Graphs of subgroups of free groups

0.06 MB·English

by  Larsen Louder

GRAPHS OF SUBGROUPS OF FREE GROUPS
LARSENLOUDERANDD.B.MCREYNOLDS
9
ABSTRACT. Weconstructanefficientmodelforgraphsoffinitelygeneratedsubgroupsoffree
0
groups. Using this we give a very short proof of Dicks’s reformulation of the strengthened
0
HannaNeumannConjectureastheAmalgamatedGraphConjecture. Inaddition,weanswera
2
questionofCullerandShalenonranksofintersectionsinfreegroups. Thelatterhasalsobeen

n doneindependentlybyR.P.KentIV.
a
J

5
2
1. INTRODUCTION

]
R Onepurpose ofthisarticleistoinvestigate theinterplay betweenthejoinandintersection ofa
G pair offinitely generated subgroups ofafreegroup. Ourmain result, Theorem 2.4, isaminor
generalization oftheconstruction ofthefirstauthorfrom[4],andproducesasimplemodelfor
.
h
analyzing intersections and joins. We use this technique to give a quick proof of a theorem
t
a of Dicks [2]. Another application of Theorem 2.4 is an answer to an unpublished question of
m
Culler and Shalen [1]. This has been done independently by Kent [3]. Explicitly, the result is
[
thefollowingtheorem.

1
Theorem 1.1. Let G=H ∗ H be a graph of free groups such that each H has rank 2. If
v 1 M 2 i
G։F thenM iscyclicortrivial.
4 3
7
Onecanderiveupperboundsontherankoftheintersection givenlowerboundsontherankof
7
thejoin. Thishasalsobeenobserved inthenicearticle ofKent[3],wheresomeupperbounds
3
. are explicitly computed. The proof of Theorem 1.1 presented here differs only slightly from
1
0 his. In the broadest terms, the two articles share with most papers in the subject an analysis
9 of immersions of graphs, a method that dates back to Stallings [5]. Specifically, Kent uses
0
directly the topological pushout of a pair of graphs along the core of their pullback, a graph
:
v whichappearshereastheunderlying graphofareduced graphofgraphs.
i
X
Acknowledgements. The authors are very grateful to Richard Kent for many discussions on
r
this topic, in particular those regarding Theorem 1.1. The first author thanks the California
a
Institute of Technology for its hospitality during a visit when this work began. The second
author would like to thank Ben Klaff for bringing the question of Culler and Shalen to his
attention. Finally, many thanks to the referee for several useful comments and suggestions,
especially amuchsimplifiedproofofLemma2.2.
2. GRAPHS OF GRAPHS
A graph of graphs is a finite graph of spaces such that all vertex spaces are combinatorial
graphs and all edge maps are embeddings. Below are some simple operations on graphs of
graphs. All vertices and edges are indicated by lower case letters, and their associated spaces
willbedenotedbythecorresponding letterinuppercase. Wewillnotkeeptrackoforientation
heredespiteitsoccasional importance—wetrustthereadertosortoutthissimplematterwhen
itarises. LetX beagraphofgraphswithverticesv andedgese .
i j
BothauthorsweresupportedinpartbyNSFpostdoctoralfellowships.
Graphsofsubgroupsoffreegroups 2
(M1) Making vertex and edge spaces connected: LetV ,...,V be the connected compo-
i,1 i,ni
nents of the vertex spaceV associated to the vertex v of the underlying graph G, and
i i
E ,...,E bethe connected components ofthe edgespace E . Weconstruct anew
j,1 j,mi j
graph ofgraphs asfollows. First, webuild theunderlying graph. Foreach iand j,we
take a collection of vertices v and edges e , one for each connected component of
i,k j,l
eachvertexspaceandedgespace,respectively. Welabelv withV ande withE ,
i,k i,k j,l j,l
and attach e tov if the image of E inV iscontained inV . The attaching maps
j,l i,k j,l i i,k
for this graph of graphs are the obvious ones. If e is adjacent to v , then we attach
j,l i,k
anendofE ×I toV bytheinclusion map.
j,l i,k
(M2) Removingunnecessary vertices: IfV isavertexspacewithexactlytwoincident edges
suchthatbothinclusionsareisomorphisms,weremovevandregardthepairofincident
edgesasasingleedge. IfV hasoneincidentedgeandtheinclusionisanisomorphism,
weremoveV andtheincident edge.
(M3) Removing isolated edges: Ifavertex spaceV hasanedgeethatisnottheimageofan
edgefromanincidentedgespace,weremoveefromV.
(M4) Collapsing freeedgesorvertices: WecallanedgeeofavertexspaceV freeifitisthe
imageofonlyoneedgefromthecollection ofincidentedgespaces,saye′⊂E. Inthis
case,weremoveeande′ fromV andE. IfavertexspaceV isapointandhasonlyone
incident edgespace, weremovevandtheincident edge.
Agraphofgraphsisreducedifanyapplicationoftheseoperationsleavesthespaceunchanged.
Notice that any graph of graphs can be converted to a reduced graph of graphs by greedily
applying(M1)through(M4). Theremainingrequisiteoperationsongraphsofgraphsareblow
upsandblowdownsatavertex.
(M5) For a vertex spaceV, divide the incident edge spaces into two classes E ,...,E and
1 n
E ,...,E ,andletV (V ,resp.) betheunionoftheimagesofE,i≤n(i>n,resp.).
n+1 m 1 2 i
WhenV ∩V isnon-trivial, wereplaceV byV ⊔V and introduce anew edge v ∩v
1 2 1 2 1 2
with the edge graphV ∩V . Next, weattach E toV for i≤n, E toV for i>n, and
1 2 i 1 i 2
thenewlyintroduced edgespaceV ∩V toV andV viatheinclusion maps.
1 2 1 2
(M6) Blowup: Weblowupavertexbyapplying(M5). Wepasstoconnectedcomponentsof
the newly created vertex and edge spaces via (M1). Finally, wepass to the associated
reduced graphofgraphsusing(M2).
(M7) Blow down: Let E be an edge space of a graph of graphs. If the two embeddings of
E have disjoint images, that is, i(E)∩t (E)= 0/, then we remove the edge e of the
underlying graphandidentify thetwoendpoints ofe. Finally, thegraphcarried bythe
new vertex is the one obtained by identifying the vertex space(s) at the ends of e by
setting i(f)=t (f),where f iseitheravertexoranedgeofE.
Remark. Notice that if X has no free or isolated edges, then the space obtained by blowing
up a vertex with an application of (M6) also has no free or isolated edges. Also, whenV is
connected, itfollowsthatV ∩V isnon-trivial andthus(M5)isapplicable.
1 2
The horizontal subgraph of a graph of graphs is the graph obtained by restricting vertex and
edgespacestovertices. Themid-graphofagraphofgraphsisthegraphobtainedbyrestricting
vertex and edge spaces to midpoints of edges. These two subgraphs are denoted G and G ,
H M
respectively. Note that neither of these graphs is necessarily connected. If X is reduced, then
G and G do not have any valence one vertices. Conversely, if either one of them has a
M H
valence one vertex, then there must be a free edge or vertex in X. If there are isolated edges,
thenacomponent ofG isapoint. Inparticular, ifX isreduced, theneverycomponent ofG
M M
hasnontrivial fundamental group.
Graphsofsubgroupsoffreegroups 3
Lemma2.1. Blowingupandblowingdownarehomotopyequivalences.
This follows easily upon observing that if two of the edges introduced during a blowup are
both adjacent to a vertex introduced during the blowup, then the images of the edge spaces
they carry are disjoint. That the remaining moves, other than (M3), preserve the homotopy
types of X, G and G , is clear. Note that (M3) only serves to remove trivial components of
H M
G .
M
Itisimportant toknowwhentoblowupX. Thefollowinglemmaachievesthis.
Lemma 2.2. Let D be a connected graph with a collection C = {D }m of (not necessarily
i i=1
distinct) connected subgraphs. If every edge of D is contained in at least two D and m>3,
i
then after relabeling the D , there is a partition C ={D ,...,D } and C ={D ,...,D }
j 1 1 n 2 n+1 m
of C such that at least two D in C intersect nontrivially and at least two D in C intersect
i 1 i 2
nontrivially.
Proof. Itsufficestofinddistinct A,B,C,D∈Csuch thatA∩B6=0/ andC∩D6=0/. Ifalltriple
intersections are empty then Chas at mosttwo elements by connectivity ofD . LetA,B,C∈C
such that A∩B∩C6=0/. Since D is connected, there is some D meeting, again without loss,
C. (cid:3)
Remark. Notice that ifV is a vertex space of a reduced graph of graphs X with at least four
incidentedgespaces, thenwecanuseLemma2.2toensurethat(M5)isapplicable.
Let X be a reduced graph of graphs such that all vertex and edge spaces are connected. The
space X has an underlying graph that we shall denote by G (X). Let m(X) be the highest
U
valenceofvertexofG (X),n(x)thenumberofverticeswithvalencem(X),andc (G (X))the
U U
Eulercharacteristic ofG (X). ThecomplexityofX isthelexicographically ordered 3–tuple
U
c(X):=(c (G (X)),m(X),n(X)).
U
Wecall ablowupofavertex vusingtwosetsofedge spaces satisfying Lemma2.2nontrivial.
Ournextlemmajustifiesthisterminology.
Lemma2.3. LetX bereduced andm(X)>3. IfX′ isobtained from X viaanontrivial appli-
cationof(M6)toavertexvwithvalencem(X),thenc(X′)<c(X).
Proof. Let {v} be the vertices of X′ introduced during a blow up of X at the vertex v. These
i
vertices musthavevalenceatleasttwo,asotherwise X hasafreeedgeandisnotreduced. We
assume contrary to the claim that c(X)=c(X′). If the Euler characteristics of the underlying
graphs of X and X′ are equal, then the subgraph B spanned by the edges associated to the
connectedcomponents ofV ∩V mustbeatree. Firstobservethatitisconnectedasotherwise
1 2
V couldnothavebeenconnected. Second,ifBisnotatree,thentheEulercharacteristic ofthe
underlying graphmustdecrease. AsBisatreewehave
1 (cid:229) 1
(1) 1− valence(v)= 1− valence(v) .
i
2 (cid:18) 2 (cid:19)
i
If both m(X′)=m(X) and n(X′)=n(X), then all but one of the vertices v has valence two
i0
since there are no valence one vertices making a positive contribution to the sum on the right
handsideof(1). Therefore,everycomponentofV (thealternativeishandledidentically)isthe
1
image of exactly one incident edge space from one element of the partition of edges incident
tov. However,thisisimpossible sincetheblowupX′ wasassumedtobenontrivial. (cid:3)
Wearenowreadytostateourmainresult.
Graphsofsubgroupsoffreegroups 4
Theorem2.4. EverygraphofgraphsX suchthatnoconnected componentofG isatreecan
M
be converted to a reduced graph of graphs X′ all of whose vertex groups have valence three.
Thereisahomotopyequivalence (X′,G ′ ,G ′ )→(X,G ,G ).
H M H M
The corank of a group G is the maximal rank of a free group that it maps onto and will be
denoted bycr(G). BeforeprovingTheorem2.4,afewremarksareinorder. First,observe that
ifX isreduced,thenthenaturalmapp (X)→p (G (X))issurjective. Second,thecomplexity
1 1 U
of all graphs of graphs homotopy equivalent to X is bounded below by (1−cr(p (X)),3,0).
1
Thatsaid,wenowgiveaproofofTheorem2.4.
ProofofTheorem2.4. Firstweapply(M4)untiltherearenofreeedges. Thisdoesnotchange
thehomotopytypeofthetriple(X,G ,G ). Therearenoisolatededgessinceeachcomponent
H M
of G is assumed to have nontrivial fundamental group. Next, we pass to connected compo-
M
nentsofedgeandvertexspacesandthenpasstotheassociated reduced graphofgraphsbyre-
movingvalencetwovertexspaces. LetX beareducedgraphofgraphs,andconsiderasequence
{X} starting with X such that X is obtained from X by nontrivially blowing up a maximal
i i i−1
valence vertex. Since all the X are homotopy equivalent and the maps p (X)→p (G (X)),
i 1 i 1 U i
i>0, are surjective, c(X)≥(1−cr(p (X)),3,0). According to Lemma2.3, c(X)>c(X ).
i 1 i i+1
Sincethecomplexity isbounded below,forsomen,X hasonlyvalencethreevertices. (cid:3)
n
A graph of graphs represents a graph of free groups when the e –neighborhood of G is a
M
productI×G . InthiscasetherearetwonaturalimmersionsG →G inthesenseofStallings
M M H
[5]. Moreover,thereisanimmersionG →G . Wesaysuchagraphofgraphsisrepresenting.
H U
Conversely, suppose that G=D (H ,...,H ,M ,···,M) is a graph of free groups with vertex
1 k 1 l
groups H, edge groups M , and that there is a map g : G→F which embeds each H. Let i
i j i j
andt j bethetwoinclusionmapsMj֒→Hi(j)andMj֒→Ht(j). RepresentFasthefundamental
group of a marked labeled graph R with one vertex, and find immersions of marked labeled
graphs h : G → R representing g | , and m : G → R representing g | . We choose the
i Hi Hi j Mj Mj
notationG inanticipationofthefactthattheyaretheconnectedcomponentsofthehorizontal
Hi
subgraph ofthegraphofgraphsunderconstruction.
Theimmersionm jfactorsthroughh i(j)andh t(j)viaij: G Mj →G Hi(j) andt j: G Mj →G Ht(j). We
construct aspace X by taking the G ×I asedge spaces, taking the G as vertex spaces, and
Mj Hi
usingasattachingmapsi : G ×{0}→G andt : G ×{1}→G . Leta : G ×I→
j Mj Hi(j) j Mj Ht(j) j Mj
G Mj bethe projection tothefirstfactor. Sinceh i(j)◦ij =m j and h t(j)◦ij =m j there isawell
definedmapp : X →Rwhichrestricts toh andagreeswith m ◦a .
i j j
We now endow X with the structure of a graph of graphs. Let b be the base point of R. Let
V =p −1(b) and E =p −1(m ), where m is the midpoint of an edge e of R. Each edge e of
l l l l l
R induces two maps of E toV, each of which is an embedding. That these are embeddings
l
can be seen as follows. If one fails to be injective on vertices of E , then some G → R is
l Hi
not an immersion. If it is injective on vertices but not on edges, then some G →R is not
Mj
immersed. Thus,wemayusethisdatatoendow X withthestructure ofagraphofgraphs. By
Theorem2.4,wecanrepeatedly blowupX untilweproduceagraphofgraphsX′ allofwhose
verticeshavevalencethree. If(M3)iseverappliedintheprocess,thenitmustbethatsomeM
i
wastrivial.
Remark. Let w be a vertex of a vertex spaceV of a graph of graphs X. It follows that w is
a vertex of G and the valence of w in G is exactly the number of edge graphs incident toV
H H
whose images contain w. IfX is reduced andV hasvalence three, then there mustbe avertex
ofV which is contained in the image of all three incident edge graphs. Moreover, if G has a
M
valencethreevertexw,thentheimagesofwinG musteachhavevalencethreeinG .
H H
Graphsofsubgroupsoffreegroups 5
ProofofTheorem1.1. We begin by representing G։F by a map from a graph of graphs X
3
to a bouquet of three circles R. Note that since G։F , at least one of the maps H →F is
3 i 3
injective. Iftheotherfailstobeinjective,theresultisimmediate. Consequently,wearereduced
to the case when both are injective and thus the construction above can be implemented. By
blowingX up,wemayassumethatX hasonlyvalencethreevertices. ThemapG։F factors
3
through the map G∼=p (X)։p (G ), and the rank of G must be either 3 or 4. If the latter
1 1 U U
holds, then M is trivial and the theorem holds. If G has rank 3, since all vertices of G are
U U
valencethree,theremustbeexactlyfour. Bytheremarkabove,ifG hasavalencethreevertex,
M
then the map from the set of valence three vertices of G to the set of valence three vertices
H
of G cannot be injective. Since the twocomponents of G each have fundamental group F ,
U H 2
they have 2 valence three vertices apiece. However, this implies the contradiction that G has
U
atmost3valence threevertices. Thus,G hasvertices ofvalence atmosttwoandsohasrank
M
atmostone,asclaimed. (cid:3)
Remark. By[4], M is contained in the subgroup generated by a basis element in at least one
ofH orH .
1 2
Other inequalities of this type are easily obtained through an analysis of a reduced valence
three graph of graphs representing the intersection. In particular, special cases of the Hanna
Neumann conjecture can be verified with this analysis. For explicit inequalities, we refer the
readertoKent[3]whohasalsoderivedthem.
3. INTERSECTIONS OF SUBGROUPS OF FREE GROUPS
Let H and H be subgroups of a fixed free group F. If G=D (H ,H ;M ), a graph of free
1 2 1 2 j
groups with two vertex groups {H}, edge groups {M }, with no monogons and a a map
i j
p : G։ F embedding each of the factors H, then the vertex spaces of a graph of graphs X
i
representing D arebipartite.
Agraph ofgraphs issimple-edged ifnovertex space has abigon. Torelate reduced graphs of
graphs to intersections of free groups we need to understand what happens when a graph of
graphs X as above is not simple-edged. Let p and q be the midpoints of a pair of offending
edges, G =G (X)the underlying graph of X,and givethe edges of G distinct oriented labels.
U
The labeling of G induces labelings of G and G . Let G ′ be the labeled graph obtained
M Hi M
by identifying p and q. By folding the labeled graph G ′ (see for instance [5]), we obtain a
M
labeled graph G with fundamental group K =p (G ,p). Folding endows G with a pair of
K 1 K K
immersionsn : G →G . Inaddition,thereisanimmersionh : G →G andtheedgemap
i K Hi j Mj K
G →G isjustn ◦h .
Mj Hi i j
Wemustconsider twocaseswithregardto pandqafterfolding. Namely,themidpoints pand
q are either in the same component of G or in distinct components of G . We address the
M M
latter firstand assume, without loss of generality, that G and G are the components ofG
M1 M2 M
containing pandq. ComputethefundamentalgroupsofG andG withrespecttotheimages
H1 H2
of p (which coincide with the images of q). From this we see that p (H )∩p (H ) contains
1 2
p (M ) and p (M ). If p (M ) 6< p (M ) and p (M ) 6< p (M ), then each inclusion p (M) ֒→
1 2 1 2 2 1 i
p (H )∩p (H )isproperandtheimageofthefundamentalgroupofG ,computedwithrespect
1 2 K
to the image of p, is precisely hM ,M i. If neither h nor h is an isomorphism of labeled
1 2 1 2
graphs, thenK properly contains M andM . Intheeventweareinthefirstcase, without loss
1 2
ofgenerality, weshall assume that p,q are contained inG . Weidentify the vertices p and q
M1
ofG andthenfoldtoobtainalabeledgraphG . Asbefore,theimmersionG →G factors
M K M1 Hi
Graphsofsubgroupsoffreegroups 6
throughtheinducedimmersionG →G . InthiscaseG →G cannotbeanisomorphismof
K Hi M K
graphsandH ∩H properly containsM .
1 2 1
Let X be a reduced simple-edged graph of graphs with underlying graph G = G (X). Let
U
X(G ) be the collection of reduced simple-edged graphs of graphs with underlying graph G . If
X,X′∈X(G ),thenX ≤X′ifthereisamapofgraphsofspacesX →X′suchthatallrestrictions
to vertex and edge spaces are embeddings. We can restrict to the subcollection X (G ) such
X
that for each X′ ∈ X (G ) there is a map X → X′ and the map G (X)→ G (X′) is a graph
X H H
isomorphism. Clearly X (G )has amaximal elementY. Tolink reduced simple-edged graphs
X
of graphs to the strengthened Hanna Neumann conjecture, weonly need to observe that since
X is simple-edged, each component of G (X) is an embedded subgraph of G (Y) (i.e. the
M M
fundamental groups ofcomponents ofG (X)arefreefactors ofthe respective components of
M
G (Y)).
M
Thestrengthened HannaNeumannconjecture thenimpliesthatifGisasaboveandtheassoci-
atedgraphofgraphsissimple-edged, then
c (G )c (G )+c (G )≥0
H1 H2 M
The equivalence of the amalgamated graph conjecture and the strengthened Hanna Neumann
conjecture of [2] follows immediately from the observation that the vertex and edge spaces
of a representing simple-edged graph of graphs can be written as in the statement of Dicks’
theorem. Weleave the details ofthe construction ofthis correspondence to thereader, though
westateaversionoftheequivalence forcompleteness.
Let X be a simple-edged reduced graph of graphs all of whose vertices are valence three that
represents ahomomorphism D (H ,H ,M )→F. Letv bethevertices ofG (X),andforeach
1 2 j i U
i, let D be the intersection of the images of the three edge spaces incident to v. Finally, let D
i i
bethe disjoint union ofthe D , S =D ∩G and S =D ∩G , and m bethe number ofedges
i 1 H1 2 H2
inD .
Theorem3.1.
c (H )c (H )+(cid:229) c (M)= 1|S |˙|S |−1m
1 2 i 1 2
4 2
i
Theproof isstraightforward. Thenumber ofvalence three vertices ofG is|S |, G hasonly
Hi i Hi
valencetwoorvalencethreevertices,andtheEulercharacteristicofH istherefore−1|S |. The
i 2 i
Euler characteristic of each G is computed in the same manner. m is the number of valence
Mj
three vertices ofG . Inthis formulation, theamalgamated graph conjecture simply states that
M
if one is given a reduced simple-edged representing graph of graphs whose horizontal graph
hastwocomponents, thentherighthandsideoftheaboveequality isnonnegative.
REFERENCES
[1] M.CullerandP.B.Shalen,Four-freegroupsandhyperbolicgeometry,preprint2008.
[2] W.Dicks,EquivalenceofthestrengthenedHannaNeumannconjectureandtheamalgamatedgraphcon-
jecture,Invent.Math.117(1994),373–389.
[3] R.P.KentIV,Intersectionsandjoinsoffreegroups,toappearinAlgebr.andGeom.Topol..
[4] L.Louder,KrulldimensionforlimitgroupsIII:Scottcomplexityandadjoiningrootstofinitelygenerated
groups,preprint2006.
[5] J.R.Stallings,Topologyoffinitegraphs,Invent.Math.71(1983),551–565.