-
Chapter 9
Bass-Serre theory
In this Chapter we introduce the notion of graphs of groups and
their funda-mental group. Then we discuss the structure theorem of
the Bass-Serre theory,i.e. the duality of graphs of groups and
simplicial group actions on trees. For amore detailed account of
Bass-Serre theory please refer to the article of H. Bass[?] or the
book of J.P. Serre [Ser]. We further introduce the simple notions
ofthe core and the free decomposition of a graph of groups.
9.1 Automorphisms of trees
In this section we briey study automorphisms of trees. Let T be
a tree. Wewill say that an automorphism Aut (T ) inverts no edges
if (e) = e1 forall e ET . From now on all automorphisms have this
property.
Put := min{d(w,(w) |w V T}. If = 0 then (v) = v for somev V T .
In this case we say that is an elliptic automorphism. If > 0
thenwe say that is a hyperbolic automorphsim. The following lemma
describes thedynamics of a hyperbolic automorphism.
Lemma 9.1 Let T be a tree and be a hyperbolic automorphism of T
. Put = . Then the following hold:
1. There exists a unique bi-innite -invariant line Y T and acts
on Yby translation along .
2. d(v, z(v)) = |z| + 2d(v, Y ) for all v V T and z Z 0, in
particulard(v, (v)) = if and only if v V Y .
Proof Choose v V T such that d(v, (v)) = . Put Y = zZ
[z(v), z+1(v)].
In order to show that Y is a bi-innte line it clearly suces to
show that[z1(v), z(v)] [z(v), z+1(v)] = z(v) for all z Z. This is
clearly equiva-lent to showing that [v, (v)] [(v), 2(v)] = (v).
83
-
84 CHAPTER 9. BASS-SERRE THEORY
Suppose that [v, (v)] [(v), 2(v)] = (v), i.e. that [v, (v)]
[(v), 2(v)]contains a vertex w dierent form (v). This clearly
implies that w and 1(w)lie in [v, (v)]. Thus d(w,(w)) = d(w,1(w))
< d(v, (v) = ( which contra-dicts the minimality of .
v
(v)
2(v)w
1(w)
It is now clear that acts as a translation by on Y . Let v be an
arbitrary
vertex of T and w be its projection to Y , i.e. the unique
vertex w V Y suchthat d(v, Y ) = d(v, w). It is now clear that the
segment [v, (v)] equals theunion [v, w] [w,z(w)] [z(w), z(v)].
v z(v)
w z(w)Y
This implies that d(v, (v)) = d(v, w) + d(w,z(w)) + d(z(v),
z(w)) =d(v, w)+ |z|+d(z(v), z(w)) = |z|+2d(v, w) = |z|+2d(v, Y ).
The unique-ness of the set Y is obvious; thus the lemma is
proven.
The proof of the above lemma gives the following simple
characterization ofhyperbolic automorphisms:
Lemma 9.2 Let be an automorphism of a tree T . Then is
hyperbolic if andonly if there exists a vertex v V T such that [v,
(v)] [(v), 2(v)] = (v).Proof If is hyperbolic then any vertex of
the -invariant line has this property.If a vertex wiht above
properties exists then if follows from the proof of the abovelemma
that acts on the line
zZ[z(v), z+1(v)] as a non-trivial translation.
In particular xes no vertex and is therefore hyperbolic.
The next lemma tells us that the product of two elliptic
isometries is hyper-bolic unless it is elliptic for the obvious
reason.
Lemma 9.3 Let T be a tree and , Aut T be elliptic automorphisms
withoutcommon xed point. Then is hyperbolic.
Proof Choose a xed point x of and a xed point y of such that
d(x, y) isminimal. This clearly implies that
[x, y] ([x, y]) = [x, y] [(x), (y)] = [x, y] [x, (y)] = x
-
9.2. GROUP ACTIONS ON TREES 85
and that
[x, y] ([x, y]) = [x, y] [(x), (y)] = [x, y] [(x), y] = yfor =
1, see Figure 9.1.
x y
(x) = (x)
1(x)
1(y)
11(y)
Figure 9.1: and are elliptic with xed points x and y,
respectively.
It follows immediately that [11(y), x] [x, (x)] = x which
impliesthat is hyperbolic by Lemma 9.2.
9.2 Group actions on trees
Let G be a group and T a simplicial tree. We say that G acts on
T or that Tis a G-tree if G acts on T from the left by tree
automorphisms that invert noedges. Mory precisely we have a map (G)
Aut T such that for any g Gand v V T and e ET we have
gv = (g)(v) and ge = (g)(e) = e1.Let T be a G-tree. We say that
g G is elliptic if (g) is elliptic and we say
that g is hyperbolic if (g) is hyperbolic.
For any g G we put |g| = min{d(v, gv) | v V T}. We call |g| the
transla-tion length of g. Clearly g is hyperbolic i |g| > 0. For
any hyperbolic elementg we further denote the unique g invariant
bi-innte line by Ag, also called theaxis of g. The following
observation is a simple consequence of Lemma 9.1.
Lemma 9.4 Let T be a G-tree and g G be a hyperbolic element.
Then thefollowing hold:
1. g is of innite order.
2. d(v, gv) = |g| if and only if v Ag.3. |gz| = |z||g| for all z
Z.We next observe that most interesting group actions on trees
always have
hyperbolic elements.
-
86 CHAPTER 9. BASS-SERRE THEORY
Lemma 9.5 Let G be a nitely generated group and T be a G-tree
such thatevery g G is elliptic. Then G acts with a global xed
point, i.e. Gx = x forsome x T .
Proof Let {g1, . . . , gn} be a generating set of G. By
assumption each gi iselliptic. Thus Ti := {x T | gix = x} T is a
non-empty subtree of T for1 i n. It follows from Lemma 9.3 that Ti
Tj = for all 1 i, j nas otherwise gigj is hyerbolic contradicting
the hypothesis that all elements areelliptic. It is a simple
exercise to verify that nitely many subtree of a giventree with
pairwise non-trivial intersection have non-trivial intersection.
Thusi=1,...,n
Ti = , i.e. G = g1, . . . , gn has a xed point.
If we drop the assumption that G is nitely generated then the
conclusion ofLemma 9.5 remains no longer true, see Lemma 9.9 below.
The following lemmatells us that groups that act on trees often
contain free subgroups, the proofuses a variation of the famous
ping pong argument.
Lemma 9.6 Let T be a G-tree and g, h G be hyperbolic such that
the length ofAgAh is strictly less than max(|g|, |h|). Then g and h
generate a free subgroupof rank 2.
Proof Let : F (g, h) G be the extension of the map g g and h h.
Thisalso gives an action of F (g, h) on T via wx = (w)x for every w
F (g, h). Tosee that g and h generate a free subgroup of G we have
to show that the kernelof is trivial. To do this it clearly suces
to show that every w F (g, h) actsnon-trivially on T .
Let w F (g, h). If w is a power of g or h then the assertion is
trivial. Thuswe can assume that after conjugation w = gz1hz
1 . . . gzkhzk with zi, zi = 0 for
1 i k. Let p be a vertex of the segment Ag Ah. We show by
induction onk 0 that p = wp if k 1 and that
[p, wp] Ah Ag Ah.
which clearly implies the claim.
Ag
Ah
q q wpwp g
z1 q
wphz1q hz1wp
Figure 9.2: Two hyperbolic elements g and h wit Ag Ah
bounded
-
9.2. GROUP ACTIONS ON TREES 87
For k = 0 there is nothing to show as w = 1 implies that [p, wp]
= [p, p] AgAh. Suppose that k 1, i.e. that w = gz1hz1w with w =
gz2hz2 . . .gzkhzk .By induction we have [p, wp] Ah Ag Ah, in
particular the projection q ofwp to Ah lies in Ag Ah.
As |h| is assumed to be greater than the length of Ag Ah this
implies thatthe projection q of hz
1wp to Ag lies in Ag Ah. It follows that the projection
gz1 q of gz1hz1wp = wp to Ag does not lie in in Ag Ah. This
proves the
claim. .
As Ag = Agn for any n 1 and |gn| = n|g| for all n we get the
followingimmediate application.
Corollary 9.7 Let T be a G-tree and g, h G such that Ag Ah is
bounded.Then there exist numbers n,m N such that gn and hm generate
a free subgroupof rank 2.
The following lemma tells us that there are ve types of actions;
the mostinteresting one being the hyperbolic actions.
Lemma and Denition 9.8 Let G be a group and T a G-tree. Then one
ofthe following holds:
1. G acts trivially on T , i.e. Gv = v for some vertex v V T .
We say thatthe action is elliptic.
2. There are two hyperbolic elements g and h such that Ag Ah is
bounded.In this case some powers of gn and hm of g and h generate a
free group.We say that the action is hyperbolic.
3. T contains an invariant bi-innite G-invariant line Y and G
acts on Yby translations. In this case we say that the action is
cyclic.
4. T contains an invariant bi-innite G-invariant line Y and G
acts on Y bytranslations and reections. In this case we say that
the action is dihedral.
5. The action is not elliptic, cyclic or dihedral and there
exists an inniteray R in T such that gRR is again an innite ray for
all g G. In thiscase we say that the action is parabolic.
Proof Case A: Suppose rst that all elements of G are elliptic.
If G is nitelygenerated then we are in case 1 by Lemma 9.5. Thus we
can assume that G isnot nitely generated. We show that the action
is either elliptic or parabolic.
Suppose that the action is not elliptic. We construct a ray R
such thatRgR is an innite ray for all g G. Choose an element g0 and
a point x0 Tsuch that gx0 = x0. Then we show inductively that there
exist elements gi andpoints xi for all i 1 such that the following
hold:
1. For each i 1 the element gi is such that gixi1 = xi1.
-
88 CHAPTER 9. BASS-SERRE THEORY
2. For each i 1 the point xi is such that gjxi = xi for 0 j i
and thatd(xi1, xi) is minimal.
Note that these choices are always possible as each nitely
generated sub-group has a xed point by Lemma 9.5 and we assume that
G has no xed point.It is clear that the innite union
R := [x0, x1] [x1, x2] [x2, x3] . . .denes a ray in T . To see
that the action of G on T is parabolic it suces toshow that any g G
xes all but nitely many vertices of R. Suppose that this
x0 x1 xi xi+1 = gi+1xi+1y = gy
py
Figure 9.3: The xed point sets of g and gi+1 are disjoint as gxi
= xi = gi+1xi
does not hold for some g G. Choose a vertex y such that gy = y
let py bethe vertex of R that is in minimal distance from y. Choose
a vertex xi of Rsuch that gxi = xi and that d(x0, xi) > d(x0,
py). Such a vertex must exist aswe assume that innitely many
vertices of R are not xed by g. It follows thatneither gi+1 nor g x
xi and their xed point sets lie in dierent components ofT {xi}.
Thus ggi+1 is hyperbolic by Lemma 9.5, a contradiction.
Case B: Suppose that G contains hyperbolic elements. If there
exists aG-invariant line then we are clearly in situation 3 or 4.
Suppose now that theaction is neither elliptic nor cyclic, dihedral
or parabolic. We rst show thatthere exist elements g, h G such that
Ag Ah is bounded.
Suppose that Ag Ah is unbounded for all hyperbolic g, h G. We
rstshow that there exist g, h such that R = Ag Ah is a ray. If no
such elementsexist then all hyperbolic element have the same axis A
which must then also bexed by the elliptic elements as Ahgh1 = hAg
and we are in situation 3 or 4.
Ag Ahgh1
Ah
Figure 9.4: The axes of the elements g, h and f = hgh1 intersect
in the ray R
Thus there are g, h G such that R = Ag Ah is a ray. Put f =
hgh1. Itis clear that Ag Af and Ah Af are also infnitite rays that
intersect R in an
-
9.2. GROUP ACTIONS ON TREES 89
innite ray, i.e. R = AgAhAf is an innite ray. As we assume that
the actionis not parabolic it follows that gR R is not an innite
ray for some g G. Itfollows that gR R is bounded. Note that gR =
Aggg1 Aghg1 Agf g1 .It is further clear that one the axis Aggg1 ,
Aghg1 or Agf g1 , say Aggg1 , hasbounded intersection with R. It
follows that Aggg1 has bounded intersectionwith Ag, Ah or Af , thus
the rst assertion of (2) follows. The remaining claimnow follows
from Lemma 9.6.
Note that an action of a non-nitely generated group G on a tree
T canbe parabolic even if all elements are elliptic. In this
situation the proof ofLemma 9.8 implies that G =
nNStab vn if R = [v0, v1] [v1, v2] . . . is a ray
such that gR R is a ray for all g G. There is partial
converse:
Lemma 9.9 Any non-nitely generated countable group G admits a
parabolicaction on a tree such that any g G is elliptic.
Proof As G is countable there exists an innite strictly
ascending sequence ofsubgroup (Un) such that G =
nNUn. G now acts on a graph with vertex set
consisting of cosets, namely {gUn | g G,n N}. Furthermore two
cosets g1Unand g2Um are joined by an edge i |n m| = 1 and g1Un g2Um
or g2Um g1Un. G acts on this graph from the left in the natural way
by g gUn = (gg)Un.As any g G is containd in some Un it follows that
all elements are elliptic. Itis further easily veried that the
graph is a tree.
Some groups cannot act non-trivially on a tree; we say that such
a grouphas the xed point property or simply say that it has
property FA. Lemma 9.5clearly implies the following.
Lemma 9.10 Finite groups have property FA.
We further say that a group G has property AR if any G-tree is
eitherelliptic, cyclic or dihedral. Recall further that a group G
is noetherian if thereis no innite strictly ascending sequence of
subgroups.
Lemma 9.11 Let G be a group. Then the following are
equivalent:
1. G is noetherian.
2. Every subgroup of G is nitely generated.
3. Every subgroup of G has property AR.
Proof (1)(2) It is clear that all subgroups of Noetherian groups
are nitelygenerated as any non-nitely generated group contains an
innite strictly as-cending sequence of subgroups.
(2)(3) Suppose that some subgroup U does not have property AR.
Thusthere exists a hyperbolic or a parabolic U -tree T . If the
action is hyperbolic thenU contains a free group and therefore also
a non-nitely generated free subgroup
-
90 CHAPTER 9. BASS-SERRE THEORY
contradicting our assumption. Assume now that there exists a
parabolic U -treeT . Choose a ray R = [x0, x1] [x1, x2] . . . such
that R uR is a ray for allu U . By assumption the group U :=
nNStab xn is nitely generated, thus
there exists some N N such that U = Stab xn for n N . Let
further u Ube a hyperbolic element of minimal translation length.
Note that R Au isan innite ray. It follows that any element of U
can be written as uzu withz Z and u U . Thus Au is U -invariant,
i.e. the action is not parabolic, acontradiction.
(3)(1) Suppose that U0, U1, . . . is an innite strictly
ascending sequence ofsubgroups. We can assume that all Ui are
countable. It follows that the groupU =
nNUn is countable but not nitely generated as any nite set of
elements
lies in some Un. As every countable non-nitely generated
subgroup admits aparabolic action by Lemma 9.9 this implies that U
does not have property ARcontradicting our assumption.
9.3 The universal covering tree
In this section we show how to construct the universal covering
tree of a graph ofgroups and describe the action of the fundamental
group of the graph of groupson this tree. We discuss two dierent
ways of doing this, namely vie graphs ofspaces and their universal
covering space and combinatorialy using equvialenceclasses of
A-graphs. The rst one is more intuitive, the second one is
howevertechnically easier to handle.
Construction B Let A be a graph of groups with base vertex v0 V
A. Wedene an equivalence relation on the set of A-paths originating
at v0 by sayingthat p p if
1. p and p are both A-path from v0 to v for some v V A.2. p pa
for some a Av.
For an A-path p from v0 to v, we shall denote the -equivalence
class ofp by pAv. As in the proof of the reduced form theorem we
see that we eachequivalence class contains a unique representative
of type
a0, e1, a1, . . . , ak1, ek, 1
with ai Rei+1 where Re is a set of left coset representatives of
e(Ae) in A(e),i.e. A(e) =
rRer e(Ae).
We now dene the graph (A, v0) as follows.
1. The vertices of (A, v0) are -equivalence classes of A-paths
originating atv0. Thus each vertex of (A, v0) has the form pAv,
where p is an A-pathfrom v0 to a vertex v V A. We put x0 = 1Av0
.
-
9.3. THE UNIVERSAL COVERING TREE 91
2. Two vertices x, x of (A, v0) are connected by an edge f = (x,
e, x) with(f) = x, (f) = x and e EA if we can express x, x as x =
pAv, x =paeAv , where p is an A-path from v0 to v, a Av, (e) = v
and (e) = v.
3. The involution on the set of edges is dened the obvious way
by putting(v, e, v)1 = (v, e1, v).
If a0, e1, a1, . . . , ak1, ek, ak and a0, e1, a
1, . . . , a
k1, ek , a
k are reduced rep-
resentatives for vertices x and x and k k then it follows from
Proposition 6.11that x and x are joined by an edge if and only if k
= k + 1 and
a0, e1, a1, . . . , ak1, ek , ak a0, e1, a1, . . . , ak1, ek ,
ak .
The edge joining x and x is then (x, ek, x).
Theorem 9.12 The graph (A, v0) is a tree. It is called the
universal coveringtree or the Bass-Serre tree of (A, v0).
Proof Let q = f1, . . . , fk be a non-trivial closed path in (A,
v0); it clearly sucesto show that q is not cyclically reduced, i.e.
that either fi+1 = f
1i for some
1 i k 1 or that f1 = f1k . Put yi1 = (fi) for 1 i k and
representyi by a reduced path pi for 0 i k 1. Choose i such that
the length of pi ismaximal. After a cyclic permutation of the path
q we may assume that i = 0.
As yi is represented by the reduced path pi = a0, e1, a1, . . .
, ak1, ek, ak andthe length of pi is maximal it follows from the
remark before the Theorem thatboth yi1 and yi+1 are represented by
the path a0, e1, a1, . . . , ak2, ek1, ak1,i.e. that yi1 = yi+1. It
further follows that yi1 is joined to yi by the edgefi = (yi1, ek,
yi) and yi+1 it joined to yi by the edge f1i+1 = (yi+1, ek, yi)
=(yi1, ek, yi) = fi which proves the assertion.
We give two examples of universal covering trees, namely the
free productZ2 Z3 and the Baumslag Solitar group BS(1, 2) which is
a HNN-extension.Example 1: The graph of groups A corresponding to
the free product Z2 Z3consist of two vertices v0 and v1 with vertex
groups Av0 = Z2 and Av1 = Z3and a single edge pair {e, e1} with (e)
= (e1) = v0, (e) = (e1) = v1and Ae = 1.
Example 2: The graph of groups A corresponding to the group
BS(1, 2)consists of a single vertex v0 with vertex group Av0 = Z
and a single edge pair{e, e1} with Ae = Z and boundary
monomorphisms e : Z Z, z z ande : Z Z, z 2z.
We next show that the fundamental group G = 1(A, v0) acts in a
natu-ral way on T = (A, v0). In the theory of covering spaces this
corresponds tothe action of the fundamental group of a space on the
universal covering viaDecktransformations.
-
92 CHAPTER 9. BASS-SERRE THEORY
Figure 9.5: The Bass-Serre tree of Z2 Z3
If g = [q] G (where q is an A-path from v0 to v0) and u = pAv
(where p isan A-path from v0 to v V A), then we dene
g u = [q] pAv := qpAv.
This action is cleary well-dened on the set of vertices of (A,
v0) and it pre-serves the adjacency relation. Thus G has a
canonical simplicial action without
inversions on (A, v0). The following is immediate:
Lemma 9.13 Consider the action of 1(A, v0) of (A, v0).
1. The stabilizer of the vertex x = pAv is pAvp1.
2. The stabilizer of the edge (pAv, e, paeAv) is
p(ae(Ae)a1)p1.
9.4 The structure theorem
In the above section we have constructed a tree on which the
fundamental groupof a given graph of groups acts. In this section
we show that for any group Gand G-tree T we can associate a graph
of groups. The structure theorem of theBass-Serre theory says that
these two constructions are mutual inverses.
Suppose we have a G-tree T . Recall that we always assume that G
acts onT by tree-automorphisms that do not invert edges. In
particular we have that
-
9.4. THE STRUCTURE THEOREM 93
Figure 9.6: The Bass-Serre tree of BS(1, 2)
g(e1) = (ge)1, g(e) = (ge), g((e) = (ge) and ge = e1 for all g G
ande ET .
We show how to construct a graph of groups A with underlying
graph A =G\T . Let Y be a maximal subtree of A. It is clear that Y
can be lifted to T , i.e.that there exists an injective graph
morphism i : Y T such that i = idYwhere : T G\T is the canonical
projection. We can clearly extend i tothe set EA such that for
every edge e EA EY either i((e)) = (i(e)) ori((e)) = (i(e)).
The resulting map will however not respect the graph structure
of A unlessY = A, i.e. unless A is a tree. It follows that we can
assume that i((e)) =(i(e)) for some e EA EY if A is not a tree
(Figure 9.7).
ie
(e) i((e))
i(e) (i(e))
Figure 9.7: The lift a maximal subtree Y of A to T
We then dene Av := StabGi(v) for v V Y = V A and Ae := StabGi(e)
for
-
94 CHAPTER 9. BASS-SERRE THEORY
e EA. The boundary monomorphisms e are dened as follows, recall
that itsuces to dene the e as e = e1 for all e EA.
1. e is the inclusion map if (i(e)) = i((e)).
2. If (i(e)) = i((e)) then we choose an element ge G such that
(i(e)) =gei((e)) and dene e : Ae A(e) by e(g) = g1e gge.
For those e EA EY with (i(e)) = i((e)) we put ge = g1e1 . As
anumber of arbitrary choices were made during the construction of A
the resultis not unique. It is however not dicult to verify that
any two graphs of groupsobtained this way are equivalent where we
say two graphs of groups A and Aare equivalent and write A A if
there exists a graph isomorphism f : A A,isomorphisms e : Ae Af(e)
for all e EA, isomorphisms v : Av Af(v) forall v V A and elements
xe A(e) such that (e)(e(g)) = x1e f(e)(e(g))xefor any e EA and g
Ae.
Let A, Y be as above and v0 V A. We will show that there exists
an isomor-phism : 1(A, v0) G and -equivariant graph isomorphism f :
(A, v0) T .
For any v A let pv = ev1, . . . , evmv be the unique reduced
path in Y Ajoining v0 and v and pv = 1, ev1, 1, . . . , 1, evmv , 1
the associated A-path. Recallthat 1(A, v0) is generated by the [p1v
Avpv] for v V A together with theelements he := [p(e), e, p
1(e)] for e EA EY . We dene a homomorphism
: 1(A, v0) G
by [p1v avpv] av for all av Av and he ge for e EAEY . It is
immediatethat this maps extends to a homomorphism. We further
dene
f : (A, v0) T
by f(pvAv) = i(v) and extending this map equivariantly to set of
all verticesby putting f(g pvAv) = (g)f(pvAv) for all g 1(A, v0)
and v V A. Thismap is clearly well-dened and extends to the set of
edges as adjacent verticesget mapped to adjacent vertices. Note
further that it is immediate that isbijective on edge and vertex
stabilizers.
Theorem 9.14 (The structure theorem) The pair (, f) is an
isomorphismof group actions on tree. Thus is an isomorphism, f is a
tree isomorphismand
f(gx) = (g)f(x)
for any g 1(A, v0) and x (A, v0)
Proof The equivariance follows from the denition of and f . Thus
it sucesto show that f and are bijective.
-
9.5. MINIMAL ACTIONS AND THE CORE OF A GRAPH OF GROUPS95
To see that is surjective we have to verify that
G = H := vV Y
Av {ge | e EA EY }.
Note that H contains Stab v for all v V Y . To see that G = H it
thereforesuces to show thatHV Y = V T . Suppose thatHV Y = V T . As
T is connectedand both HV Y and V T are H-invariant it follows that
there exist adjacentvertices v V Y and w V T HV Y , thus there
exists an edge e1 with(e1) = v and (e1) = w. By construction of Y
and H there exists an edgee2 with (e2) = v and w
= (e2) HV Y that is Stab v-equivalent to e1,i.e. there exists a
g Stab v such that ge1 = e2. This however implies thatgw = w HV Y
as Stab v H, a contradiciton.
The surjectivity of together with the denition of f clearly
implies thesurjectivity of f . To see that f is injective it suces
to verify that f is locallyinjective. If f is not locally injective
then f identies two edges e1 and e2 withv := (e1) = (e2), in
particular ge1 = e2 for some g Stab v Stab e1. Note(g) Stab f(e1) =
Stab f(e2) contradicting the fact that is bijective onedge
stabilizers. Thus f is a bijection.
It remains to verify that is injective. Let g 1(A, v0) 1. If g
acts witha xed point that (g) = 1 as is bijective on stabilizers.
If g is hyperbolicthen gv = v for some vertex v and therefore
(g)f(v) = f(gv) = f(v) as f isinjective. Thus (g) acts
non-trivially on T which implies that (g) = 1.
9.5 Minimal actions and the core of a graph ofgroups
We will often restrict ourselves to situations where the action
of a group G ona tree T is minimal, i.e. where T contains no proper
subtree that is invariantunder the action of G. If G acts without
xed point of T this is simply achievedby replacing T with the
minimal G-invariant subtree of T . If the action is ellipticwe
replace T with an arbitrary vertex of T that is xed by G. We say
that agraph of groups is minimal if the corresponding action of the
fundamental groupon the Bass-Serre tree is minimal.
Given a minimal graph of groups A, the structure theorem makes
it easyto determine of what type the action of the fundamental
group on the univer-sal covering tree is. Proposition 9.15 below
implies in particular that in theabove examples the action of Z2 Z3
is hyperbolic and the action of BS(1, 2) isparabolic.
To simplify the statement we restrict attention to reduced
graphs of groups.We say that a graph of groups A is reduced if for
every vertex v V A of valence2 and non-loop edge e with (e) = v we
have that e is not surjective. If agraph of groups is not reduced
and v and e are as before then we say that thepair (v, e) is
inessential.
-
96 CHAPTER 9. BASS-SERRE THEORY
Note that any nite graph of groups can be easily modied to be
reducedby repeatedly collapsing inessential pairs: Suppose that (v,
e) is an inessentialpair. We then replace the graph of groups A by
the graph of groups A obtainedfrom A by omitting the edge e and
vertex v, putting (f) = (f1) = (e) forall e with (e) = v and
putting e = e1 = e 1e f . It is easily seen thatthis procedure does
not change the fundamental group of the graph of groups.
Proposition 9.15 Let A be a minimal reduced graph of groups. Let
G =1(A, v0) and T = (A, v0). The action of G on T is
1. elliptic i the graph A underlying A consists of a single
vertex.
2. cyclic i A consists of a single loop-edge and both boundary
monomor-phisms are surjective. Thus there is a short exact sequence
1 Av G Z 1.
3. dihedral i A consists of a single non-loop edge and the image
of bothboundary monomorphisms is of index 2 in the vertex group.
Thus G =A C B wih |A : C| = |B : C| = 2.
4. parabolic i A consists of a single loop edge and one boundary
monomor-phism is surjective.
5. hyperbolic otherwise.
Proof Part (1) is trivial. Suppose now the action is cyclic. The
minimalityimplies that T is a line and that G acts on T by
translation, it follows that G\Tis a circle. It is further clear
that any element that xes a vertex already xesall of T thus all
boundary monomorphisms are surjective. As we assume thatA is
reduced this implies that the loop G\T consistes of a single
loop-edge.
Suppose next that the action is dihedral. Thus T is a line and G
acts bytranslations and reection. In particular G\T is a segment,
i.e. has verticesv0, . . . , vk and edges e1, . . . , ek such that
(ei) = vi1 and (ei) = vi for 1 i k. Note that any element that
stabilizes an edge stabilizes all of T , that thestabilizer of T is
of index two in the stabilizer of those vertices that along
whichreections occurs and coincides with the stabilizer of the
remaining vertices.As the vertices v1, . . . , vk1 correspond to
vertices without reections it followsthat the reducedness
assumption implies that they cannot exist, i.e. that k = 1.This
proves the claim.
We conclude by investigating the parabolic case with ray R. If
all elementsare elliptic then R gets embedded into G\T under the
quotient map and itis easily veried that the graph of groups A
cannot be reduced. Thus we canassume that G contain a hyperbolic
element. Choose a hyperbolic element g Gof minimal translation
length. Pick vertices v, w R such that d(v, w) = |g|. Itis clear
that the segment [v, w] projects onto G\T and that the map is
injectiveexcept that v is being identied with w. This is true as G
must be generatedby g and Stab v. The assertion now follows
easily.
-
9.5. MINIMAL ACTIONS AND THE CORE OF A GRAPH OF GROUPS97
An immediate consequence is the following characterisation of
groups thathave the xed point property FA.
Lemma 9.16 Let G be group. Then G has property FA if and only if
thefollowing hold:
1. G is nitely generated.
2. G does not split as a proper amalgamated product.
3. G does not split as an HNN-extension.
Proof Note rst that if G has property FA then it must be nitely
generatedby Lemma 9.9. It further cannot split as an amalgamated
product or an HNN-extension as G would then admit a non-elliptic
action on the correspondingBass-Serre tree.
Suppose now that G fullls conditions (1)-(3) and that T is a
G-tree. Wehave to show that the action of G is elliptic; because of
Lemma 9.5 it sucesto show that every element g G is elliptic. If
some element is hyperbolicthen the action is not elliptic and G
splits as an amalgamated product or anHNN-extension by Proposition
9.15.
In the following we establish the notion of the core of a graph
of groups, asubgraph of groups that captures the essential part of
a graph of groups. Let Abe a graph of groups. We then dene
core(A) = A()
where the subgraph A is chosen as follows.
1. If there exists a vertex v V A such that the inclusion of Av
in 1(A) issurjective then is the subtree of A that is spanned by
all vertices thathave this property. Notice that in this case A
must be a tree thus is awell-dened subtree. In this case we say
that A has a simple core.
2. Otherwise we choose to be the minimal (with respect to
inclusion)subgraph of A such that the inclusion of 1(A(), v0) into
1(A, v0) issurjective for one/any v0 V. The uniqueness of is easily
veried.
We associate to any G-tree T the subtree TG T in the following
way: IfG acts without xed point we dene TG to be the minimal
G-invariant subtreeof T . If G acts with xed point then one could
choose TG to be any point thatis xed under the action of G. In
order to have a well-dened object we will inthe latter case dene TG
T to be the subtree that consists of all points thatare xed under
the action of G. It is easily veried that the tree TG has
thefollowing relationship with the core of a graph of groups.
Lemma 9.17 Let A be a graph of groups, G = 1(A, v0) and T = (A,
v0).Choose TG as above. Then core(A) is equivalent to A(G\TG).
-
98 CHAPTER 9. BASS-SERRE THEORY
The following lemma describes the relationship between a graph
of groupsand its core in somewhat more detail, it essentially means
that a graph of groupscan be obtained from its core by attaching
inessential trees.
Lemma 9.18 Let A be a graph of groups and A() its core. Suppose
that thecore is not simple.
Then for every v V AV there exists a unique edge path e1, . . .
, ek suchthat the following hold.
1. ei EA E for 1 i k.
2. (e1) V A and (ei) = (ei+1) V AV for 1 i < k and (ek) =
v.
3. ei(Aei) = A(ei) for 1 i k.
We will occasionally depict a graph of groups A by drawing A
such that theedges of , i.e. the edges corresponding to its core
are thicker. Note that inFigure 9.8 we have Av1 , Av2 Av3 and Av7 ,
Av8 , Av9 Av6
v1
v2
v3 v4
v5
v6 v7
v8
v9
Figure 9.8: A graph of groups and its core
9.6 The free decomposition of a graph of groups
If the core of a graph of group has a trivial edge group then
its fundamentalgroup decomposes as a free product. This simple
observation gives rise to thenotion a the free decomposition of a
graph of groups.
Let A be a graph of groups, E EA the set of edges with trivial
edge groupand A = AE. Let Ai A with i I for some index set I be the
componentsof A. We call the subgraphs of groups Ai = A(Ai) the free
factors of A. We willsay that a free factor is cyclic if its
fundamental group is cyclic.
Let further n be the number of edges outside a maximal tree of
the graphA obtained from A by contracting the subgraphs Ai to a
point. We call n thefree rank of A. It is easy to see that
1(A) = ri=1
1(Ai) Fn.
-
9.6. THE FREE DECOMPOSITION OF A GRAPH OF GROUPS 99
Note however that this decomposition does not necessarily
coincide with thecanonical decomposition of 1(A) into free factors
and a free group as we do notexclude that 1(Ai) is innite cyclic or
a proper free product for some i I.We call the pair
cf (A) := (r, n)
the free complexity of A.We will depict the free decomposition
of A by drawing all edges with non-
trivial edge groups as solid lines and edges with trivial edge
groups as dottedlines. We furthermore draw vertices that have
non-trivial vertex groups but areonly incident with edges with
trivial edge groups as fat dots. Clearly every fatvertex
corresponds to a free factor all by itself.
Figure 9.9: A graph of groups A with free complexity cf (A) =
(3, 2)
Occasionally we will combine this type of gure with the gures
introducedin Section 9.5, i.e. we will draw the subgraph
corresponding to a free factorwith thick and thin lines, where the
thick lines correspond to the core of thefree factor.
Figure 9.10: A graph of groups with three free factors and their
cores
-
100 CHAPTER 9. BASS-SERRE THEORY