A Subgroup of a Direct Product of Free Groups whose Dehn Function has a Cubic Lower Bound

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by  Will Dison

A SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPS
WHOSE DEHN FUNCTION HAS A CUBIC LOWER BOUND
9
WILLDISON
0
0
2 Abstract. WeestablishacubiclowerboundontheDehnfunctionofacertain
finitelypresentedsubgroupofadirectproductof3freegroups.
n
a
J

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DepartmentofMathematics

UniversityWalk
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Bristol,BS81TW
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UnitedKingdom
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. [email protected]
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1. Introduction
m
[ ThecollectionS ofsubgroupsofdirectproductsoffreegroupsissurprisinglyrich
and has been studied by many authors. In the early 1960s Stallings [13] exhibited
2
a subgroupof (F )3, where F is the rank 2 free group, as the first known example
v 2 2
of a finitely presented group whose third integral homology group is not finitely
8
5 generated. Bieri [2] demonstrated that Stallings’ group belongs to a sequence of
7 groups SBn ≤ (F2)n, the Stallings-Bieri groups, with SBn being of type Fn−1 but
0 not of type FP . (See [6] for definitions and background concerning finiteness
n
.
8 properties of groups.)
0 In the realm of decision theory, Miha˘ılova[9] and Miller [10] exhibited a finitely
7 generatedsubgroupof(F )2withundecidableconjugacyandmembershipproblems.
0 2
It is thus seen that even in the 2-factor case fairly wild behaviour is encountered
:
v amongst the subgroups of direct products of free groups.
i
X Incontrasttothis, BaumslagandRoseblade[1]showedthatinthe 2-factorcase
this wildness only manifests itself amongst the subgroups which are not finitely
r
a presented. They proved that if G is a finitely presented subgroup of F(1) ×F(2),
whereF(1) andF(2) arefreegroups,thenGisitselfvirtuallyadirectproductofat
most 2 free groups. This work was extended by Bridson, Howie, Miller and Short
[5] to an arbitrary number of factors. They proved that if G is a subgroup of a
direct product of n free groups and if G enjoys the finiteness property FP , then
n
G is virtually a direct product of at most n free groups. Further information on
the finiteness properties of the groups in class S was provided by Meinert [8] who
calculated the BNS invariants of direct products of free groups. These invariants
determine the finiteness properties of all subgroups lying above the commutator
subgroup.
Severalauthorshaveinvestigatedthe isoperimetricbehaviourofthefinitely pre-
sented groups in S. Elder, Riley, Young and this author [7] have shown that the
2000 Mathematics Subject Classification. 20F65(primary),20E05,20F06(secondary).
1
2 WILLDISON
Dehnfunction ofStallings’groupSB isquadratic. The method espousedby Brid-
3
son in [3] proves that the function n3 is an upper bound on the Dehn functions of
each of the Stallings-Bieri groups. In contrast, there have been no results which
give non-trivial lower bounds on the Dehn functions of any groups in S. In other
words,until now there has been no finitely presented subgroup of a direct product
of free groups whose Dehn function was known to be greater than that of the am-
bientgroup. The purposeofthis paper is toconstructsucha subgroup: we exhibit
a finitely presented subgroup of (F )3 whose Dehn function has the function n3 as
2
a lower bound.
Let K be the kernel of a homomorphism θ : (F )3 → Z2 whose restriction to
2
each factor F is surjective. By Lemma 3.1 below the isomorphism class of K is
2
independent of the homomorphism chosen. Results in [8] show that K is finitely
presented.
Theorem 1.1. The Dehn function δ of K satisfies δ(n)(cid:23)n3.
Note that Theorem1.1 makes no reference to a specific presentationof K since,
asiswellknown,theDehnfunctionofagroupisindependent(upto≃-equivalence)
ofthepresentationchosen. WereferthereadertoSection2forbackgroundonDehn
functions, including definitions of the symbols (cid:23) and ≃.
The organisation of this paper is as follows. Section 2 gives basic definitions
concerning Dehn functions, van Kampen diagrams and Cayley complexes. We
expect that the reader will already be familiar with these concepts; the purpose
of the exposition is principally to introduce notation. In Section 3 we define a
class of subgroups of direct products of free groups of which K is a member. The
subsequent section gives finite generating sets for those groups in the class which
are finitely generated and Section 5 proves a splitting theorem which shows how
certain groups in the class decompose as amalgamated products. We then prove
in Section 6 how, in certain circumstances, the distortion of the subgroup H in
an amalgamated product Γ = G ∗ G gives rise to a lower bound on the Dehn
1 H 2
function of Γ. Theorem1.1 followsas a corollaryofthis result when appliedto the
splitting of K given in Section 5.
2. Dehn Functions
InthissectionwerecallthebasicdefinitionsconcerningDehnfunctionsoffinitely
presented groups. All of this material is standard. For further background and a
more thorough exposition, including proofs, see, for example, [4] or [11].
Given a set A, write A−1 for the set of formalinversesof the elements of A and
writeA±1 forthesetA∪A−1. DenotebyA±∗ thefreemonoidonthesetA±1. We
refer to elements of A±∗ as words in the letters A±1 and write |w| for the length
of such a word w. Given words w ,w ∈A±∗ we write w f=reew if w and w are
1 2 1 2 1 2
freely equal and w ≡w if w and w are equal as elements of A±∗.
1 2 1 2
Definition 2.1. Let P = hA|Ri be a finite presentation of a group G. A word
w ∈ A±∗ is said to be null-homotopic over P if it represents the identity in G. A
null-P-expression for sucha wordis a sequence (x ,r )m in A±∗×R±1 suchthat
i i i=1
m
wfr=ee x r x−1.
i i i
i=1
Y
A SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPS 3
Define the area ofa null-P-expressionto be the integerm anddefine theP-area of
w, written AreaP(w), to be the minimal area taken over all null-P-expressions for
w.
TheDehnfunctionofthepresentationP,writtenδP,isdefinedtobethefunction
N→N given by
±∗
δP(n)=max{AreaP(w) : w ∈A , w null-homotopic, |w|≤n}.
Although the Dehn functions of different finite presentations of a fixed group
may differ, their asymptotic behaviour will be the same. This is made precise in
the following way.
Definition2.2. Givenfunctionsf,g :N→N,writef (cid:22)gifthereexistsaconstant
C >0 such that f(n)≤Cg(Cn+C)+Cn+C for all n. Write f ≃g if f (cid:22)g and
g (cid:22)f.
Lemma 2.3. If P1 and P2 are finite presentations of the same group then δP1 ≃
δP .
2
For a proof of this standard result see, for example, [4, Proposition 1.3.3].
A useful tool for the study of Dehn functions is a class of objects known as
van Kampen diagrams. Roughly speaking, these are planar CW-complexes which
portray diagrammatically schemes for reducing null-homotopic words to the iden-
tity. Such diagrams, whose definition is recalled below, allow the application of
topologicalmethods to the calculationofDehnfunctions. Forbackgroundandfur-
ther details see, for example, [4, Section 4]. For the definition of a combinatorial
CW-complex see, for example, [4, Appendix A].
Definition 2.4. A singular disc diagram ∆ is a finite, planar, contractible combi-
natorial CW-complex with a specified base vertex ⋆ in its boundary. The area of
∆,writtenArea(∆), isdefinedto be the number of2-cellsofwhich∆is composed.
The boundary cycle of ∆ is the edge loop in ∆ which starts at ⋆ and traverses ∂∆
in the anticlockwise direction. The interior of ∆ consists of a number of disjoint
open 2-discs, the closures of which are called the disc components of ∆.
Each1-cellof∆hasassociatedto ittwodirectededgesǫ andǫ ,withǫ−1 =ǫ .
1 2 1 2
Let DEdge(∆) be the set of directed edges of ∆. A labelling of ∆ over a set A is
a map λ:DEdge(∆)→A±1 such that λ(ǫ−1)=λ(ǫ)−1. This induces a map from
the set of edge paths in ∆ to A±∗. The boundary label of ∆ is the word in A±∗
associated to the boundary cycle.
Let P =hA|Ri be a finite presentation. A P-van Kampen diagram for a word
w ∈A±∗ is a singular disc diagram ∆ labelled over A with boundary label w and
such that for each 2-cell c of ∆ the anticlockwise edge loop given by the attaching
map of c, starting at some vertex in ∂c, is labelled by a word in R±1.
Lemma 2.5 (Van Kampen’s Lemma). A word w ∈A±∗ is null-homotopic over P
if and only if there exists a P-van Kampen diagram for w. In this case the P-area
of w is the minimal area over all P-van Kampen diagrams for w.
For a proof of this result see, for example, [4, Theorem 4.2.2].
Associated to a presentation P = hA|Ri of a group G there is a standard
combinatorial 2-complex KP with π1(KP) ∼= G. The complex KP is constructed
by taking a wedge of copies of S1 indexed by the letters in A and attaching 2-cells
indexed by the relations in R. The 2-cell corresponding to a relation r ∈R has |r|
4 WILLDISON
edgesandis attachedby identifying its boundary circuitwith the edge pathin K1
P
along which one reads the word r.
The Cayley 2-complex associated to P, denoted by Cay2(P), is defined to be
theuniversalcoverofKP. IfonechoosesabasevertexofCay2(P)torepresentthe
identity element of G then the 1-skeleton of this complex is canonically identified
with the Cayley graph of P. Given a P-van Kampen diagram ∆ there is a unique
label-preservingcombinatorialmapfrom∆toCay2(P)whichmapsthebasevertex
of ∆ to the vertex of Cay2(P) representing the identity.
3. A Class of Subgroups of Direct Products of Free Groups
Inthissectionweintroduceaclassofsubgroupsofdirectproductsoffreegroups
of which the group K defined in the introduction will be a member. We first fix
some notation which will be used throughout the paper. Given integers i,m ∈ N,
let F(i) be the rank m free group with basis e(i),...,e(i). Given an integer r ∈N,
m 1 m
let Zr be the rank r free abelian group with basis t ,...,t .
1 r
Given positive integers n,m ≥ 1 and r ≤ m, we wish to define a group Kn(r)
m
to be the kernel of a homomorphism θ : F(1)×...×F(n) → Zr whose restriction
m m
to each factor F(i) is surjective. For fixed n, m and r, the isomorphism class of
m
the group Kn(r) is, up to an automorphism of the factors of the ambient group
m
F(1) × ...× F(n), independent of the homomorphism θ. This is proved by the
m m
following lemma.
Lemma 3.1. Let F be a rank m free group. Given a surjective homomorphism
φ:F →Zr, there exists a basis e ,...,e of F so that
1 m
t if 1≤i≤r,
i
φ(e )=
i
(0 if r+1≤i≤m.
Proof. The homomorphism φ factors through the abelianisation homomorphism
Ab:F →A, where A is the rank m free abelian group F/[F,F], as φ=φ¯◦Ab for
somehomomorphismφ¯:A→Zr. Sinceφ¯issurjective,AsplitsasA ⊕A whereφ¯
1 2
is an isomorphism on the first factor and 0 on the second factor. There thus exists
a basis s ,...,s for A so as
1 m
t if 1≤i≤r,
φ¯(s )= i
i
(0 if r+1≤i≤m.
We claimthat the s lift under Ab to a basisfor F. To see this let f ,...,f be
i 1 m
any basis for F and let f¯,...,f¯ be its image under Ab, a basis for A. Let ρ ∈
1 m
Aut(A)bethechangeofbasisisomorphismfromf¯,...,f¯ tos ,...,s . Itsuffices
1 m 1 m
toshowthatthislifts underAbtoanautomorphismofF. Butthisis certainlythe
case since Aut(A) ∼= GLm(Z) is generated by the elementary transformations and
each of these obviously lifts to an automorphism. (cid:3)
Definition 3.2. For integers n,m ≥ 1 and r ≤ m, define Kn(r) to be the kernel
m
of the homomorphism θ :F(1)×...×F(n) →Zr given by
m m
t if 1≤j ≤r,
θ(e(i))= j
j (0 if r+1≤j ≤m.
A SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPS 5
Note that Kn(1) is the nth Stallings-Bieri group SB .
2 n
By a result in Section 1.6 of [8], if r ≥ 1 and m ≥ 2 then Kn(r) is of type
m
Fn−1 but not of type FPn. In particular the group K ∼= K23(2) defined in the
introduction is finitely presented.
4. Generating Sets
We give finite generating sets for those groups Kn(r) which are finitely gener-
m
ated. We make use of the following notational shorthand: given formal symbols x
and y, write [x,y] for xyx−1y−1 and xy for yxy−1.
Proposition 4.1. If n≥2 then Kn(r) is generated by S ∪S ∪S where
m 1 2 3
S ={e(1) e(j) −1 : 1≤i≤r,2≤j ≤n},
1 i i
S ={e(j)(cid:0): r+(cid:1) 1≤i≤m,1≤j ≤n},
2 i
S ={ e(1),e(1) : 1≤i<j ≤r}.
3 i j
Proof. Partition S as S′ ∪(cid:2)S′′ wher(cid:3)e
2 2 2
S′ ={e(j) : r+1≤i≤m,2≤j ≤n}
2 i
and
S′′ ={e(1) : r+1≤i≤m}.
2 i
Project Kn(r) ≤ F(1) ×...×F(n) onto the last n−1 factors to give the short
m m m
exact sequence 1→K1(r) →Kn(r) →F(2)×...×F(n) →1. Note that S ∪S′
m m m m 1 2
projects to a set of generators for F(2)×...×F(n) and that K1(r) is the normal
m m m
closure in F(1) of S′′ ∪S . If ζ ∈ F(1) and w ≡ w(e(1),...,e(1)) is a word in the
m 2 3 m 1 m
generators of F(1) then
m
ζw =ζw“e(11)(e(12))−1,...,e(m1)(e(m2))−1”.
Thus S ∪S′ ∪S′′∪S generates Kn(r). (cid:3)
1 2 2 3 m
5. A Splitting Theorem
The followingresultgivesanamalgamatedproductdecompositionofthe groups
Kn(r) in the case that r =m. Note that with slightly more work one could prove
m
amoregeneralresultwithoutthisrestriction. Weintroducethefollowingnotation:
given a collection of groups M,L ,...,L with M ≤ L for each i, we denote by
∗ 1 k i
k (L ; M) the amalgamated product L ∗ ...∗ L .
i=1 i 1 M M k
Theorem 5.1. If n ≥ 2 and m ≥ 1 then Kmn(m) ∼= ∗mk=1(Lk; M), where M =
Kmn−1(m) and, for each k =1,...,m, the group Lk ∼=Kmn−1(m−1) is the kernel of
the homomorphism
θ :F(1)×...×F(n−1) →Zm−1
k m m
given by
t if 1≤j ≤k−1,
j
θ (e(i))= 0 if j =k,
k j 
tj−1 if k+1≤j ≤m.

6 WILLDISON
Proof. Projecting Kn(m) onto the factor F(n) gives the short exact sequence 1→
m m
Kn−1(m) → Kn(m) → F(n) → 1. This splits to show that Kn(m) has the
m m m m
structure of an internal semidirect product M ⋊Fˆm(n) where Fˆm(n) ∼= Fm(n) is the
subgroup of F(n−1)×F(n) generated by
m m
e(n−1) e(n) −1, ..., e(n−1) e(n) −1.
1 1 m m
Since the action by conjugatio(cid:0)n of(cid:1)e(n−1) e(n) −1(cid:0)on M(cid:1) is the same as the action
k k
of e(n−1), we have that
k (cid:0) (cid:1)
Kn(m)=M ⋊Fˆ(n)
m m
∼=∗m M ⋊ e(n−1) e(n) −1 ; M
k=1 k k
∼=∗m (cid:16)M ⋊De(n−1)(cid:0) ; M(cid:1) .E (cid:17)
k=1 k
Define a homomorphism p :F((cid:16)1)×..D.×F(n−E1) →(cid:17)Z by
k m m
1 if j =k,
p e(i) =
k j (0 otherwise,
(cid:0) (cid:1)
and note that L ∩kerp is the kernel Kn−1(m) of the standard homomorphism
k k m
θ : F(1) ×...×F(n−1) → Zm given in Definition 3.2. Considering the restriction
m m
of p to L gives the short exact sequence 1 → Kn−1(m) → L → Z → 1 which
k k m k
demonstrates that L =Kn−1(m)⋊he(n−1)i. (cid:3)
k m k
6. Dehn Functions of Amalgamated Products
In this section we will be concerned with finitely presented amalgamated prod-
ucts Γ = G ∗ G where H, G and G are finitely generated groups and H is
1 H 2 1 2
proper in each G . Suppose each G is presented by hA |R i, with A finite. Note
i i i i i
that we are at liberty to choose the A so as each a∈A represents an element of
i i
G rH. Indeed, since H is proper in G there exists some a′ ∈A representing an
i i i
elementofG rH andwecanreplaceeachotherelementa∈A bya′aifnecessary.
i i
±∗
LetB be afinitegeneratingsetforH andforeachb∈B choosewordsu ∈A
b 1
and v ∈ A±∗ which equal b in Γ. Define E ⊂ (A ∪A ∪B)±∗ to be the finite
b 2 1 2
collection of words {bu−1,bv−1 : b∈B}. Then, since Γ is finitely presented, there
b b
exist finite subsets R′ ⊆R and R′ ⊆R such that Γ is finitely presented by
1 1 2 2
′ ′
P =hA ,A ,B|R ,R ,Ei.
1 2 1 2
Theorem 6.1. Let w ∈ A±∗ be a word representing an element h ∈ H and let
1
u∈A±∗ and v ∈A±∗ be words representing elements α∈G rH and β ∈G rH
1 2 1 2
respectively. If [α,h]=[β,h]=1 then
AreaP([w,(uv)n])≥2ndB(1,h)
where dB is the word metric on H associated to the generating set B.
Proof. Let ∆ be a P-van Kampen diagram for the null-homotopic word [w,(uv)n]
(see Diagram 1). For each i = 1,2,...,n, define p to be the vertex in ∂∆ such
i
that the anticlockwise path in ∂∆ from the basepoint around to p is labelled by
i
the word w(uv)i−1u. Similarly define q to be the vertex in ∂∆ such that the
i
A SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPS 7
u v u v u v
q q q
1 2 n
w w
p p p
1 2 n
u v u v u v
Figure 1. The van Kampen diagram ∆
anticlockwise path in ∂∆ from the basepoint around to q is labelled by the word
i
w(uv)nw−1(uv)i−nv−1. WewillshowthatforeachithereisaB-path(i.e. anedge
path in ∆ labelled by a word in the letters B) from p to q .
i i
We assume that the reader is familiar with Bass-Serre theory, as exposited in
[12]. Let T be the Bass-Serre tree associated to the splitting G ∗ G . This
1 H 2
consists of an edge gH for each coset Γ/H and a vertex gG for each coset Γ/G .
i i
The edge gH has initial vertex gG and terminal vertex gG . We will construct a
1 2
continuous (but non-combinatorial)map ∆→T as the composition of the natural
map ∆→Cay2(P) with the map f :Cay2(P)→T defined below.
There is a natural left action of Γ on each of Cay2(P) and T and we construct
f to be equivariant with respect to this as follows. Let m be the midpoint of the
edge H of T and define f to map the vertex g ∈ Cay2(P) to the point g·m, the
midpoint of the edge gH. Define f to map the edge of Cay2(P) labelled a ∈ A
i
joining vertices g and ga to the geodesic segment joining g ·m to ga·m. Since
a 6∈ H this segment is an embedded arc of length 1 whose midpoint is the vertex
gG . Define f to collapsethe edge inCay2(P)labelledb∈B joining verticesg and
i
gbtothe pointg·m=gb·m. This iswelldefinedsincegH =gbH. This completes
the definition of f on the 1-skeleton of ∆; we now extend f over the 2-skeleton.
Let c be a 2-cell in Cay2(P) and let g be some vertex in its boundary. Assume
thatcismetrisedsoastobeconvexandletlbesomepointinitsinterior. Theform
of the relations in P ensures that the boundary label of c is a word in the letters
A ∪B for some i and so every vertex in ∂c is labelled gg′ for some g′ ∈G . Thus
i i
f as so far defined maps ∂c into the ball of radius 1/2 centred on the vertex gG ;
i
we extend f to the interior of c by defining it to map the geodesic segment [l,p],
wherep∈∂c,tothegeodesicsegment[gG ,f(p)]. Thisisindependentofthevertex
i
g ∈ ∂c chosen and makes f continuous since geodesics in a tree vary continuously
with their endpoints. We now define f¯:∆→T to be the mapgivenby composing
f with the label-preserving map ∆→Cay2(P) which sends the basepoint of ∆ to
the vertex 1∈Cay2(P).
Sincewcommuteswithuandvwehavethatf¯(p )=w(uv)i−1u·m=(uv)i−1u·m=
i
f¯(q ); define S to be the preimage under f¯of this point. By construction, the im-
i
age of the interior of each 2-cell in ∆ and the image of the interior of eachA -edge
i
is disjoint from f¯(p ). Thus S consists of vertices and B-edges and so finding a
i
B-path from p to q reduces to finding a path in S connecting these vertices. Let
i i
s and t be the vertices of ∂∆ immediately preceding and succeeding p in the
i i i
8 WILLDISON
boundary cycle. Unless h = 1, in which case the theorem is trivial, the form of
theword[w,(uv)n],togetherwiththenormalformtheoremforamalgamatedprod-
ucts, implies that all the vertices p , s and t lie in the boundary of the same disc
i i i
componentD of∆. Furthermore,since uandv arewordsin the letters A andA
1 2
respectively,the points f(s ) and f(t ) are separatedin T by f(p ). Thus s and t
i i i i i
are separated in D by S and so there exists an edge path γ in S from p to some
i i
other vertex r ∈ ∂D. Since γ is a B-path it follows that the word labelling the
i i
sub-arcoftheboundarycycleof∆fromp tor representsanelementofH,and,by
i i
considering subwords of [w,(uv)n], we see that the only possibility is that r = q .
i i
Thus, for each i=1,...,n, the path γ gives the required B-path connecting p to
i i
q . We choose each γ to contain no repeated edges.
i i
For i 6= j, the two paths γ and γ are disjoint, since if they intersected there
i j
would be a B-path joining p to p and thus the word labelling the subarc of the
i j
boundary cycle from p to p would represent an element of H. Observe that no
i j
twoedgesin anyofthe paths γ ,...,γ lie inthe boundaryofthe same2-cellin∆
1 n
since each relation in P contains at most one occurrence of a letter in B. Because
the word labelling ∂∆ contains no occurrences of a letter in B the interior of each
edgeofapathγ liesintheinteriorof∆andthusintheboundaryoftwodistinct2-
i
cells. Since eachpath γ contains no repeated edges we therefore obtain the bound
i
n
Area(∆)≥ 2|γ |. But the wordlabelling eachγ is equalto h in Γ andso the
i=1 i i
length of γi is at least dB(1,h) whence we obtain the required inequality. (cid:3)
P
We are now in a position to prove the main theorem.
Proof of Theorem 1.1. Toavoidexcessivesuperscriptswechangenotationandwrite
x ,y for the generators of F(i), i=1,2.
i i 2
ByProposition4.1andTheorem5.1wehavethatK ∼=K23(2)∼=L1∗ML2 where,
as subgroups of F(1) ×F(2), L = K2(1) is generated by A = {x x−1,y ,y },
2 2 1 2 1 1 2 1 2
L ∼= K2(1) is generated by A = {x ,x ,y y−1} and M = K2(2) is generated
2 2 2 1 2 1 2 2
by B = {x x−1,y y−1,[x ,y ]}. To obtain the generating set for L we have here
1 2 1 2 1 1 2
implicitly usedthe automorphismofF(1)×F(2) whichinterchangesx with y and
2 2 i i
realises the isomorphism between L and K2(1).
2 2
For n ∈ N, define h to be the element [xn,yn] ∈ K2(2) and define w to be
n 1 1 2 n
the word[(x x−1)n,yn]∈A±∗ representingh . Note thath commutes withboth
1 2 1 1 n n
y ∈ A and x ∈ A so by Theorem 6.1 the word [w ,(y x )n], which has length
2 1 2 2 n 2 2
16n, has area at least 2ndB(1,hn). We claim that dB(1,hn)≥n2.
Suppose that in F(1) × F(2) the element h is represented by a word w ≡
2 2 n
w(x x−1,y y−1,[x ,y ]) in the generators B. Let k be the number of occurrences
1 2 1 2 1 1
of the third variable in the word w. We will show that k ≥n2.
Observe that, as a group element, the word w(x x−1,y y−1,[x ,y ]) is equal to
1 2 1 2 1 1
thewordw(x ,y ,[x ,y ])w(x−1,y−1,1).Thuswehavethat[xn,yn]isfreelyequal
1 1 1 1 2 2 1 1
tow(x ,y ,[x ,y ])andthatw(x−1,y−1,1),andthusw(x ,y ,1),isfreelyequalto
1 1 1 1 2 2 1 1
theemptyword. Itfollowsthat[xn,yn]canbeconvertedtotheemptywordbyfree
1 1
expansions, free contractions and deletion of k subwords [x ,y ]. Hence [xn,yn] is
1 1 1 1
a null-homotopic word over the presentation P = hx ,y |[x ,y ]i with P-area at
1 1 1 1
most k. But P presents the rank 2 free abelian group, and basic results on Dehn
functions give that [xn,yn] has area n2 over this presentation. Thus k ≥n2. (cid:3)
1 1
A SUBGROUP OF A DIRECT PRODUCT OF FREE GROUPS 9
NotethattheaboveproofalsoshowsthatK2(2)hasatleastquadraticdistortion
2
in each of K2(1) and F(1) ×F(2). In fact it can be shown that the distortion is
2 2 2
precisely quadratic.
Acknowledgements. I wouldliketo thank mythesis advisor,MartinBridson,for
his many helpful comments made during the preparation of this article.
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WillDison,DepartmentofMathematics,UniversityWalk,Bristol,BS81TW,United
Kingdom
E-mail address: [email protected]