Table Of ContentMETABELIAN GROUPS WITH QUADRATIC DEHN
FUNCTION AND BAUMSLAG-SOLITAR GROUPS
YVES DE CORNULIER AND ROMAIN TESSERA
Abstract. Weprovethatacertainclassofmetabelianlocallycompactgroups
have quadratic Dehn function. As an application, we embed the solvable
Baumslag-Solitargroupsinfinitelypresentedmetabeliangroupswithquadratic
1
Dehn function. Also, we prove that Baumslag’s finitely presented metabelian
1
groups,inwhichthelamplightergroupsembed,havequadraticDehnfunction.
0
2
n 1. Introduction
a
J
The Dehn function of a finitely presented group is a natural invariant, which
3
from the combinatorial point of view is a measure of the complexity of the word
2
problem, andfromthegeometricpointofviewcanbeinterpretedasanisoperime-
]
try invariant. We refer the reader to Bridson’s survey [8] for a detailed discussion.
R
G Given a group with a solvable word problem, it is natural to wonder whether it
can be embedded into a group with small Dehn function; a major result in this
.
h
context is a characterization of subgroups of groups with at most polynomial
t
a
Dehn function as those groups with NP solvable word problem [9]. It is then
m
natural to ask for which groups this can be improved. Word hyperbolic groups
[
are characterized as those with linear Dehn function and there are many known
3
stringentrestrictionsontheir subgroups; forinstance, theirabeliansubgroups are
v
8 virtually cyclic. In particular, they cannot contain a solvable Baumslag-Solitar
4 group
1
0 BS(1,n) = (cid:104)t,x| txt−1 = xn(cid:105)
.
3 for any integer n with |n| ≥ 1. The next question is whether the group is
0
embeddable into a finitely presented group with quadratic Dehn function. We
0
1 obtainhereapositiveanswerforthesolvableBaumslag-Solitargroups. Thegroup
:
v BS(1,n) is well-known to have an exponential Dehn function whenever |n| ≥ 2,
i as for instance it is proved in [14] that any strictly ascending HNN-extension of
X
a finitely generated nilpotent group has an exponential Dehn function.
r
a
Theorem 1. The solvable Baumslag-Solitar group BS(1,n) can be embedded into
a finitely presented metabelian group with quadratic Dehn function.
Date: November 26, 2010.
2000 Mathematics Subject Classification. 20F65 (primary); 20F69, 20F16, 53A10
(secondary).
1
2 YVES DE CORNULIER AND ROMAIN TESSERA
This answers a question asked in [9]. It was previously known, by the general
result mentioned above, that it can be embedded into a “big” finitely presented
groupwithpolynomialDehnfunction[9]. ThenBS(1,n)wasprovedtobeembed-
dable in a finitely presented metabelian group with at most cubic Dehn function
in [5]. Theorem 1 is optimal, because as we mentioned above, BS(1,n) cannot be
embedded into a word hyperbolic group, and all other groups have a quadratic
lower bound on their Dehn function [19]. Quadratic Dehn function groups enjoy
some special properties not shared by all polynomial Dehn function groups, for
instance they have all asymptotic cones simply connected [20]. After Gromov
[13], many solvable groups were proved to have quadratic Dehn function (see for
instance [2, 12, 21]).
Our construction of a finitely presented metabelian group with quadratic Dehn
function in which BS(1,n) embeds, is elementary and uses representation by
matrices; the proof involves an embedding as a cocompact lattice into a group of
matrices over a product of local fields. The theorem then follows from a general
result (Theorem 3.1), allowing to prove that various metabelian groups, made up
from local fields, have quadratic Dehn function. Another application concerns
a finitely presented metabelian group Λ , introduced by G. Baumslag [6] as an
p
instance of a finitely presented metabelian group in which the lamplighter group
(Z/pZ)(cid:111)Z embeds as a subgroup (see Example 3.3).
Theorem 2. Baumslag’s finitely presented metabelian group Λ has quadratic
p
Dehn function.
A polynomial upper bound was recently obtained by Kassabov and Riley [17].
We hope that the method developed in the paper will convince the reader that
the study of the Dehn function of discrete groups is indissociable from its study
for general locally compact groups, a fact more generally true in the problem of
quasi-isometry classification of solvable groups.
Organisation. In the next section, we describe our embedding of the Baumslag-
Solitargroup. InSection3, westateourfundamentalresult, namelyTheorem3.1.
Section4isdedicatedtoanimportanttechnicallemmawhosebasicideaismainly
due to Gromov. Finally, Section 5 contains the proof of Theorem 3.1.
Acknowledgement. We thank Mark Sapir for suggesting us this problem and
for valuable discussions. We are indebted to the referee for many corrections and
clarifications.
2. Construction of the embedding
Our results use, in an essential way, a notion of Dehn function not restricted
to discrete groups. Let G be any locally compact group. Recall [1, Section 1.1]
that G is compactly presented if for some/any compact generating symmetric set
S and some k (depending on S), there exists a presentation of the abstract group
METABELIAN GROUPS WITH QUADRATIC DEHN FUNCTION 3
G with S as set of generators, and relators of length ≤ k. If G is discrete, this
amounts to say that G is finitely presented. If (S,k) is fixed, a relation means
a word w in the letters of S which represents the trivial element of G; its area
is the least m such that w is a product, in the free group, of ≤ m conjugates of
relations of length ≤ k. The Dehn function of G is defined as
δ(n) = sup{area(w)|w relation of length ≤ n}.
The precise value of δ(n) depends on (S,k), but not the ≈-asymptotic behavior,
where u(n) ≈ v(n) if for suitable positive constants a ,...,b independent of n
1 4
a u(a n)−a n−a ≤ v(n) ≤ b u(b n)+b n+b , ∀n ≥ 0.
1 2 3 4 1 2 3 4
In the combinatorial point of view, the Dehn function of a finitely presented
group is a measure of the complexity of its word problem. In the geometric point
of view, compact presentability means simple connectedness at large scale [13,
1.C ], andtheDehnfunctionappearsasaquantifiedversion, andanupperbound
1
on the Dehn function is often referred as an “isoperimetric inequality”.
For any non-zero integer n ∈ Z−{0}, the solvable Baumslag-Solitar group can
be described as
BS(1,n) = Z[1/n](cid:111)Z,
where Z acts on Z[1/n] by multiplication by n.
(cid:18) (cid:19) (cid:18) (cid:19)
n 0 2 1
Consider the two commuting matrices A = , B = (B can be
0 n 1 1
replaced by any matrix in GL (Z) with two real eigenvalues not of modulus one).
2
Define the group
Γ = Z[1/n]2 (cid:111) Z2.
n (A,B)
Clearly, Γ is finitely generated, since we can “go up” in Z[1/n]2 by conjugating
n
by A. Moreover, it contains an obvious copy of BS(1,n), namely (Z[1/n]×{0})(cid:111)
(Z×{0}).
Theorem 2.1. The group Γ is finitely presented with quadratic Dehn function.
n
Corollary 2.2. The group BS(1,n) can be embedded into a finitely presented
group with quadratic Dehn function.
Theorem 2.1 is obtained by working inside a more convenient group, which
contains Γ as a cocompact lattice. Let Q denote the p-adic field, and define
n p
the ring Q as the direct product of all Q when p ranges over the set of distinct
n p
prime divisors of n. Then the natural diagonal embedding of Z[1/n] into Q ⊕R
n
has discrete cocompact image (check it as an exercise or see [23, Chap. IV, §2]),
and the corresponding embedding of Z[1/n]k into Qk ⊕Rk is equivariant for the
n
natural actions of GL (Z[1/n]). Accordingly, Γ stands as a cocompact lattice in
k n
the locally compact group
(Q2 ⊕R2)(cid:111)Z2,
n
4 YVES DE CORNULIER AND ROMAIN TESSERA
where the action is still defined by the same pair of matrices (A,B) (viewed
as matrices over the ring Q × R). Now we use the fact that the (asymptotic
n
behavior of the) Dehn function is a quasi-isometry invariant (see [3] for a proof
in the discrete setting; the same proof working in the general case), so Γ has the
n
same Dehn function as this larger group.
Decompose R2 = V ⊕ V along the eigenspaces of B, so that B dilates V
+ − +
and contracts V . Observe that
−
• BA−1 contracts both V and Q2
− n
• B−1A−1 contracts both V and Q2
+ n
• A−1 contracts both V and V .
+ −
Thus, for every pair among Q2, V and V , there is a common contraction of
n + −
the acting group. This phenomenon is enough to prove that the Dehn function
is quadratic, and Theorem 2.1 is a consequence of a more general result, which
is the object of the next section (and will be proved in the last section).
Remark 2.3. It was observed in [11, Section 9] that the asymptotic cone of
BS(1,n) (|n| ≥ 2) or SOL (K) is (for any choice of an ultrafilter) bilipschitz
3
homeomorphic to the Diestel-Leader R-graph
{(x,y) ∈ T×T : b(x)+b(y) = 0},
where T is the universal complete R-tree everywhere branched of degree 2ℵ0 and
b any Busemann function on T. The same reasoning shows that any asymptotic
cone of Γ is bilipschitz homeomorphic to
n
{(x,y,z) ∈ T3 : b(x)+b(y)+b(z) = 0}
(which is also bilipschitz homeomorphic to the asymptotic cone of the SOL (K)
5
for any local field K, as follows from the final remarks in [11, Section 9]).
3. A general result and further comments
By local field we mean a non-discrete locally compact normed field, see [23].
Consider a semidirect product
m
(cid:77)
G = V (cid:111)A,
i
i=1
whereAisafinitelygeneratedabeliangroupofrankdandwhereV isA-invariant
i
and is isomorphic to a product of local fields, on which A acts by scalar multipli-
cation.
Theorem 3.1. Assume that for every pair i,j, there exists an element of A
acting by strict contractions on both V and V (i.e. acting by multiplication by
i j
an element of norm < 1 on each factor). Then G is compactly presented with
quadratic Dehn function if d ≥ 2, and linear Dehn function if d = 1.
METABELIAN GROUPS WITH QUADRATIC DEHN FUNCTION 5
Remark 3.2. The consideration of common contractions is ubiquitous in this
context, see for instance [7, 1].
Example 3.3. Let K be any local field. Let SOL (K) be the semidirect
2d−1
product of Kd by the set of diagonal matrices with determinant of norm one. It
has a cocompact subgroup of the form Kd(cid:111)Zd−1, obtained explicitly by reducing
to matrices whose diagonal entries are all powers of a given element of K of norm
(cid:54)= 1, so we can apply Theorem 3.1, which yields that SOL (K) has quadratic
2d−1
Dehn function whenever 2d−1 ≥ 5. This was proved by Gromov when K = R
[13, 5.A ]. As a further application, let p be prime and consider the finitely
9
presented group
Λ = (cid:104)a,s,t| ap, [s,t], [at,a], as = ata(cid:105),
p
which was introduced by Baumslag [6] as a finitely presented metabelian group
containing a copy of the lamplighter group (Z/pZ) (cid:111) Z. Let F ((t)) denote the
p
field of Laurent series over the finite field F . The following proposition is very
p
standard, but we have no reference for a complete proof. As a consequence we
deduce that Λ has quadratic Dehn function.
p
Proposition 3.4. For any prime p, the group Λ embeds as a cocompact lattice
p
into SOL (F ((u))).
5 p
Proof (sketched). Let F [X,X−1] be the ring of Laurent polynomials over F .
p p
Consider the group
Ω = F [X,X−1,(1+X)−1](cid:111)Z2,
p p
where the generators of Z2 act by multiplication by X and 1+X respectively. We
have an obvious homomorphism Λ → Ω mapping a to the unit element of the
p p
ring F [X,X−1,(1+X)−1] and (t,s) to the canonical basis of Z2; it is essentially
p
contained in Baumslag’s proof [6] that this is an isomorphism.
Consider the embedding
σ : F [X,X−1,(1+X)−1] → F ((u))3
p p
(P ,P ,P ) (cid:55)→ (P (u),P (u−1),P (u−1))
1 2 3 1 2 3
This is an embedding as a cocompact lattice: this can be checked by hand (see
[18, Proof of Prop. 3.4]), or follows from general results [16].
If we make the two generators of Z2 act on F ((u))3 by the diagonal matrices
p
(u,u−1,u−1) and (u+1,u−1 +1,u),
this makes σ a Z2-equivariant embedding. Moreover, denoting by D1(K) the
3
group of 3×3 matrices with determinant of norm one, it is readily seen that this
embedding of Z2 into D1(F ((u))) has discrete and cocompact image. Therefore
3 p
the embedding σ extends to a discrete cocompact embedding of Ω into
p
F ((u))3 (cid:111)D1(K) = SOL (F ((u))). (cid:3)
p 3 5 p
6 YVES DE CORNULIER AND ROMAIN TESSERA
Remark 3.5. If n ≥ 2 is not prime, replacing, in the argument, F ((u)) by
p
(Z/nZ)((u)), we readily obtain that Λ has quadratic Dehn function as well.
n
However, the argument does not apply for n = 0, because Λ contains a copy of
0
the wreath product Z(cid:111)Z and therefore does not stand as a discrete linear group
over a product of local fields, and actually Kassabov and Riley [17] proved that
Λ has exponential Dehn function.
0
Remark 3.6. It is natural to ask whether there is a group with some embedding
into a discrete group with polynomial Dehn function, but not into one with
quadratic Dehn function. The answer is positive, as M. Sapir showed to the
authors: consider a problem in ntime(n3) (that is, solvable in cubic time by a
non-deterministic Turing machine), but not in ntime(n2). Such problems exist
by [15, p. 76] or [4, p. 69-70]. By [9], there exists a finitely presented group
Γ with word problem in ntime(n3) but not ntime(n2), and again by [9] there
exists a finitely presented group Λ with polynomial Dehn function containing Γ
as a subgroup. However, Γ cannot be embedded into a finitely presented group
with quadratic Dehn function, since otherwise, using [22, Theorem 1.1] its word
problem would be in ntime(n2). Still, it would be interesting to have an example
of more geometrical nature.
4. Reduction to special words
In this section, we prove Proposition 4.3, which reduces the computation of
the Dehn function to its computation for words of a special form. The basic idea
is due to Gromov [13, p. 86].
Lemma 4.1. Let u : R → R be a family of functions, indexed by integers
k >0 >0
k ≥ 1, satisfying lim u (x) > 0 for all k. Assume that
x→∞ k
zu (y)
k
−→ 0 uniformly in y ≥ 1,k ≥ 1.
uk(yz) z→∞
Consider a function f : R → R , locally bounded and positive constants
>0 >0
c ,c ,x , such that for all k ≥ 1,x ≥ x
1 2 0 0
f(x) ≤ u (x)+c kf(c x/k).
k 1 2
Then, for some constants A,x and some k ≥ 1, we have
1
f(x) ≤ Au (x), ∀x ≥ x .
k 1
For example, u (x) = a xα, for α > 1 and arbitrary constants a > 0, satisfy
k k k
the assumption.
Proof. Set η = 1/(2c c ). There exists, by the assumption, ε > 0 (we choose
1 2 0
ε ≤ 1/2) such that for all z ≥ 1/ε , y,k ≥ 1 we have zuk(y) ≤ η. Therefore,
0 0 u (yz)
k
for all x,ε > 0 such that xε ≥ 1 and ε ≤ ε we have uk(xε) ≤ η (as we check by
0 εu (x)
k
METABELIAN GROUPS WITH QUADRATIC DEHN FUNCTION 7
setting y = xε and z = ε−1). Taking ε = c /k, we get, for k ≥ c /ε and for all
2 2 0
x ≥ ε−1
(cid:16)c x(cid:17) c η
2 2
u ≤ u (x).
k k
k k
We now fix k ≥ c /ε (so ε is fixed as well and ε ≤ ε ≤ 1/2). We let x ≥
2 0 0 1
max(ε−1,x k) be large enough so that inf u (x) > 0. Therefore, since f is
0 x≥x1 k
locally bounded, there exists A ≥ 2 such that for all x ∈ [x ,ε−1x ] we have
1 1
f(x) ≤ Au (x).
k
Now let us prove that f(x) ≤ Au (x) for all x ≥ x , showing by induction on
k 1
n ≥ 1 that f(x) ≤ Au (x) for all x ∈ [x ,ε−nx ]. It already holds for n = 1;
k 1 1
suppose that the induction is proved until n−1 ≥ 1. Take x ∈ [ε1−nx ,ε−nx ].
1 1
By induction hypothesis, we have f(x(cid:48)) ≤ Au (x(cid:48)) for all x(cid:48) ∈ [x ,εx]. We have
k 1
f(x) ≤ u (x)+c kf(c x/k)
k 1 2
≤ u (x)+c kAu (c x/k)
k 1 k 2
≤ u (x)(1+c Ac η)
k 1 2
≤ u (x)(1+A/2) ≤ Au (x). (cid:3)
k k
If x is real, we denote by (cid:98)x(cid:99) = sup(]−∞,x]∩Z) and (cid:100)x(cid:101) = inf([x,+∞[∩Z)
its lower and upper integer parts.
Let G be a locally compact group generated by a compact symmetric subset
S. Let F be the nonabelian free group over S. For w,w(cid:48) ∈ F , we write w ≡ w(cid:48)
S S
if w and w(cid:48) represent the same element of G.
Let F be a set of words in S. We write F[k] the set of words obtained as the
concatenation of ≤ k words of F. We can define the restricted Dehn function
δ (n) as the supremum of areas of null-homotopic words in F (say sup∅=0). We
F
say that F is efficient if there exists a constant C such that for every word w
in S, there exists w(cid:48) ∈ F such that w(cid:48) ≡ w and |w(cid:48)| ≤ C|w| . Also, if x is a
S S
nonnegative real number, δ(x) can obviously be defined as the supremum of areas
of loops of length ≤ x (so δ(x) = δ((cid:98)x(cid:99))).
Lemma 4.2. Suppose that F is efficient. Then for any k ∈ Z and n ∈ R
>0 >0
we have
(cid:16) (cid:108)n(cid:109)(cid:17) (cid:16) (cid:108)n(cid:109)(cid:17)
δ(n) ≤ kδ (C +1) +δ Ck ;
F[k]
k k
in particular for n ≥ k we have
(cid:18) (cid:19)
2n
δ(n) ≤ kδ (C +1) +δ (2Cn).
F[k]
k
Proof. Consider a loop γ of length ≤ n and cut it into k segments [a ,a ] of
i i+1
length ≤ (cid:100)n/k(cid:101) (see Figure 1). Set b = a−1a ; there exists b(cid:48) ≡ b with b(cid:48) ∈ F
i i i+1 i i i
and b(cid:48) ≤ C(cid:100)n/k(cid:101). Set γ = b(cid:48)b−1, so γ is null-homotopic.
i i i i i
Thus the loop γ has been decomposed into k loops γ ,...,γ of length ≤
1 k
(C+1)(cid:100)n/k(cid:101) and the loop γ(cid:48) defined by the word b(cid:48) ...b(cid:48), of length ≤ Ck(cid:100)n/k(cid:101),
1 k
8 YVES DE CORNULIER AND ROMAIN TESSERA
a
1
γ
γ 2
1 a
2
a
0
γ’
γ
a
k-1
Figure 1. The path γ cut into k segments.
lying in F[k]. Accordingly
k
(cid:88)
area(γ) ≤ area(γ )+area(γ(cid:48))
i
i=1
≤ kδ((C +1)(cid:100)n/k(cid:101))+δ (Ck(cid:100)n/k(cid:101)). (cid:3)
F[k]
Proposition 4.3. Suppose that F is efficient. Suppose that for some ζ > 1, for
all k, there is a constant a such that we have δ (n) ≤ a nζ for all n. Then
k F[k] k
there exists a constant C(cid:48) such that δ(n) ≤ C(cid:48)nζ for all n.
Proof. Set u (x) = a (2Cx)ζ, where C is given by the definition of efficiency of
k k
F. By Lemma 4.2, we have
(cid:18) (cid:19)
(C +1)2n
δ(n) ≤ u (n)+kδ
k
k
for all large n. The conclusion then follows from Lemma 4.1. (cid:3)
METABELIAN GROUPS WITH QUADRATIC DEHN FUNCTION 9
Remark 4.4. If C is the constant given in the definition of efficiency, we have
the following general inequality
δ (n) ≤ kδ ((2C +1)n).
F[k] F[3]
Indeed, take a loop in F[k] of size n, defined by a word u ...u . The endpoint
1 k
of the path u ...u is at distance at most n/2 from the origin, and can therefore
1 i
be joined to the origin by a path s in F of size ≤ Cn/2. The loop µ defined by
i i
s , s and u lies in F[3], and the original loop is filled by the k loops µ ,...,µ ,
i−1 i i 1 k
yielding the inequality.
In particular, in the hypotheses of Proposition 4.3 it is enough to check the
case k = 3.
5. Proof of Theorem 3.1
Let us turn back to the group G of Theorem 3.1. Refining the decomposition
(cid:76)V if necessary, we can suppose that each V is a local field K on which A (cid:39) Zd
i i i
acts by scalar multiplication. Fix a multiplicative norm on each K , and let S
i i
be the one-ball in K . Let T be a symmetric generating set in A, and let | · |
i
denote the word length in A with respect to T. Let F denote the set of words of
the form
(cid:32) (cid:33)
m
(cid:89)
t v t−1 t,
i i i
i=1
where v ∈ S and t ,t are words in the letters of T.
i i i
Lemma 5.1. Let G = (cid:76)m K (cid:111)A be a group as in Theorem 3.1, with each K
i=1 i i
a local field with an action by scalar multiplication. Replace the last assumption
(on pairs (i,j)) by the weaker assumption that for every i there exists an element
of A acting on K by contractions. Then F is efficient.
i
Proof. Let t ∈ T act on K by multiplication by λ (t) ∈ K∗. Define c > 1 by
i i i
c = min (max |λ (t)|) and C = max(2,max |λ (t)|).
i t i t,i i
(cid:83)
We first claim that every word of length ≤ n in the generating set S ∪ T
i
represents an element xt = ((x ),t) of (cid:76)K (cid:111) A with (cid:107)x (cid:107) ≤ Cn and |t| ≤ n.
i i i
This is checked by induction on n. If we multiply on the right by an element of
T, the induction step works trivially. Let us look when we multiply on the right
by an element v of S . Then xtv = x(tvt−1)t. Then in K , (cid:107)tvt−1(cid:107) ≤ Cn, so
i i
(cid:107)x+tvt−1(cid:107) ≤ Cn +Cn ≤ Cn+1.
Now consider such an element ((x ),t), with (cid:107)x (cid:107) ≤ Cn and |t| ≤ n and write
i i
it as an element of F of length ≤ Kn for some constant K. By definition of c,
we can write x = t v t−1 with t a word on the alphabet T of length at most
i i i i i
(cid:100)log (Cn)(cid:101) ≤ n(cid:100)logC/logc(cid:101)
c
and(cid:107)v (cid:107) ≤ 1aletterofS . Sotheelementisrepresentedbytheword((cid:81)mt v t−1)t,
i i 1 i i i
which belongs to F and has length ≤ n(2m(cid:100)logC/logc(cid:101)+1). (cid:3)
10 YVES DE CORNULIER AND ROMAIN TESSERA
Proof of Theorem 3.1. The group G has an obvious retraction onto A (cid:39) Zd,
and therefore its Dehn function is bounded below by that of Zd, which is linear
if d = 1 and quadratic otherwise.
Let us now prove the upper bound on the Dehn function. As we mentioned
above, we can suppose that each V is a local field K with action of A by scalar
i i
(cid:83)
multiplication. Take S = T ∪ S as set of generators. Define R as the set of
i
relators consisting of
• finitely many defining relations of A with respect to T;
• relations of length 4 of the form tst−1 = s(cid:48), t ∈ T, s,s(cid:48) ∈ S for some i;
i
• relations of length 4 of the form [s,s(cid:48)] = 1 when s ∈ S ,s(cid:48) ∈ S ;
i j
• relations of length 3 of the form ss(cid:48) = s(cid:48)(cid:48) when s,s(cid:48),s(cid:48)(cid:48) ∈ S .
i
The proof that follows will show that this is a presentation of G and that the
corresponding Dehn function is quadratically bounded.
Denote by α the corresponding area function with respect to R. Define
c(m ,m ) = α(m−1m )
1 2 1 2
as the corresponding bi-invariant [0,∞]-valued distance. (The claim that R is a
set of defining relators of G is equivalent to showing that c takes finite values on
pairs (m ,m ) of homotopic [i.e. m ≡ m ] words). It is useful to think of c as
1 2 1 2
the cost of going from m to m . Recall that we write equality of words as = and
1 2
equality in G as ≡. By the triangular inequality together with bi-invariance, we
get the following useful “substitution inequality”, which we use throughout
α(xyz) ≤ α(xy(cid:48)z)+c(y,y(cid:48)).
Claim 1. Under the assumptions of Lemma 5.1, there exists a constant C such
that for all i, if v,w ∈ S and if s is a word of length ≤ n with respect to T and
i
svs−1 ≡ w (i.e. s·v = w for the given action), then c(svs−1,w) ≤ Cn2.
Indeed,wecanwrites ≡ twitht = t t awordoflengthninT,whereallletters
1 2
in t contract K and all letters in t dilate K . Clearly, for every right terminal
2 i 1 i
segment τ of t (made of the j last letters in t), τvτ−1 represents an element w
j j
of S . Therefore an immediate induction on the j provides c(τvτ−1,w ) ≤ |τ|, so
i j
c(tvt−1,w) ≤ n. Now A has a quadratic Dehn function, so c(s,t) ≤ C n2, hence
1
c(svs−1,tvt−1) ≤ 2C n2 and thus
1
c(svs−1,w) ≤ n+2C n2 ≤ C n2
1 2
and the claim is proved.
By assumption (of the theorem), there exist elements s contracting both K
ij i
and K . We can suppose that all s belong to T (enlarging T if necessary). If
j ij
s ∈ S, let it act on K by multiplication by λ . There exists a positive integer
i s,i
M (depending only on G and T) such that for all t ∈ T, sMt contracts both K
ij i
and K .
j
