Degree of commutativity for right-angled Artin groups

0.18 MB·

by  Motiejus Valiunas

DEGREE OF COMMUTATIVITY FOR RIGHT-ANGLED
ARTIN GROUPS
MOTIEJUS VALIUNAS
7 Abstract. The degree of commutativity of a finite group F, defined
1 as the probability that two randomly chosen elements in F commute,
0 hasbeenstudiedextensively. Recently[1]thisdefinitionwasgeneralised
2 to all finitely generated groups. In this paper the degree of commuta-

n tivity is computed for right-angled Artin groups with respect to their
a natural generating set. An additional result concerning sphere sizes of
J groupswithrationalsphericalgrowthseries(withrespecttosomefinite

6 generating set) is obtained.
1

]
R
G
1. Introduction
.
h
t
a Let F be a finite group. The degree of commutativity of F is defined by
m
(x,y) F2 [x,y] = 1
[ (1) dc(F) = |{ ∈ | }|,
F 2

1 | |
v i.e. the probability that two elements of F chosen uniformly at random
4 commute. In [1], Antol´ın et al. generalise this definition to infinite finitely
7
generated groups. For this, let G be a group which has a finite generating
3
4 set X. For any element g G, let g = g X be the length of g with respect
0 to X. For any n Z , le∈t | | | |
0
. ∈ ≥
1
(2) B (n) = g G g n
0 G,X X
{ ∈ || | ≤ }
7
be the ball in G with respect to X of radius n, and let
1
:
v (3) S (n)= g G g =n
G,X X
i { ∈ | | | }
X
be the sphere in G with respect to X of radius n. One writes B (n) or
G
r
B(n) for the ball (and S (n) or S(n) for the sphere) if the generating set
a G
or the group itself is clear. This was used in [1] to define the degree of
commutativity for the group G with respect to X as
(x,y) B (n)2 [x,y] = 1
G,X
(4) dc (G) = limsup |{ ∈ | }|.
X B (n)2
n G,X
→∞ | |
Note that if G is finite then for any generating set X one has B (N) = G
G,X
for some N, so this definition agrees with (1).
Date: January 17, 2017.
2010 Mathematics Subject Classification. 20P05.
Key words and phrases. Right-angled Artin groups, degree of commutativity,
growth series.
1
2 MOTIEJUSVALIUNAS
Themaininterestofthispaperisthedegreeofcommutativity ofright-angled
Artin groups (often abbreviated as RAAGs). These are groups associated
with a finite simple graph Γ and defined by the presentation
(5) G = V(Γ) [x,y] = 1 for all xy E(Γ) .
Γ
h | ∈ i
If A V(Γ) is any subset, then one can consider the subgroup G of G
Γ(A) Γ
⊆
generated by A, called the special subgroup of G associated with A.
Γ
This class of groups includes all groups built from the infinite cyclic group
Z by taking direct and free products. In particular, G is a non-trivial
Γ
free (resp. direct) product of its special subgroups if and only if the graph Γ
(resp.its complement Γc)is notconnected. However, this class also includes
non-cyclic groups that cannot be decomposed as a non-trivial direct or free
product, e.g. when Γ is an n-gon for n 5 [4].
≥
Note that G is abelian if and only if Γ is complete. If this is not the case,
Γ
then x,y G (for some x,y V(Γ) not joined by an edge) is a free
Γ
h i ≤ ∈
group of rank 2, as can be shown by considering reduced words in G [6]. In
Γ
particular, G is an exponentially growing group. One would expect such
Γ
a group to be highly non-abelian, which is confirmed by the main result of
this paper:
Theorem 1. Let Γ be a finite graph. Then
1 if Γ is complete,
(6) dc (G ) =
V(Γ) Γ
(0 otherwise.
To prove this one may use results on the growth of G . In particular,
Γ
consider the spherical growth series for G , which is defined for a general
Γ
group G with a finite generating set X by
∞
(7) s (t)= tgX = S (n)tn;
G,X | | G,X
| |
g G n=0
X∈ X
one writes s (t) for s (t). It was shown by [2] that in fact for any
Γ GΓ,V(Γ)
finite graph Γ, the function s is rational. This implies some nice properties
Γ
on the growth of G ; in particular:
Γ
Proposition 2. Let G be an infinite group with a finite generating set X
such that s (t) is a rational function. Then there exist constants α Z ,
G,X >0
∈
λ [1, ) and D > C > 0 such that
∈ ∞
(8) Cnα 1λn S (n) Dnα 1λn
− G,X −
≤ | | ≤
for all n 1.
≥
Together with an explicit form of centralisers in G , describedin [6], Propo-
Γ
sition 2 can be used to prove Theorem 1.
The paper is structured as follows. Section 2 applies to all infinite groups
with rational spherical growth series and is dedicated to a proof of Proposi-
tion 2. Section 3 is used to prove Theorem 1.
DEGREE OF COMMUTATIVITY FOR RAAGS 3
Acknowledgements. I would like to thank my PhD supervisor, Armando
Martino, without whose guidance this paper would not have been possible. I
would also like to thank Charles Cox and Enric Ventura for valuable discus-
sions.
2. Groups with rational growth series
This section provides a proof of Proposition 2. Let G be an infinite group,
and suppose that the spherical growth series of G with respect to a finite
generating set X is a rational function. In particular, the spherical growth
series is
∞ p(t)
(9) s(t) = s (t) = S(n)tn =
G,X
q(t)
n=0
X
where S(n) = S (n) = S (n):= S (n), and
G G,X G,X
| |
r r˜
(10) q(t) = q tc (1 λ t)αi and p(t)= p tc˜ (1 λ˜ t)α˜i
0 i 0 i
− −
i=1 i=1
Y Y
are non-zero polynomials with no common roots, with α ,α˜ Z for all i.
i i >0
Since the series (S(n)) is growing at most exponentially, ∈s(t) is analytic
∞n=0
(and so continuous) at 0, so one has
p
(11) 1 = S(0) = lims(t)= 0 limtc˜ c
−
t 0 q0 t 0
→ →
and so c = c˜ and p = q . Thus, without loss of generality, c = c˜= 0 and
0 0
q = p = 1.
0 0
Coefficients of such a series are described in [5, Lemma 1]; in particular, it
follows that
r αi 1
−
(12) S(n)= b njλn
i,j i
i=1 j=1
X X
for n large enough, with b = 0 for all i.
i,αi−1 6
Now consider the terms of (12) that give a non-negligible contribution to
S(n) for large n. In particular, one may assume without loss of generality
that
(13) λ := λ = λ = = λ > λ λ λ
| 1| | 2| ··· | k˜| | k˜+1|≥ | k˜+2| ≥ ··· ≥ | r|
for some k˜ r and that
≤
(14) α := α = α = = α > α α α
1 2 ··· k k+1 ≥ k+2 ≥ ··· ≥ k˜
for some k k˜. Note that one must have λ 1: otherwise s(t) is analytic
at 1 (as its≤radius of convergence is λ 1 > ≥1) and so the series S(n)
− n
converges, contradicting the fact that G is infinite.
P
For n Z , define
0
∈ ≥
k
(15) c = b exp(iϕ n)
n j,α 1 j
−
j=1
X
4 MOTIEJUSVALIUNAS
where λ = λexp(iϕ ) for some ϕ R/2πZ, for 1 j k. It follows that
j j j
∈ ≤ ≤
(16) S(n) = nα 1λn(c +o(1))
− n
as n . In particular, since S(n) (0, ) R for all n, it follows that
→ ∞ ∈ ∞ ⊆
(17) liminf (c ) 0 and lim (c ) = 0.
n n
n ℜ ≥ n ℑ
→∞ →∞
It is clear that
S(n) k
(18) limsup b ,
n nα−1λn ≤ | j,α−1|
→∞ Xj=1
which shows existence of the constant D in Proposition 2; in order to prove
the Proposition, it is enough to show that liminf S(n)/(nα 1λn) > 0.
n −
→∞
However, this bound does not follow solely from the fact that s(t) is a
rational function, as the example below shows.
Example 1. Consider the case p(t) = 12t2(1 2t)+(1+2t)(1 2t+4t2)
− −
and q(t) = 1 5t+12t2 16t3 +8t4 = (1 t)(1 2t)(1 2ωt)(1 2ω¯t),
−th − − − − −
where ω is a 6 primitive root of unity. Then λ = 2 and α = 1, and [5,
Lemma 1] can be used to calculate
(19) S(n) = c 2n +1
n
where
0, n 0 (mod 6)
≡
2, n 1 (mod 6)
(20) c = 4 2ωn 2ω¯n =  ≡ ±
n
− − 6, n 2 (mod 6)
 ≡ ±
8, n 3 (mod 6)
≡

Note that with these values of p andq one has S(n) Z for all n and
 >0
S(0) = 1, as required. But as c = 0 for infinitely ma∈ny values of n, one
n
also would have liminf S(n)/(nα 1λn) = 0 with this s(t). The aim is
n −
now to show that submu→lti∞plicativity of the sequence (S(n)) gives some
∞n=0
restrictions on the series s(t) so that this cannot happen.
As the b are non-zero and the ϕ are distinct, given (17) the following
j,α 1 j
−
result seems highly likely:
Lemma 3. The numbers c are real, and for some constant δ > 0, the set
n
(21) n Z c δ
0 n
{ ∈ ≥ | ≥ }
is relatively dense in [0, ).
∞
However, the author has been unable to come up with a straightforward
proofofLemma3withoutusingsomeadditionaltheoryon‘quasi-periodicity’
of the sequence (c ) . One such proof is provided at the end of this sec-
n ∞n=0
tion.
Assuming the Lemma, one can find an N Z such that for all n, there
>0
∈
exists a β = β 0,...,N with c δ. Define
n n+β
∈ { } ≥
(22) P := max λ βS(β) 0 β N ,
−
{ | ≤ ≤ }
DEGREE OF COMMUTATIVITY FOR RAAGS 5
and let M Z be such that for all n M, one has
>0
∈ ≥
δ
(23) S(n) nα 1λn c
− n
≥ − 2
(cid:18) (cid:19)
(such an M exists by (16)). Then submultiplicativity of sphere sizes implies
that for all n M,
≥
δ δ
(n+β )α 1λn+βn c (n+β )α 1λn+βn
(24) 2 n − ≤ n+βn − 2 n −
(cid:18) (cid:19)
S(n+β ) S(n)S(β ) S(n)Pλβn.
n n
≤ ≤ ≤
It follows that
δ δ
(25) S(n) (n+β )α 1λn nα 1λn
n − −
≥ 2P ≥ 2P
for n M, showing that
≥
S(n) δ
(26) liminf > 0,
n nα 1λn ≥ 2P
→∞ −
whichshows existence of theconstant C > 0inProposition 2. Thusinorder
to prove Proposition 2 it is now enough to prove Lemma 3.
Proof of Lemma 3. To prove the Lemma, one may employ a digression into
a certain class of functions from R to C, called ‘uniformly almost periodic
functions’. The theory for these functions is presented in a book by Besi-
covitch [3].
Let f : R C be a function. Given ε > 0, define the set E(f,ε) R to be
the set of→all numbers τ R (called the translation numbers for f⊆belonging
∈
to ε) such that
(27) sup f(x+τ) f(x) ε.
x R| − |≤
∈
The function f is said to be uniformly almost periodic (u.a.p.) if, for any
ε > 0, the set E(f,ε) is relatively dense in R, i.e. the inclusion E(f,ε) ֒ R
→
is a quasi-isometry. It is easy to see that any periodic function is u.a.p., and
that every continuous u.a.p. function is bounded.
Now note that the function
c :R C
→
k
(28)
t b exp(iϕ t)
j,α 1 j
7→ −
j=1
X
is a sum of continuous periodic functions, and so is a continuous u.a.p.
function by [3, Section 1.1, Theorem 12]. By definition, c = c(n) for any
n
n Z .
0
∈ ≥
The aim is to show that the function c¯: t c( t ) is also u.a.p. For this,
7→ ⌊ ⌋
note that c is everywhere differentiable and the derivative c(t) is a sum of
′
continuous periodic functions, so is continuous and u.a.p. – in particular,
it is bounded, by some R > 0, say. For a given ε (0,R), set a constant
∈
6 MOTIEJUSVALIUNAS
M := ε/ 2sin πε and define f : R R by f(t) = Msin(πt). It is easy
2R →
to check that
(cid:0) (cid:0) (cid:1)(cid:1)
ε ε ε
(29) E f, n ,n+ .
2 ⊆ − 2R 2R
(cid:16) (cid:17) n[∈Zh i
For any τ R, define n = τ + 1 Z to be the nearest integer to τ. Pick
∈ τ 2 ∈
τ E f, ε E c, ε – then c(x+τ) c(x) ε for all x R, and, by
(29∈), τ n2 ∩ ε , s2oinpar(cid:4)ticu|lar(cid:5) c(x+−τ) c(|x≤+n2 ) ε for∈all x Rby
the ch|o(cid:0)i−ce oτ(cid:1)f| ≤R.2RT(cid:0)hus(cid:1)c(x+n ) |c(x) ε−for all xτ | ≤R,2i.e. n E∈(c,ε).
τ τ
| − | ≤ ∈ ∈
But by [3, Section 1.1, Theorem 11], the set E f, ε E c, ε is relatively
dense, hence (by the previous paragraph) so is the 2set∩E(c,ε)2 Z. However,
for any n E(c,ε) Z and any x R one has (cid:0) (cid:1) (cid:0) ∩(cid:1)
∈ ∩ ∈
(30) c¯(x+n) c¯(x) = c( x+n ) c( x ) = c( x +n) c( x ) ε
| − | | ⌊ ⌋ − ⌊ ⌋ | | ⌊ ⌋ − ⌊ ⌋ | ≤
and so E(c,ε) Z E(c¯,ε) Z. Itfollows that E(c¯,ε) Z is relatively dense
∩ ⊆ ∩ ∩
(and so the function c¯:t c( t ) is u.a.p.).
7→ ⌊ ⌋
Now recall that (17) provides constraints for limits of sequences ( (c )) and
n
ℜ
( (c )): namely,
n
ℑ
(31) liminf (c ) 0 and lim (c ) = 0.
n n
n ℜ ≥ n ℑ
→∞ →∞
Itisnothardtoseethatc R foralln: indeed,ifeither (c ) = δ < 0
n 0 n
or (c ) = δ > 0 for som∈e n≥then the fact that the set Eℜ(c¯,δ/2)− Z is
n
|ℑ | ∩
relatively dense contradicts (31). Similarly, if c > 0 for some N then the
N
set E(c¯,δ) Z is a relatively dense set contained in n Z c(n) δ ,
∩ { ∈ | ≥ }
where δ = c /2. To prove Lemma 3 it is therefore enough to show that the
N
sequence (c ) is not identically zero.
n ∞n=0
Now recall that the sequence (c ) is defined by
n
k
(32) c = b exp(iϕ n),
n j,α 1 j
−
j=1
X
and suppose for contradiction that c = 0 for all n Z , and in particular
n 0
∈ ≥
for 0 n k 1. This is saying that Mv = 0, where
≤ ≤ −
1 1 1
···
exp(iϕ ) exp(iϕ ) exp(iϕ )
1 2 k
(33) M =  ... ... ·.·.·. ... 
 
exp(iϕ1)k−1 exp(iϕ2)k−1 exp(iϕk)k−1
 ··· 
 
and
b
1,α 1
−
b
2,α 1
(34) v =  .− .
.
.
 
bk,α 1
 − 
 
Thus M has a zero eigenvalue and so detM = 0. But Mt is a Vandermonde
matrix with pairwisedistinct rows, sodetM = 0. Thisgives a contradiction
which completes the proof. 6 (cid:3)
DEGREE OF COMMUTATIVITY FOR RAAGS 7
3. Degree of commutativity
The aim of this section is to prove Theorem 1. For this, let Γ be a finite
graph. If Γ is complete, then G is a free abelian group and so dc (G )= 1
Γ X Γ
for any generating set X. Hence one may assume without loss of generality
that Γ is not complete. In particular, G = G is a non-abelian (moreover,
Γ
exponentially growing) group. One thus aims to show that dc (G) = 0.
V(Γ)
Suppose for contradiction that dc (G ) > 0. That means that for some
V(Γ) Γ
constant ε> 0, one has
C (g) B(n)
G
(35) | ∩ | ε
B(n)2 ≥
g B(n)
∈X
for infinitely many values of n, where C (g) denotes the centraliser of an
G
element g G and B(n) = B (n) = B (n) := B (n) for X = V(Γ).
G G,X G,X
∈ | |
In the proof minimal length conjugates of elements in G will be considered.
In particular, by [6, Section II, Proposition] any g G has a unique conju-
∈
gate g˜ G, called the cyclic reduction of g, such that g = p 1g˜p and such
∈ −g g
that g˜ is minimal among all the conjugates; moreover, for such elements
| |
we have g = 2p + g˜. Such an element g˜ is called cyclicly reduced, and
g
| | | | | |
a subset A V(Γ) is called the support of g˜, written supp(g˜), if the set of
⊆
letters used in (any) word of length g˜ representingg˜is precisely A. Finally,
| |
for any subset A V(Γ) one may define its link to be
⊆
(36) linkA= x V(Γ) xy E(Γ) for all y A .
{ ∈ | ∈ ∈ }
The proof now can be divided into the following steps:
(i) showing that one may restrict their consideration of centralisers of
elements g G only to elements such that the support of their cyclic
∈
reduction is some fixed subset A V(Γ);
⊆
(ii) provingthatagenericelementg Gwithsupp(g˜)= Ahas p ‘small’;
g
(iii) showing,basedonstep(ii),thatB∈ (n)andB | | (n)are
GΓ(A),A GΓ(linkA),linkA
bothboundedaway from zero by constant multiples of B (n) for
GΓ,V(Γ)
n arbitrarily large;
(iv) obtaining a contradiction based on the fact that G G is a
Γ(A) Γ(linkA)
×
special subgroup of G (by definition of link).
Γ
Before carrying on with the proof, consider the sequence (d ) where
n ∞n=0
(x,y) B (n)2 [x,y] = 1
G,V(Γ)
(37) d := |{ ∈ | }|.
n B (n)2
G,V(Γ)
One aims to show that d 0 as n . Note that for many groups of
n
→ → ∞
exponential growth, including all the non-elementary hyperbolic groups [1],
thesequence(d ) convergestozeroexponentially. However,thefollowing
n ∞n=0
example shows that this is not the case for some right-angled Artin groups.
The result of Theorem 1 may be therefore more delicate than one might
think.
8 MOTIEJUSVALIUNAS
Example 2. Suppose Γ is a complete bipartite graph K , i.e. Γ has vertex
m,m
set
(38) V(Γ) = x ,...,x ,y ,...,y
1 m 1 m
{ }
and edge set
(39) E(Γ) = (x ,y ) 1 i,j m .
i j
{ | ≤ ≤ }
In this case G = F F (direct product of two free groups of rank m) and
Γ m m
×
so one can calculate sphere sizes in G and its special subgroups easily. Note
Γ
that clearly (by the definition of link) every element of G commutes with
Γ(A)
every element of G . Now consider the case where A = x ,...,x
Γ(linkA) 1 m
{ }
and so linkA= y ,...,y . It follows that
1 m
{ }
(40) (x,y) B(n)2 [x,y] = 1 B (n) B (n).
{ ∈ | } ⊆ GΓ(A) × GΓ(linkA)
An explicit computation shows that
m(2m 1)n 1
(41) B (n) = B (n) = − −
GΓ(A) GΓ(linkA) m 1
−
and
2m2n(2m 1)n
(42) B (n) = − +e (2m 1)n +e
GΓ (m 1)(2m 1) 1 − 2
− −
where e ,e are some constants that depend on m. It follows that
1 2
(43) d BGΓ(A)(n)BGΓ(linkA)(n) 2m−1 2
n ≥ B (n)2 ∼ 2mn
GΓ (cid:18) (cid:19)
as n . In particular, the sequence (d ) converges to zero only poly-
→ ∞ n ∞n=0
nomially for G .
Γ
One now may carry out steps (i)–(iv) above as follows. For step (i), note
that (35) can be rewritten as
C (g) B(n)
G
(44) | ∩ | ε
B(n)2 ≥
A V(Γ) g B(n)
⊆X ∈X
supp(g˜)=A
and so (44) holds for infinitely many n. But as Γ is finite, there are only
2V(Γ) < subsets of V(Γ), thus in particular there exists a subset A
| |
∞ ⊆
V(Γ) such that
C (g) B(n)
(45) | G B(∩n)2 | ≥ 2−|V(Γ)|ε
g B(n)
∈X
supp(g˜)=A
holds for infinitely many n. This completes step (i).
For step (ii), one may use Proposition 2. In particular, as G has rational
sphericalgrowthseriesby[2],thePropositionsaysthatthereexistconstants
α Z , λ 1, C = C > 0 and D = D > C such that
>0 G G
∈ ≥
(46) Cnα 1λn S(n) Dnα 1λn
− −
≤ ≤
DEGREE OF COMMUTATIVITY FOR RAAGS 9
for all n 1. As it is also assumed that G has exponential growth, one has
≥
λ > 1. It is easy to show that in this case
Dλ
(47) Cnα 1λn <B(n) < nα 1λn
− −
λ 1
−
for all n 1.
≥
Now one can bound the number of terms in (45) corresponding to elements
g G with p large (even without requiring supp(g˜) = A). Indeed, as any
g
∈ | |
g G can be written as g = p 1g˜p with g = 2p + g˜, (46) and (47)
∈ −g g | | | g| | |
imply
1 C (g) B(n) g B(n) p > s
G g
| ∩ | |{ ∈ | | | }|
B(n) B(n) ≤ B(n)
g B(n)
∈X
pg>s
| |
n
⌊2⌋ S(i)B(n 2i)
(48)
−
≤ B(n)
i=s+1
X
n−1
D 1 α−1λ−n2 + D2λ ⌊ 2 ⌋ i(n−2i) α−1λ−i.
≤ C 2 C(λ 1) n
(cid:18) (cid:19) − i=s+1(cid:18) (cid:19)
X
Thefirsttermof thesumabove clearly tendsto zero, and thesecond term is
bounded above by the infinite sum ∞i=s+1iα−1λ−i, which tends to zero as
s since the series iα 1λ i converges. Hence there exists a value of
→ ∞ i − − P
s Z whichensuresthattherighthandsidein(48)islessthan2 V(Γ) 1ε
0 −| |−
∈ ≥ P
for n large enough. This means that
C (g) B(n)
G
(49) | ∩ | ε˜
B(n)2 ≥
g B(n)
∈X
supp(g˜)=A
pg s
| |≤
for infinitely many n (where ε˜:= 2 V(Γ) 1ε > 0), completing step (ii).
−| |−
For step (iii), note that one may write
C (g) B(n) g B(n) supp(g˜) = A, p s
G g
| ∩ | |{ ∈ | | |≤ }|
B(n)2 ≤ B(n)
g B(n)
∈X
(50) supp(g˜)=A
pg s
| |≤
C (g) B(n)
G
max | ∩ | g B(n),supp(g˜)= A, p s
× B(n) ∈ | g|≤
(cid:26) (cid:12) (cid:27)
(cid:12)
where both terms in the productare bou(cid:12)nded above by 1. It follows by (49)
(cid:12)
that both
g B(n) supp(g˜)= A, p s
g
( ) |{ ∈ | | |≤ }| ε˜
∗ B(n) ≥
and
C (g) B(n)
G
( ) max | ∩ | g B(n),supp(g˜) =A, p s ε˜
† B(n) ∈ | g| ≤ ≥
(cid:26) (cid:12) (cid:27)
(cid:12)
hold for infinitely many n.
(cid:12)
(cid:12)
10 MOTIEJUSVALIUNAS
We claim now that ( ) and ( ) imply that subgroups G and G
Γ(A) Γ(linkA)
∗ †
(respectively) are visible in G .
Γ
For ( ), note that the set in the numerator consists of elements g B (n)
G
∗ ∈
which have an expression g = p 1g˜p with p B (s) and g˜ B (n).
−g g g ∈ GΓ ∈ GΓ(A)
It follows that
(51) g B(n) supp(g˜) = A, p s B (s)B (n)
|{ ∈ | | g|≤ }| ≤ GΓ GΓ(A)
and so ( ) implies that
∗
( ) ε˜ BGΓ(A)(n) 1
∗∗ B (s) ≤ B (n) ≤
GΓ GΓ
for infinitely many n, where the second inequality comes from the fact that
B (n) B (n).
GΓ(A) ⊆ GΓ
For ( ), one needs to consider forms of centralisers of elements g G with
† ∈
supp(g˜) = A. Firstly, fix an element g G with supp(g˜)= A and note that
∈
one clearly has C (g) = p 1C (g˜)p , so if p s then one has
G −g G g | g|≤
(52) C (g) B(n) C (g˜) B(n+2s).
G G
| ∩ | ≤ | ∩ |
Secondly, let A ,...,A A form a partition of A such that the graphs
1 k
⊆
Γ(A )c are precisely the connected components of the graph Γ(A)c. Then
i
the Centraliser Theorem [6] says that one has
(53) C (g˜)= h h G
G 1 k Γ(linkA)
h i×···×h i×
where the h G are some cyclicly reduced elements with supp(h ) = A .
i i i
∈
Now pick any vertices a A for 1 i k, and let
i i
∈ ≤ ≤
(54) A˜:= a 1 i k linkA.
i
{ | ≤ ≤ }∪
Define the homomorphism ϕ :G C (g˜) by extending the map
Γ(A˜) → G
ϕ : A˜ C (g˜),
|A˜ → G
(55) a h for 1 i k,
i i
7→ ≤ ≤
x x for x linkA.
7→ ∈
It is easy to see that ϕ is a well-defined isomorphism. The aim is now to
show that if C (g˜) is equipped with the restriction of word metric in G,
G
then ϕ 1 is a 1-Lipschitz map, i.e. that one has ϕ 1(h) h for each
− | − |A˜ ≤ | |V(Γ)
h C (g˜). Indeed, let h C (g˜); since any vertex in A = supp(h ) is
G G i i
∈ ∈
connected by an edge to any vertex of A for j = i or of linkA, it follows
j
6
that one has an expression
(56) h= hβ1 hβkh¯
1 ··· k
for some β Z and h¯ G . Since the A are disjoint from each other
i Γ(linkA) i
∈ ∈
and from linkA, one can choose words representing h¯ and the hβi so that
i
(56) gives a graphically reduced expresion for h, i.e. the corresponding word
representing h has length h. In particular, one has
| |
(57) h = hβ1 + + hβk + h¯ .
| | 1 ··· k
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)