Horocyclic products of trees

0.54 MB·English

by  Laurent Bartholdi

HOROCYCLIC PRODUCTS OF TREES
LAURENT BARTHOLDI, MARKUS NEUHAUSER AND WOLFGANG WOESS
6 Abstract. Let T1,...,Td be homogeneous trees with degrees q1 +1,...,qd +1 3,
0 respectively. For each tree, let h : T Z be the Busemann function with respect≥to a
j
0 →
fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of
2
T1,...,TdisthegraphDL(q1,...,qd)consistingofalld-tuplesx1 xd T1 Tdwith
n h(x1)+ +h(xd)=0, equipped with a naturalneighbourhood·r·e·lati∈on. I×n·t·h·×e present
a ···
paper,we explorethe geometric,algebraic,analytic andprobabilistic propertiesofthese
J
7 graphs and their isometry groups. If d = 2 and q1 = q2 = q then we obtain a Cayley
1 graph of the lamplighter group (wreath product) Zq Z. If d = 3 and q1 = q2 = q3 = q
then DL is the Cayley graph of a finitely presented≀group into which the lamplighter

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G −
graph of a finitely presented group. This group is of type Fd−1, but not Fd. It is not
h. automatic, but it is an automata group in most cases. On the other hand, when the
at qj do not all coincide, DL(q1,...,qd) is a vertex-transitive graph, but is not the Cayley
m graph of a finitely generated group. Indeed, it does not even admit a group action with
finitely many orbits and finite point stabilizers. The ℓ2-spectrum of the “simple random
[
walk” operator on DL is always pure point. When d = 2, it is known explicitly from

1 previous work, while for d = 3 we compute it explicitly. Finally, we determine the
v Poisson boundary of a large class of group-invariant random walks on DL. It coincides
7 with a part of the geometric boundary of DL.
1
4
1 Contents
0
6 1. Introduction 1
0 2. Isometry groups 5
/
h 3. Cayley graphs 11
t 4. The DL complex 17
a
m 5. The spectrum of simple random walk 24
6. The Poissonboundary of random walk 35
:
v References 40
i
X
r
a
1. Introduction
Let X be a locally finite, infinite, connected graph without loops. We write x y
∼
if x,y are neighbours (connected by an edge), and deg(x) for the number of neighbours
of x. Suppose that X is written as a disjoint union of non-empty sets H , k Z (the
k
∈
horocycles), where each element in H has neighbours both in H and in H , but none
k k−1 k+1
in any other H . (This condition is tailored to our purposes and can be generalized.) The
l
associated surjection h : X Z, where h(x) = k if x H , is a graph homomorphism
k
→ ∈
of X onto the two-way-infinite path Z. We call it a Busemann function, although this
terminology is justified completely only in specific cases (see below), and we say that
Date: 17 January 2006.
2000 Mathematics Subject Classification. 05C50,20E22, 47A10, 60B15.
Key words and phrases. restricted wreath product, trees, horocycles, Diestel-Leader graph, growth
function, normal form, Markov operator, spectrum.
Supported by FWF (Austrian Science Fund) project P15577.
1
2 L. Bartholdi, M. Neuhauser and W.Woess
(X,h) is a Busemann pair. Now let X ,...,X be a family of such graphs with associated
1 d
Busemann functions h : X Z (we use the same symbol h for each of them). Then their
j
→
horocyclic product is
d
(1.1) X = x x X X : h(x )+ +h(x ) = 0
h j 1 d 1 d 1 d
{ ··· ∈ ×···× ··· }
j=1
Y
with neighbourhood
x x y y
1 d 1 d
(1.2) ··· ∼ ··· ⇐⇒
there are i = j such that x y , x y and x = y for all k = i,j.
i i j j k k
6 ∼ ∼ 6
In particular, one must have h(x ) h(y ) = h(y ) h(x ) = 1. Thus, x = x x
i i j j 1 d
− − ± ··· 7→
H(x) = h(x ),...,h(x ) is a graph homomorphism of X onto the simplicial lattice
1 d h j
A = k = (k ,...,k ) Zd : k + +k = 0 . Inthe latter, two points areneighbours
d−1 (cid:0){ 1 d (cid:1)∈ 1 ··· d } Q
if they differ by a vector e e , where i = j and e Zd is the unit vector with a 1 in its
i j i
− 6 ∈
i-th coordinate.
There is an analogous construction for groups, compare with Kaimanovich and
Woess [23, p.356]. Let Γ ,...,Γ be topological (e.g. in particular, discrete) groups,
1 d
each one equipped with a continuous homomorphism h : Γ Z (or R; we again use
j
→ →
the same symbol h for each of them). Then their horocyclic product is
d
(1.3) Γ = g g Γ Γ : h(g )+ +h(g ) = 0 ,
h j 1 d 1 d 1 d
{ ··· ∈ ×···× ··· }
j=1
Y
which is a closed subgroup of the direct product of the Γ . For finitely generated groups
j
Γ , this kind of construction was used for example by Bridson [10]. However, our
j
approach has a different “history”, and below, the groups will be non-discrete isometry
groups of homogeneous trees. Here, horocyclic products of groups will arise as isometry
(automorphism) groups of horocyclic products of graphs.
If (X ,h) and (X ,h) are two Busemann pairs, then a Busemann isometry g from the
1 2
first to the latter is a graph isomorphism g : X X such that x h(gx ) h(x ) is
1 2 1 1 1
→ 7→ −
constant. We write h(g) for this constant. In particular, the group Aut(X,h) of a given
Busemann pair (X,h) consists of all Busemann isometries X X. Given (X ,h) as
j
→
d d
above (j = 1,...,d), the group Aut(X ,h) acts on X by graph isometries via
h j h j
j=1 j=1
Q Q
(1.4) gx = (g x ) (g x ), where g = g g and x = x x .
1 1 d d 1 d 1 d
··· ··· ···
ThebasicexampleofaBusemannpairariseswhentheunderlying graphisatree T,that
is, a connected graph without cycles, where 2 deg(x) < for every vertex x. There
≤ ∞
are several choices (one for each element in the boundary of the tree, see below) to equip
the edge set of T with an orientation such that each vertex x has a unique predecessor
x− and deg(x) 1 successors y T such that y− = x. Then it is easily understood
− ∈
that in the induced half-order, the ancestor relation 4, any two vertices x,y T have a
∈
maximal common ancestor xfy. If o T is a reference vertex (origin), then we define
∈
h(x) = d(x,ofx) d(o,ofx), where d( , ) denotes the usual graph metric. Then (T,h)
− · ·
is the typical example of a Busemann pair.
Horocyclic productsof trees 3
In the present paper, we shall deal with homogeneous trees T = T , where each vertex
q
has degree q+1 (q 2). In this case, the horocyclic structure (i.e., the ancestor relation)
≥
is unique up to isomorphism. We write DL(q ,...,q ) for the horocyclic product of the
1 d
trees T = T ,...,T = T . The “DL” stands for Diestel and Leader, who were
1 q1 d qd
the first [16] to introduce the graph DL(2,3) in attempt to answer a question raised by
Woess [33, 31]: “is there a locally finite vertex-transitive graph which is not quasi-
isometric with a Cayley graph of some finitely generated group ?” (Recall that a graph
is called vertex-transitive if its isometry group acts transitively on the vertex set.) A
very recent announcement of Eskin, Fisher and Whyte [18] confirms that the graphs
DL(q ,q ) (q = q ) are such examples.
1 2 1 2
6
The purpose of this paper is to present a picture of the many interesting features of the
graphs DL(q ,...,q ).
1 d
In 2, we first recall in more detail the horocyclic structure of the homogeneous tree T
q
§
and the group Aff(T ) = Aut(T ,h) of all its Busemann self-isometries. The latter group
q q
has been called the affine group of the tree by analogy with the affine group over R acting
onthehyperbolicupperhalfplane. WedeterminethefullisometrygroupAut(DL)ofDL =
d
DL(q ,...,q ). We prove that it is a finite extension of the group = Aff(T ). The
1 d A h qj
j=1
latter acts transitively on DL and is amenable as a locally compact, toQtally disconnected
group with the topology of pointwise convergence.
If the q do not all coincide, we show that this group is also non-unimodular (i.e.,
j
the left Haar measure is not right-invariant). Consequently, by a theorem of Soardi
and Woess [31], the graph DL is non-amenable, i.e., it satisfies a strong isoperimetric
inequality (the Cheeger inequality). We also conclude that Aut(DL) cannot have a co-
compact lattice, thatis, thereisnodiscrete (closed) subgroupthatactsonDLwithfinitely
many orbits. In particular, if the q do not all coincide, then DL is vertex-transitive, but
j
is not the Cayley graph of a finitely generated group.
In 3, we study DL (q) = DL(q,...,q), the horocyclic product of d copies of T . We
d q
§
use an approach that is reminiscent of the method for constructing lattices in Lie groups
over local fields, as outlined on the first page of the book by Margulis [25]. When
q = p p is the factorization of q as a product of prime powers, and p d 1 for
1 r ι
··· ≥ −
all ι 1,...,r , the graph DL (q) is a Cayley graph of a group of affine matrices over
d
∈ { }
a ring of Laurent polynomials whose coefficients come from a suitable finite ring. There
is some degree of freedom in the choice of the ring of coefficients. When d = 2 or d = 3,
we can take the ring Z = Z/(qZ) of integers modulo q, and for d = 2 this is a way
q
to describe the lamplighter group Z Z, while for d = 3 we obtain a finitely presented
q
≀
group into which the lamplighter group embeds. This group has appeared in previous
work by Baumslag [5] and others. In general, DL (q) is quasi-isometric with DL (qs) for
d d
every s 1. Thus, DL (q) is always quasi-isometric with a Cayley graph of some finitely
d
≥
generated group, while on the other hand, [16] and [18] suggest that the vertex-transitive
graph DL(q ,...,q ) is not quasi-isometric with any Cayley graph when the q do not all
1 d j
coincide.
4 L. Bartholdi, M. Neuhauser and W.Woess
In 4, we consider DL(q ,...,q ) as a (d 1)-dimensional cell complex and explore
1 d
§ −
its homotopy type, which is that of a union of countably many (d 1)-spheres glued
−
together at a single point. This should be compared with a deep theorem of Bestvina
and Brady [6]. Thus, when DL is the Cayley graph of a group, then this group is of
type F , but not of type F , and in particular it is finitely presented when d 3. We
d−1 d
≥
deduce that, for each d, the lamplighter group can be embedded in a metabelian group of
type F . In general, it is known [9] that every metabelian group embeds in a metabelian
d
group of type F , while embeddability in F for larger d is conjectured.
3 d
In 5, we turn our attention to a more analytic-probabilistic object. Simple random
§
walk onany locally finite, connected graphX is theMarkov chain whose transition matrix
P = p(x,y) is given by
x,y∈X
(cid:0) (cid:1)
1/deg(x), if y x,
(1.5) p(x,y) = ∼
0, otherwise.
(
P acts on functions f : X R by
→
(1.6) Pf(x) = p(x,y)f(y).
y
X
In our case, deg( ) = (d 1)(q + + q ) is constant, and we are interested in the
1 d
· − ···
spectrum of P on the space ℓ2(DL) of all square-summable complex functions on DL. The
spectral radius ρ(P) is equal to 1 if and only if q = = q . As a set, spec(P) is an
1 d
···
interval contained in [ ρ(P)/(d 1), ρ(P)], and with the exception of a “degenerate”
− −
case, it coincides with the latter. In particular, for DL (q) the spectrum of P is the
d
same as the spectrum of the projection of P on the lattice A . The latter spectrum
d−1
is absolutely continuous. On the other hand, for arbitrary q ,...,q , the spectrum of
1 d
P on DL(q ,...,q ) is pure point: there is an orthonormal basis of ℓ2(DL) that consists
1 d
of finitely supported eigenfunctions of P. This extends previous results regarding the
lamplighter group and the basic Diestel-Leader graphs DL(q ,q ), see Grigorchuk and
1 2
Z˙uk [20], Dicks and Schick [15] and Bartholdi and Woess [3]. For the case d = 2,
the eigenvalues and eigenfunctions were computed explicitly in those references. Here,
we present explicit computations for d = 3 and DL (q), while the general case seems
3
intractable (except numerically).
Finally, in 6, we study the general class of random walk on DL whose transition matrix
§
is irreducible, invariant under the group , and has finite first moment. Using results of
A
Cartwright, Kaimanovich and Woess [14]andBrofferio [11]concerning random
walks on Aff(T ), we show that those random walks on DL converge almost surely to the
q
geometricboundaryofDL. The latteristheidealboundaryadded toDLwhen considering
the closure of DL in T , where T is the well-known end compactification of T . We
i i i i
then use the ray criterion of Kaimanovich, see [23], to prove that the active part of the
Q
boundary (i.e., the suppobrt of the lbimit distribution of the random walk) is the “largest
possible” model for distinguishing limit points of the random walk: it is the Poisson
boundary.
Horocyclic productsof trees 5
This paper has become rather long andtouches quite different aspects. We have consid-
ered the possibility of splitting it in two parts, but concluded that this would contradict
its “exploratory” spirit.
2. Isometry groups
We start with a picture of the homogeneous tree T in horocyclic layers, since it will
2
be useful throughout the paper to keep this description in mind. Note that the negative
direction is “upwards” in the picture.
ω
HHHHH...−−−01321 ........0.......................................................◦...............0........................................................................1....................................................................................................................................0...................................0.......................................................1.............................................................................................................................................1.....................................................................................................................................0...........................................................................................................................................................0............................................................................1....................................................1...............................0....................................................................................................................................................................................0.........................................................................................1...................................................................................................................................1......................................................................................................................................0.....................................................................................................................................0.................................0........................................................................................................................1...........................................................................................................................................................................................................................................................1...............................0...........................................0..................................................................1............................................................................................................................................................................................1..................................................................................................................................... ......
. . .
. . .
. . .
∂∗T
Figure 1
Along with that picture comes a more detailed description of the geometry of T = T .
q
Any pair of vertices is connected by a unique geodesic path xy whose length (number
of edges) is the distance d(x,y). A geodesic ray is a one-sided infinite geodesic path (iso-
metric embedding of a half-line graph). Two rays are called equivalent, if their symmetric
difference (as sets of vertices) is finite. An end of T is an equivalence class of rays. The
boundary ∂T of T is the set of ends of T. For each ξ ∂T and each x T there is a
∈ ∈
unique geodesic ray xξ that represents ξ and starts with x. We choose an origin (root)
o T and write x = d(x,o). If v,w T = T ∂T then we can define their confluent
∈ | | ∈ ∪
c(v,w) as the last common element on ow and oz, a vertex of T unless w = z ∂T.
∈
With the ultrametric b
q−|c(w,z)|, if w = z
θ(w,z) = 6
0, if w = z,
(
T becomes a compact space.
b
6 L. Bartholdi, M. Neuhauser and W.Woess
We now select an end ω ∂T and write ∂∗T = ∂T ω . Given ω, we can define
∈ \ { }
the predecessor x− of x T as the neighbour of T that lies on xω. Thus, the ancestor
∈
relation is
(2.1) x 4 y x yω,
⇐⇒ ∈
and for x,y in general position, xfy is the maximal common ancestor, as explained in 1.
§
We write u(x,y) = d(x,xf y). Then the horocycle index of x (the Busemann function
with respect to ω) is
h(x) = u(x,o) u(o,x),
−
and the k-th horocycle is H = x T : h(x) = k . In particular,
k
{ ∈ }
(2.2) d(x,y) = u(x,y)+u(y,x) and h(x) h(y) = u(x,y) u(y,x).
− −
As in Figure 1, we can label the edges of T with the elements of Z , such that the edges
q
between a vertex and its q successors carry distinct labels, and such that on the geodesic
from any vertex to ω, only finitely many labels are non-zero. This labelling will be used
several times in the sequel.
For T = T , its affine group Aff(T ) is the stabilizer of ω in Aut(T ). It is an amenable
q q q
and non-unimodular closed subgroup of Aut(T ) that acts transitively on T , and all
q q
its elements are Busemann isometries. We have h(g) = h(go) for g Aff(T ). See
q
∈
Cartwright, Kaimanovich and Woess [14] for more details about the structure of
Aff(T ).
q
We shall need some basic facts about the modular function of an isometry group of a
locally finite graph X which is closed with respect to pointwise convergence. For more
details, see Trofimov [32] and Woess [33]. If Γ Aut(X) is such a group, and x X,
≤ ∈
then Γ denotes the stabilizer of x in Γ, while Γx is the orbit of x under Γ. Since Γ is
x
locally compact, it carries a left Haar measure dg. The modular function ∆ on Γ is the
unique multiplicative homomorphism Γ R which satisfies
+
→
∆(g ) f(gg )dg = f(g)dg
0 0
ZΓ ZΓ
for every g Γ and every continuous, compactly supported function f on Γ. Inserting
0
∈
for f the indicator function of Γ (which is an open, compact subgroup of Γ), one finds
x
the formula
(2.3) ∆(g) = Γ (gx) / Γ x
x gx
| | | |
for g Γ and for arbitrary x X, where Γ y is the (finite) number of elements in the
x
∈ ∈ | |
Γ -orbit of y; see e.g. [32, 33]. In particular, one has the following.
x
(2.4) Lemma. If Γ acts transitively on X then Γ is unimodular if and only if Γ y =
x
| |
Γ x for some ( every) x X and all its neighbours y.
y
| | ⇐⇒ ∈
In the sequel, we fix integers q ,...,q 2, and write o for the originof T = T , while
1 d ≥ j j qj
the symbol o will be used for the origin o = o o of DL = DL(q ,...,q ). If x,y DL,
1 d 1 d
··· ∈
then we say that a neighbour y of x has type e e , if y− = x and y = x−. In this case,
i− j i i j j
x is a neighbour with type e e of y. We write N (x) for the set of neighbours with
j i i,j
−
type e e of x.
j i
−
Horocyclic productsof trees 7
(2.5) Proposition. The group
d
= (q ,...,q ) = Aff(T )
A A 1 d h qj
j=1
Y
acts transitively on DL = DL(q ,...,q ) by (1.4). It is amenable. Furthermore, is
1 d
A
unimodular if and only if q = = q .
1 d
···
Proof. Let x = x x be in DL. Then there are g Aff(T ) such that g o = x ,
1··· d j ∈ qj j j j
j = 1,...,d. Settingg = g g asin(1.4),wegetg ,since h(g ) = h(x ) = 0.
1··· d ∈ A j j j j
Thus, go = x, and the action is transitive. Amenability of follows from the fact that it
A P P
is a closed subgroup of the direct product of the amenable groups Aff(T ).
qj
Regarding unimodularity, let x be in DL. By construction, must map every neigh-
x
A
bour y of x of type e e to a neighbour of x of the same type, and every permutation
j i
−
of this type can be achieved. Now x has exactly q neighbours of type e e . Therefore,
j j i
−
y = q , and (by exchanging x y and i j) x = q . If we vary i,j (i = j)
x j y i
|A | ↔ ↔ |A | 6
and apply Lemma 2.4, then we see that our group is unimodular if and only if all q
j
coincide. (cid:3)
Besides theelements of , theremaybefurther isometries ofDL. LetS = S(q ,...,q )
1 d
A
be the group of all permutations σ of 1,...,d such that q = q for all j. Then S
σ(j) j
{ }
acts on DL by
(2.6) σx = xσ−1(1) xσ−1(d),
···
that is, σ permutes identical trees in the horocyclic product. Thus, S also acts on by
A
group automorphisms (σ,g) gσ = σgσ−1. If all q are distinct, then S(q ,...,q ) is of
j 1 d
7→
course trivial. We shall prove the following.
(2.7) Theorem. The full isometry group of DL(q ,...,q ) is the semidirect product of S
1 d
with with respect to the action (σ,g) gσ,
A 7→
Aut(DL) = S⋉ .
A
Thus, Aut(DL) is amenable, and it is unimodular if and only if all q coincide.
i
For the proof, we need a description of the (graph-theoretical) link N(x) of a vertex
x DL, that is, the subgraph of DL spanned by the neighbours of x. Under the graph
∈
homomorphism H : DL A , where H(x) = k = h(x ),...,h(x ) , the link N(x)
d−1 1 d
→
maps onto the link N(k) in the lattice A . The latter link has (d 1)-cliques (complete
d−1 (cid:0) − (cid:1)
graphs on d 1 vertices) as its building blocks. Namely, for i 1,...,d , write
− ∈ { }
S+(k) = k+e e : j = i and S−(k) = k+e e : j = i .
i { i − j 6 } i { j − i 6 }
Each of those spans a complete subgraph of N(k). We have N(k) = d S+(k) =
i=1 i
d S−(k) ,
i=1 i S
SSi+(k)∩Sj+(cid:1)(k) = Si−(k)∩Sj−(k) = ∅, and Si+(k)∩Sj−(k) = {k+ei −ej} (i 6= j).
We write S±(x) for the set of all points in N(x) which are mapped to S±(k) by H. Note
i i
that S+(x) S−(x) = N (x). The edges in N(x) are as follows.
i ∩ j i,j
8 L. Bartholdi, M. Neuhauser and W.Woess
(1) If y,z S+(x) then there are j,k = i such that H(y) = k + e e and H(z) =
∈ i 6 i − j
k+e e . In this case, there is an edge between y and z if and only if j = k and y = z ,
i k i i
− 6
in which case z is a neighbour of type e e of y (i.e., z− = y and y− = z ). Thus, the
j − k j j k k
subgraph of N(x) that is mapped onto an edge in S+(k) is the graph D(q ,q ) consisting
i i i
of q independent edges with their endpoints; see Figure 2a.
i
(2) If y,z S−(x) then there are j,k = i such that H(y) = k + e e and H(z) =
∈ i 6 j − i
k + e e . In this situation, there is an edge between y and z if and only if j = k;
k i
− 6
furthermore, z is a neighbour of type e e of y (i.e., z− = y and y− = z ). Thus, the
k − j k k j j
subgraph of N(x) that is mapped onto the edge [k+e e ,k+e e ] in S+(k) is the
j − i k − i i
complete bipartite graph K(q ,q ); see Figure 2b.
j k
•••.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................•••............ ••..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................•••........................
Figure 2a: D(3,3) Figure 2b: K(2,3)
Figure 3a shows the link of a vertex of DL(2,2,3). When d 3, the link is connected.
≥
When d = 2, it consists of q +q isolated points, and in this case, it will be more useful
1 2
to consider the 2-link N (x) spanned by all points at distance 1 and 2 from x. Each of
2
the q neighbours v of type e e of x = x x is connected by an edge to each of the
1 1 2 1 2
−
q 1 points x y , where y = x is a sibling of x in T , that is, y− = x−. In turn, there
2 − 1 2 2 6 2 2 2 2 2
is an edge between each of those neighbours v of x and its q neighbours of the same type
1
e e . Exchanging the role of e and e , one finds the other part of N (x). See Figure
1 2 1 2 2
−
3b, where we have also drawn the edges from x to its neighbours in dotted lines.
S+(x)
SS21+−•.......................................(..(.............................................................................x..x.................................................................................................................)..)........................................................................................................................................•...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................••..............................................................................................................................................................................................................•.............................................................................................................................................•..•.........................................................................................................................................................................................................................................................................................................................3.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................••..................................................................................................................................•..............................................................................................................................................................................••.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................•....................................................................................................................................................................................SS.............................................................................................................................12.............+−..................................................................................................................•.......((..................xx)) •.....................................•..........................................................................................................................................................•.........................................................................................................................................................................•...........................................................•..................................................•.•.....................................................................................................................................................................................................................•.••..........................................................................................................................................................................................................................................................••..........................................•....................x.....•........................................................................................................••........................................................................................................................................................................................................................•............................................................•......................................•............................................................................................................................••..................................................................................................................................•.......
S−(x)
3
Figure 3a: N(x) in DL(2,2,3) Figure 3b: N (x) in DL(3,2)
2
(2.8) Lemma. For every x DL and every g in the stabilizer Aut(DL) of x, there is
x
∈
σ S such that for all i,j (i = j), we have gNi,j(x) = Nσ−1(i),σ−1(j)(x).
∈ 6
Proof. Our g acts as a graph isometry on N(x), and also on N (x). If d = 2 (see Figure
2
3b) then g must permute the two connected components of N (x). This permutation
2
must be trivial unless q = q . Thus, the statement follows when d = 2.
1 2
If d 3, then by the above, each S−(x) is a complete (d 1)-partite subgraph of
≥ i −
N(x). That is, its vertex set is partitioned into the sets N (x) having cardinality q ,
j,i j
Horocyclic productsof trees 9
j = i, such that every pair of vertices in N (x) N (x), k = j, is connected by an
j,i k,i
6 × 6
edge, while there are no edges between different vertices within each N (x). Also, S−(x)
j,i i
is a maximal complete (d 1)-partite subgraph of N(x) (i.e., it is not contained in any
−
bigger complete (d 1)-partite subgraph). This property must be preserved by isometries
−
of N(x), and gS−(x) must again be a maximal complete (d 1)-partite subgraph of
i −
N(x), whose d 1 classes must have the same cardinalities q as the sets N (x), j = i.
j i,j
− 6
Thus, there is a permutation σ of 1...,d such that gS−(x) = S− (x), and from
{ } i σ−1(i)
j = i : qj = qi = k = σ−1(i) : qk = qi we deduce qi = qσ−1(i) for each i. Therefore,
|{ 6 }| |{ 6 }|
σ S.
∈
For each j = i, we must have gNj,i(x) = Nk,σ−1(i)(x) and gNi,j(x) = Nl,σ−1(j)(x) for
6
some k = σ−1(i), l = σ−1(j). We still have to show that k = σ−1(j), and consequently
6 6
l = σ−1(i). Now note that with respect to its (“inner”) graph metric, N(x) has diameter
3, and that the only points at distance 3 from all y N (x) are precisely those in N (x).
i,j j,i
∈
Therefore
Nl,σ−1(j)(x) = gNi,j(x) = Nσ−1(i),k(x)
as sets, and thus l,σ−1(j) = σ−1(i),k . This completes the proof of the lemma. (cid:3)
We shall also n(cid:0)eed the fo(cid:1)llow(cid:0)ing prepa(cid:1)ratory lemma.
(2.9) Lemma. If g Aut(DL) satisfies gN (x) = N (x) for all i,j (i = j), then there
x i,j i,j
∈ 6
is h such that g h .
∈ Ax N(x) ≡ N(x)
(cid:12) (cid:12)
(As a matter of fact, it will turn out below that g itself must belong to .)
(cid:12) (cid:12)
A
Proof. For each pair (i,j), the isometry g permutes the q elements of N (x). By the
i i,j
structure of S+(x), this permutation must be independent of j (j = i). That is, there is a
i 6
permutation h among the successors of x in T such that (gy) = h y for all y N (x)
i i i i i i i,j
∈
and all j = i. This permutation can be extended to an isometry of T , again denoted h ,
i i
6
that fixes x and permutes the branches of T “below” x . Setting h = h h according
i i i 1 d
···
to (1.4), we obtain the required element of . (cid:3)
x
A
Proof of Theorem 2.7. Consider an arbitrary g Aut(DL). Set x = g−1o, where o is the
∈
root of DL. Then there is g such that x = g−1o, whence gg−1o = o. By Lemma
x ∈ A x x
2.8, there is σ S, acting on DL by (2.6), such that g′ = σ−1gg−1 Aut(DL) satisfies
∈ x ∈ o
g′N (o) = N (o) for all i,j. We claim that g′ is type-preserving, that is, for all x DL,
i,j i,j
∈
(2.10) g′N (x) = N (g′x) i,j 1,...,d (i = j).
i,j i,j
∀ ∈ { } 6
(Note that this does hold for every g′ .) Since DL is connected and (2.10) is true for
∈ A
x = o, it is sufficient to show the following.
(2.11) If (2.10) holds for x DL then it holds for every y N(x).
∈ ∈
So suppose that (2.10) holds for x. Let h be as in Lemma 2.9, associated with
x
∈ A
g′, and set g′′ = h−1g′ Then g′′v = v for every v x N(x). Let y N(x), so that
∈ { } ∪ ∈
y N (x) for some i,j. Since g′′y = y, Lemma 2.8 implies that there is τ S such that
i,j
∈ ∈
g′′Ni,j(y) = Nτ−1(i),τ−1(j)(y).
10 L. Bartholdi, M. Neuhauser and W.Woess
Since x N (y) and g′′x = x, we find that τ(i) = i and τ(j) = j. Also, if k = i,j,
j,i
∈ 6
then N (y) = N (x) is stabilized by g′′. Thus, τ(k) = k. We see that τ is the identity,
k,i k,j
and as h ,
∈ A
g′N (y) = hg′′N (y) = hN (y) = N (hy) = N (g′y),
i,j i,j i,j i,j i,j
as claimed. This proves (2.11) and consequently (2.10).
We now use (2.10) to show that g′ . Our claim is the following.
∈ A
(2.12) If x,y DL satisfy x = y for some i 1,...,d then (g′x) = (g′y) .
i i i i
∈ ∈ { }
Indeed, if this holds, then define g Aff(T ) as follows. Given x T , choose x DL
i ∈ qi i ∈ i ∈
with i-th coordinate x , and set g x = (g′x) . This is independent of the specific choice
i i i i
of x by (2.12). We therefore get g′ = g g .
1 d
··· ∈ A
(0)
For any d and i,j 1,...,d with i = j, we can define recursively N (x) = x and
∈ { } 6 i,j { }
N(k)(x) = N (y) : y N(k−1)(x) , where x DL. We observe that
i,j { i,j ∈ i,j } ∈
S v = w for all v,w N(k)(x).
j j ∈ i,j
The proof of (2.12) is different in the cases d = 2 and d 3. In both cases, we may
≥
assume without loss of generality that i = 1 in (2.12). Suppose therefore that x,y DL
∈
satisfy x = y .
1 1
Case 1: d = 2. Consider x ,y and their common ancestor u = x fy in T . Since
2 2 2 2 2 2
x = y , we have h(x ) = h(y ), whence d(x ,u ) = d(y ,u ) = h(x ) h(u ) =: k 0.
1 1 2 2 2 2 2 2 2 2
− ≥
We can find u T such that x 4 u and h(x ) h(u ) = k, so that u = u u DL
1 1 1 1 1 1 1 2
∈ − − ∈
and x,y N(k)(u). Since g′ is type-preserving, we have
∈ 2,1
g′N(k)(u) = N(k)(g′u) g′x,g′y.
2,1 2,1 ∋
Using the above observation, we get (g′x) = (g′y) , which proves (2.12).
1 1
Case 2: d 3. The subgraph of DL(q ,...,q ) spanned by v = v v DL : v =
1 d 1 d 1
≥ { ··· ∈
x is connected; indeed, it is isomorphic with DL(q ,...,q ). Thus, there is a path from
1 2 d
}
x to y in DL all whose vertices have the same first coordinate x : if v,w are successive
1
vertices on this path then w N (v) where i,j = 1, and v = w = x . But then (2.10)
i,j 1 1 1
∈ 6
implies g′w N (g′v), so that g′v and g′w differ only in the i-th and j-th coordinates.
i,j
∈
In particular, (g′w) = (g′v) , whence inductively (g′y) = (g′x) .
1 1 1 1
We conclude that g = σg′, where σ S and g′ , so that we have completed the
∈ ∈ A
description of Aut(DL).
If h and g = σg′ Aut(DL) with σ S and g′ , then ghg−1 = σ(g′hg′−1)σ−1
∈ A ∈ ∈ ∈ A
is type-preserving, so that (2.12) implies ghg−1 . It is now obvious that the factor
∈ A
group Aut(DL)/ is S. (cid:3)
A
We briefly remind the reader of the concept of amenability of a locally compact group
( existence of a finitely additive, left invariant probability measure on the group): see
≡
Paterson [27]. Recall thata locallyfinite graphX iscalled amenable, if its isoperimetric
constant is 0, that is, the number
κ = inf ∂F /Vol(F) : F X finite ,
{| | ⊂ }