → B)dν(x) =
A
µ(y → A)dν(y).
B
In other words, µ is ν-symmetric iff for all measurable ϕ : X × X → R≥0
Z
Z
ϕ(x, x0 )dν(x)dµ(x → x0 ) =
ϕ(x, x0 )dν(x0 )dµ(x0 → x).
X×X
X×X
If X is discrete, we say that µ is symmetric if this holds where ν is the counting measure
(i.e. if µ(x → {y}) = µ(y → {x}) for all x, y).
Example B.3. Let G = (V, E) be a graph where the degree of each vertex is finite. We
can then define a random walk by having µ(u → v) = d1u Nu (v), where Nu is the number
of edges between u and v (i.e. we take account of multiple and self- edges if they exist).
This walk is ν-symmetric for the measure ν(u) = du .
In particular, if G is the Cayley graph of a group Γ w.r.t. a finite symmetric generating
set S we obtain the standard random walk on Γ. This is a symmetric random walk since
the associated measure ν̄G on Γ is Γ-invariant.
Definition B.4. A codiffusion on a topological space Y is a continuous map c : M → Y
defined on a convex subset M ⊆ MY containing all the delta-measures δy such that
c(δy ) = y for every y ∈ Y and such that the pullback c−1 (y) is convex for every y ∈ Y .
Example B.5. In an affine space we have the “centre of mass”:
Z
c(σ) = ~y · dσ(~y )
Y
defined on the set the measures for which the co-ordinate functions yi are integrable.
In the case where Y is an affine inner product space, we can characterize c(σ) for some5
σ ∈ MY in a different fashion: consider the function (“Mean-Square distance from y”)
Z
2
d2σ (y) = ky − y 0 kY dσ(y 0 ),
Y
and note that
d2σ (y) =
Z
2
k(y − c(σ)) − (y 0 − c(σ))kY dσ(y 0 ) =
Y
*
= k(y −
2
c(σ))kY
+
d2σ (c(σ))
Z
+ 2 y − c(σ), c(σ) −
+
0
0
y dσ(y ) .
Y
Since c(σ) −
R
y 0 dσ(y 0 ) = 0 by definition, we find:
Y
(B.1)
2
d2σ (y) = k(y − c(σ))kY + d2σ (c(σ)).
In other words, c(σ) is the unique point of Y where d2σ (y) achieves its minimum. More
generally if Y be a metric space and σ ∈ MY , we set:
Z
def
d2σ (y) =
d2 (y, y 0 ) · dσ(y 0 ).
Y
5σ must be such that the following integral converges for all y ∈ Y .
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
19
Lemma B.6. d2σ (y) ≤ d2σ (y1 ) + d2Y (y, y1 ) + 2dY (y, y1 )dσ (y1 ) = (dY (y, y1 ) + dσ (y1 ))2 .
Proof. By the triangle inequality, dY (y, y 0 ) ≤ dY (y, y1 ) + dY (y1 , y 0 ). Squaring and integrating dσ(y 0 ) gives:
Z
d2σ (y1 ) ≤ d2Y (y, y1 ) + d2σ (y1 ) + 2dY (y, y1 ) dY (y1 , y 0 )dσ(y 0 ).
Y
R
R
Using Cauchy-Schwarz (which, for probability measures σ, reads ( f dσ)2 ≤ f 2 dσ)
completes the proof.
Corollary B.7. If dσ (y) is finite for some y ∈ Y then it is finite everywhere.
Corollary B.8. If dσ (y) is finite, then it is σ-integrable. Furthermore, for any y1 ∈ Y we
have:
Z
4d2σ (y1 ) ≥ d2σ (y)dσ(y) = d2Y (y, y 0 ) σ×σ .
Y
Proof. This follows from Lemma B.6 immediately by integrating dσ(y) and using CauchySchwarz again.
We therefore let M0 be the set of (regular Borel) probability measures on Y such that
d2σ (y) is finite. This is clearly a convex set (though it is not closed if d is unbounded).
Definition B.9. Let σ ∈ M0 . If dσ (y) has a unique minimum on Y , it is called the
(Riemannian) centre of mass of σ, denoted again by c(σ).
If Y is a CAT(0)-space, then Corollary A.8 states that, as a function of y, d2Y (y, y 0 ) is
2-convex. This property clearly also holds to d2σ (y) (seen e.g. by differentiating under the
integral sign and using the monotone convergence Theorem). The existence of c(σ) then
follows from Lemma A.4. In this case we can take M = M0 .
Remark B.10. The 2-convexity of d2σ (y) on the segment [y, c(σ)] implies the following
crucial inequality6:
(B.2)
d2σ (y) ≥ d2σ (c(σ)) + d2Y (c(σ), y)
(c.f. equation (A.1)).
We can also integrate this inequality dσ(y) to obtain the bound d2σ (c(σ)) ≤
1
2
d2Y (y, y 0 ) σ×σ .
B.2. The Heat operator. From now on let X be a topological space, (Y, d) a complete
CAT(0) space. We also fix a measure ν ∈ MX , a ν-symmetric random walk µ ∈ WX ,
and let c be the center-of-mass codiffusion on Y , defined on the convex set M0 ⊆ MY .
Our arena of play will be two subspaces of M (X, Y ), the space of measurable functions
from X to Y . The first one is a generalization of the usual L2 spaces:
Definition B.11. Let f, g : X → Y be measurable. The L2 distance between f, g (denoted
dL2 (ν) (f, g)) is given by

1/2
Z
dL2 (ν) (f, g) =  d2Y (f (x), g(x))dν(x) .
X
6This formula (and the next one) are actually equalities in the Hilbertian case – see equation (B.1).
20
LIOR SILBERMAN
This is a (possibly infinite) pseudometric. Moreover, dL2 (ν) (f, g) = 0 iff f = g ν-a.e. as
usual. Fixing f0 ∈ M (X, Y ), we set:
o
n
L2ν (X, Y ) = f ∈ M (X, Y )dL2 (ν) (f, f0 ) < ∞ .
As usual, the completeness of Y implies the completeness of L2ν (X, Y ).
Remark B.12. That L2ν (X, Y ) is a CAT(0) space follows from Y having the property.
Proof. We start with some notation. Let y1 , y2 ∈ Y , t ∈ [0, 1]. Set [y1 , y2 ]t to be the point
at distance td(y1 , y2 ) from y1 along the segment. In a CAT(0) space this is a continuous
function on Y × Y × [0, 1]. Now let f, g ∈ M (X, Y ) and define [f, g]t (x) = [f (x), g(x)]t .
Then [f, g]t is measurable as well, and clearly d2L2 (ν) (f, [f, g]t ) = t2 d2L2 (ν) (f, g) since
dY (f (x), [f, g]t (x)) = t2 dY (f (x), g(x)) holds pointwise. Thus L2ν (X, Y ) is a geodesic
space. Next, let f, g, h ∈ L2ν (X, Y ) and let u = [f, g]t . The explicit CAT(0) inequality for
h(x), f (x), g(x), u(x) reads:
d2Y (h(x), u(x)) ≤ td2Y (g(x), h(x)) + (1 − t)d2Y (f (x), h(x)) − t(1 − t)d2Y (f (x), g(x),
and integration w.r.t. dν(x) gives the explicit CAT(0) inequality of L2ν (X, Y ). This is
immediately equivalent to the general CAT(0) condition.
Definition. Let f ∈ M (X, Y ), and let τ ∈ MX . The pushforward measure f∗ τ ∈ MY
is the Borel measure on Y defined by (f∗ τ )(E) = τ (f −1 (E)).
Definition B.13. Let ε ∈ [0, 1], f ∈ M (X, Y ).
(1) The Heat operators Hµε are
(Hµε f )(x) = c (f∗ (εµ (x →) + (1 − ε) δx ))
Whenever this makes sense. In particular H = Hµ1 = H 1 is called the heat
operator. Note that by the convexity of M , H ε f are defined whenever H 1 f is.
(2) Say that a function f ∈ M (X, Y ) is (µ-) harmonic if Hf (x) is defined for all
x ∈ X and equal to f (x).
Remark B.14. If Hf (x) = f (x) then d2f∗ µ(x→) (y) and d2Y (y, f (x)) achieve their minimum at the same point, y = f (x). Thus H ε f (x) = f (x) for all 0 ≤ ε ≤ 1. In other
words, f is harmonic iff H ε f = f for all ε.
Example B.15. Consider a graph G = (V, E) with the graph metric, and let µ be the
standard random walk on G. Let f : V → R by any function. It is then clear that H1 is the
“local average” operator (the normalized adjacency matrix).
By the following Lemma (since f∗ (εµ(x →) + (1 − ε)δx ) = εf∗ (µ(x →)) + (1 −
ε)δf (x) ), H ε f is well defined for some L2ν (X, Y ) spaces.
Lemma B.16. Let f0 ∈ M (X, Y ) be a µ-harmonic, and let f ∈ L2ν (X, Y ) (around f0 ).
Then f∗ µ(x →) ∈ M for ν-a.e. x ∈ X. In particular, H ε f is defined ν-a.e.
q
a2 +b2
≤
we have for any y ∈ Y , x0 ∈ X:
Proof. Since a+b
2
2
Z
Z
Z
d2Y (y, f (x0 )) dµ(x → x0 ) ≤ 2 d2Y (y, f0 (x0 )) dµ(x → x0 )+2 d2Y (f0 (x0 ), f (x0 )) dµ(x → x0 ).
X
X
X
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
21
The first integral is finite by µ-harmonicity and Lemma B.6. Integrating the second one
dν(x) and using the fact that µ is ν-symmetric, we find:
Z
dν(x)dµ(x → x0 )d2Y (f0 (x0 ), f (x0 )) dµ(x → x0 )
X×X
Z
d2Y
=
0
0
0
Z
0
(f0 (x ), f (x )) dν(x )dµ(x → x) =
X×X
d2Y (f0 (x0 ), f (x0 )) dν(x0 )
X
=
d2L2 (ν) (f, f0 )
< ∞.
It must thus be finite ν-a.e.
Actually the same proof shows that if Hf0 is well-defined ν-a.e. then so is Hf for all
f ∈ L2ν (X, Y ) – but we should know more:
Claim B.17. Let σ1 , σ2 ∈ M0 , and let τ be a probability measure on Y × Y such that
τ (A × Y ) = σ1 (A) and τ (Y × A) = σ2 (A) for all measurable A ⊆ Y . Then
Z
d2Y (c(σ1 ), c(σ2 )) ≤
d2Y (y, y 0 )dτ (y, y 0 ).
Y ×Y
Proof. I have only managed to prove the Hilbertian case. This should hold for CAT(0)
spaces in general.
Z
Z
c(σ1 ) =
ydσ1 (y) =
ydτ (y, y 0 )
Y ×Y
Y
and
Z
c(σ2 ) =
y 0 dτ (y, y 0 ),
Y ×Y
we have by Cauchy-Schwarz:
2
kc(σ1 ) − c(σ1 )kY ≤
Z
Y ×Y
2
ky − y 0 kY dτ (y, y 0 ) ·
Z
dτ (y, y 0 ).
Y ×Y
Proposition B.18. Let f1 , f2 ∈ M (X, Y ) satisfy fi∗ µ(x →) ∈ M0 for ν-a.e. x ∈ X.
Then dL2 (ν) (H ε f1 , H ε f2 ) ≤ dL2 (ν) (f1 , f2 ). In particular, H ε : L2ν (X, Y ) → L2ν (X, Y )
is Lipschitz continuous.
Proof. Let σi = fi∗ µε (x →), and let τ = (f1 × f2 )∗ (µε (x →)) where f1 × f2 : X →
Y × Y is the product map. By the claim
Z
d2Y (H ε f1 (x), H ε f2 (x)) ≤
d2Y (f1 (x0 ), f2 (x0 )) dµε (x → x0 ).
X×X
The result now follows by integrating dν(x) and using the symmetry of µε . In particular,
we note that
dL2 (ν) (H ε f, H ε f0 ) ≤ dL2 (ν) (f, f0 )
and if, in addition, H ε f0 ∈ L2ν (X, Y ) (e.g. if f0 is harmonic) then H ε f ∈ L2ν (X, Y ) for
all f ∈ L2ν (X, Y ).
22
LIOR SILBERMAN
Continuing
in a different direction, in order for Hf (x) to be defined, it suffices to know
R
that X d2Y (y, f (x0 ))dµ(x → x0 ) is finite for some y. Thinking “locally” we make the
natural choice y = f (x), and set:
Z
def
2
|df |µ (x) =
d2Y (f (x), f (x0 )) dµ(x → x0 ).
X
The second (and more important) space of functions to be considered is
n
o
Bν (X, Y ) = f ∈ M (X, Y )|df |µ ∈ L2 (ν)
(L2 (ν) = L2ν (X, R) is the usual space of square-integrable real valued-functions on X).
By the preceding discussion if f ∈ Bν (X, Y ) then Hf (x) is defined ν-a.e. Anticipating
the following section we call Bν (X, Y ) the space of functions of finite energy.
We return now to the formula d2σ (y) ≥ d2σ (c(σ)) + d2Y (y, c(σ)) which followed from
the 2-convexity of d2σ (y). Setting σ = f∗ (µ(x →)) (so that c(σ) = Hµ f (x)) and writing
out d2σ (Hµ f (x)) in full we define:
Z
def 2
0 2
|d f |µ (x) = dσ (c(σ)) =
d2Y (Hµ f (x), f (x0 )) dµ(x → x0 )
X
(finite for any f ∈ Bν (X, Y )). Then for any measurable f
Z
2
(B.3)
d2Y (y, f (x0 ))dµ(x → x0 ) ≥ d2Y (y, Hµ f (x)) + |d0 f |µ (x)
X
(Note that Hµ f is undefined iff the LHS is infinite). We now derive an important pair of
inequalities which are a basic ingredient of the proof that our random groups have property
(T):
Lemma B.19.
2
2
|d0 f |µn (x) ≤ |df |µn (x).
Proof. Follows from equation (B.3) by replacing µ with µn (which is also ν-symmetric),
setting y = f (x) and ignoring the d2Y (y, Hf (x)) term.
Proposition B.20. Let7 Hn = Hµn . Then
Z
Z
2
dν(x)·|d(Hn f )|µ (x) ≤
dν(x) dµn+1 (x → x0 ) − dµn (x → x0 ) d2Y (Hn f (x), f (x0 )).
X
X×X
00
Proof. Set y = Hn f (x ), and integrate (B.3) w.r.t. dν(x00 )dµ(x00 → x) = dν(x)dµ(x →
x00 ) on the RHS and LHS respectively to get:
Z
Z
Z
dν(x00 )dµ(x00 → x)
dµn (x → x0 )d2Y (Hn f (x00 ), f (x0 )) ≥
dν(x)dµ(x → x00 )d2Y (Hn f (x), Hn f (x00 ))+
X×X
X
Z
+
X×X
dν(x)dµ(x → x00 )
X×X
Z
dµn (x → x0 )d2Y (Hn f (x), f (x0 )).
X
R
def
Now on the LHS, X dµ(x00 → x)µRn (x → A) = µn+1 (x00 → A). On the LHS the inner
integral does not depend on x00 and X dµ(x → x00 ) = 1. In other words:
Z
Z
dν(x00 )dµn+1 (x00 → x0 )d2Y (Hn f (x00 ), f (x0 )) ≥
dν(x)dµ(x → x00 )d2Y (Hn f (x), Hn f (x00 ))+
X×X
X×X
7In the affine case (but not in general) this equals the n-th power (iterate) of H .
µ
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
Z
23
dν(x)dµn (x → x0 )d2Y (Hn f (x), f (x0 )).
+
X×X
On the LHS we can now rename the variable Rx00 to be x. We also move the second term
def
on the RHS to the left. Finally we note that X dµ(x → x00 )d2Y (Hn f (x), Hn f (x00 )) =
2
|d(Hn f )|µ (x).
B.3. The Energy. We can rewrite the results of the previous section more concisely by
introducing one more notion.
Definition B.21. Let f ∈ M (X, Y ). The energy of f is:
Z
Z
1
1
2
E(f ) = k|df |µ kL2 (ν) =
dν(x) dµ(x → x0 )d2Y (f (x), f (x0 )).
2
2
X
X
Lemma B.22. Let f0 , f ∈ M (X, Y ) such that dL2 (ν) (f, f0 ) < ∞. If f0 has finite energy
then so does f .
R
Proof. We recall that dν(x)dµ(x → x0 )d2Y (f (x0 ), f0 (x0 )) = d2L2 (ν) (f, f0 ). Using the
inequality of the means and the triangle inequality we get:
d2Y (f (x), f (x0 )) ≤ 3d2Y (f (x), f0 (x)) + 3d2Y (f0 (x), f0 (x0 )) + 3d2Y (f0 (x0 ), f (x0 )).
Multiplying by
1
2
and integrating dν(x)dµ(x → x0 ) gives:
E(f ) ≤ 3E(f0 ) + 3d2L2 (ν) (f0 , f ).
Proposition B.23. Bν (X, Y ) is a convex subset of M (X, Y ). E(f ) is a convex function
on Bν (X, Y ).
√
Proof. We prove the stronger assertion that E is convex. Let f, g ∈ Bν (X, Y ) and let
t ∈ [0, 1]. Let ut = [f, g]t . By the convexity of the metric (Lemma A.9), we have
dY (ut (x), ut (x0 )) ≤ (1 − t) · dY (f (x), f (x0 )) + t · dY (g(x), g(x0 )).
Now since
p
2E(f ) = kdY (f (x), f (x0 ))kL2 (ν·µ) ,
the triangle inequality of L2 (ν · µ) reads:
p
p
p
E(ut ) ≤ (1 − t) E(f ) + t E(g).
This implies both that E(ut ) < ∞ (i.e. ut ∈ Bν (X, Y )) and the convexity of
√
E.
Definition B.24. Let (X, µ, ν) and (Y, d) be as usual, and let f range over Bν (X, Y ). We
define the Poincaré constants:
πn (X, Y ) =
Eµn (f )
.
Eµ (f )>0 Eµ (f )
sup
For example, in section 2.4 we saw that for a graph G with the usual random walk and a
1
Hilbert space Y one has πn (G, Y ) ≤ 1−λ(G)
where 1 − λ(G) is the (one-sided) spectral
gap of G).
24
LIOR SILBERMAN
B.4. Group actions. Let Γ be a locally compact group acting on the topological space X,
and assume ν ∈ MX is Γ-invariant. Assume that there exists a measure ν̄ on X̄ = Γ\X
such that if E ⊂ X is measurable and γE ∩ E = ∅ for all γ ∈ Γ \ {e} then ν(E) = ν̄(Ē).
Also assume that Γ\X can be covered by countably many such Ē of finite measure. In
this setup every Γ-invariant measurable function f : X → R descends to a measurable
function f¯ : X̄ → R, and we can set:
Z
2
kf : ΓkL2 (ν) =
|f¯(x̄)|2 dν̄(x̄)
X̄
2
For example, if Γ is a finite group acting freely on X then kf : ΓkL2 (ν) =
1
|Γ|
2
kf kL2 (ν)
2
where kf kL2 (ν) is the usual L2 norm on (X, ν).
We now throw in an equivariant diffusion µ ∈ WX , and a metric Γ-space Y (i.e. Γ acts
on Y by isometries). If f ∈ M Γ (X, Y ) is equivariant then Hµ f (wherever defined) is also
equivariant, |df |µ (x) is invariant and the energy is properly defined by
1
2
k|df |µ : ΓkL2 (ν)
2
We then consider the space BνΓ (X, Y ) of equivariant functions of finite energy. In this
context Lemma B.19and Proposition B.20 above read:
Eµ (f ) =
Proposition B.25. Let8 Hn = Hµn . Then
Z
Z
n+1
1
Eµ (Hn f ) ≤
dν̄(x)
dµ
(x → x0 ) − dµn (x → x0 ) d2Y (Hn f (x), f (x0 )),
2 X̄
X
and
Z
Z
1
dν̄(x)
dµn (x → x0 )d2Y (Hn f (x), f (x0 )) ≤ Eµn (f ).
2 X̄
X
Proof. Essentially the same as before, integrating dν̄ instead of dν since the integrands are
Γ-invariant in all cases.
Equation (B.3) has two more important implications:
Proposition B.26. Let f ∈ B Γ (X, Y ). Then
d2L2 (ν) (f, Hµ f ) ≤ 2Eµ (f ).
Proof. Set y = f (x)in the equation and ignore the second term on the RHS to get:
Z
|df |2µ (x) =
d2Y (f (x), f (x0 ))dµ(x → x0 ) ≥ d2Y (f (x), Hµ f (x)).
X
Integrating this dν̄(x) we obtain the desired result.
Γ
Proposition B.27. Let f ∈ B (X, Y ). Then E(Hµ f ) ≤ 2E(f ).
Proof. By the triangle inequality and the inequality of the means,
1 2
d (Hµ f (x), Hµ f (x0 )) ≤ d2Y (Hµ f (x), f (x0 )) + d2Y (f (x0 ), Hµ f (x0 )).
2 Y
Integrating dµ(x → x0 ) gives:
Z
1
2
0 2
|dHµ f |µ (x) ≤ |d f |µ (x) +
dµ (x → x0 )d2Y (f (x0 ), Hµ f (x0 )).
2
X
8In the Hilbertian case (but not in general) this equals the n-th power (iterate) of H .
µ
ADDENDUM TO “RANDOM WALK ON RANDOM GROUPS” BY M. GROMOV
25
We now integrate dν̄(x). We evaluate the second term first using the ν-symmetry of µ:
Z
Z
Z
dν̄(x)
dµ (x → x0 )d2Y (f (x0 ), Hµ f (x0 )) = dν̄(x0 )d2Y (f (x0 ), Hµ f (x0 )),
X̄
X
so that:
Z
Eµ (Hµ f ) ≤
X̄
h
i
2
dν̄(x) |d0 f |µ (x) + d2Y (f (x), Hµ f (x)) .
By Equation (B.3) we get:
Z
Z
Eµ (Hµ f ) ≤
dν̄(x)
dµ(x → x0 )d2Y (f (x), f (x0 )) = 2Eµ (f ).
X̄
X
R EFERENCES
1. P. de la Harpe and A. Valette, La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque,
no. 175, Société mathémathique de France, 1989.
2. M. Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73–146.
3. D. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Func. Anal.
Appl. 1 (1967), 63–65.
4. M. Ledoux, Concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, 2001.