Lec 11, Limit Groups, R-trees

not free. Then there is a finite set S = {s : H � Hs} of proper

epimorphisms such that:

• for all h ∈ Hom(H,F), there exists a ∈ Aut(H) such that h ◦ afactors through S.

Hs

H

F

H........................................................................................................................................................................................................... ............a

.................................................................................................................................................................................................................................

s

.................................................................................................................................................................................................................................

h

...................................................................................................................................................................................................... ............

• We will refine this statement and discuss a proof.

Theorem (Kharl, Mias,razzzz) . Suppose H is

• Areal tree ( T, dT )is a metric space such that between any two

points t, t′ ∈ T , there is a unique arc∗ from t to t′ and this arc is

the image of an isometric embedding of an interval.

Example. A finite simplicial real tree is a finite tree with each

edge identified with an interval.

Example. A countable increasing union of finite simplicial real

trees.

Example. 0-hyperbolic spaces embed into real trees.

∗the image of an embedding σ : [x, x′]→ T with σ(x) = t and σ(x′) = t′

Real trees

• An isometry η of a real tree T either elliptic or hyperbolic.

• Elliptic η fixes a point of T . The axis of η is Fix(η).

• Hyperbolic η leaves invariant an isometrically embedded R (its

axis Aη). Points on Aη are translated by

`T (η):= min {dT (t, η(t)) | t ∈ T}.

1 Isometries of real trees

• We will be interested in isometric actions of a fg group H on

T .

• The H-tree T is minimal if T contains no proper invariant

H-subtrees.

Lemma. If H is fg and T is a minimal H-tree, then T is either

a point or the union of the axes of the hyperbolic elements of

H.

•A (H)is the set of isometry classes of non-trivial minimal H-

trees.

• Gromov topology: lim{(Tn, dn)} = (T, d) iff for all finite K ⊂ T ,

ε > 0, and finite P ⊂ H there are, for all large n, subsets Kn of

Tn and bijections fn : Kn → K such that:

|d(ηfn(sn), fn(tn))− dn(ηsn, tn)| < ε

for all sn, tn ∈ Kn and η ∈ P .

• Intuitively, larger and larger pieces of the limit tree with their

restricted actions (approximately) appear in nearby trees.

2. Spaces of real trees

• We are interested in projectivized spaces of non-trivial H-trees,

i.e. (T, d) ∼ (T, λd), λ > 0.

PA(H) := A(H)/(0,∞)

• Non-trivial h ∈ Hom(H,F) gives an action of H on the Cayley

tree for F and so determines Th ∈ PA(H).

• Since the Cayley tree is a free F-tree, if η ∈ H fixes a point in

Th, then h(η) = 1.

• In particular, ker(Th) = ker(h).

• Th and Tiφ◦h are isometric where iφ denotes conjugation by

φ ∈ F.

Theorem (Paulin, Culler-Morgan). Let {hi} be a sequence in

Hom(H,F). Then {Thi} has a convergent subsequence in PA(H).

• Such limits have nice properties.

• An H-tree T is super stable if the following property holds:

if J ⊂ I are non-degenerate arcs with FixT (I) non-trivial, then

FixT (J) = FixT (I).

• An H-tree T is very small if it is non-trivial (i.e. not a point),

minimal, non-degenerate arc stabilizers are primitive abelian, and

non-degenerate tripod stabilizers are trivial.

Proposition. Suppose {hi} is a sequence in Hom(H,F), no hihas cyclic image, and T = limThi. Then:

1. T is irreducible (i.e. not linear and no fixed end);

2. Ker(T ) = ker−→ hi; and

3. T/ker(T ) is very small and super stable.

In particular, H/ker(T ) is a limit group.

Proof. We will show tripod stabilizers are trivial. The rest is

similar.

• Assume η stabilizes the endpoints of a tripod.

• Nearby Thi have tripods with endpoints nearly stabilized by η.

• So, η fixes the cone point and hi(η) = 1.

• Let H have a fixed finite generating set S. For h ∈ Hom(H,F),

‖h‖ := maxs∈S |h(s)|.

• h ∼ h′ if h′ = iγ ◦ h ◦ a where γ ∈ F and a ∈ Aut(H).

• h is shortest if h = minh′∼h ‖h′‖.

composable, and not Z. If T = limThi where {hi} is a sequence

of shortest elements, then T is not faithful.

3. Shortening

Theorem(Shortening) . Suppose H is fg, freely inde-

.

Theorem (K-M,S). Suppose H is fg and not free. Then there

is a finite set S = {s : H � Hs} of proper epimorphisms such

that:

• for all h ∈ Hom(H,F), there exists a ∈ Aut(H) such that h ◦ afactors through S.

Further, if H is a not a limit group then we may always take

a = IdH.

Proof. • If H is not a limit group then there is a finite set

{η1, . . . , ηN} such that every h ∈ Hom(H,F) kills some ηi. We

may take S = {H/〈〈η1〉〉, . . . H/〈〈ηN〉〉}.

4. Proof of Theorem (K-M,R )

• By Grushko, we may assume that H is freely indecomposable.

We may also assume that H is non-abelian.

• It is enough to show that there is S through which every short-

est h ∈ Hom(H,F) factors. Suppose not.

• Let {η1, . . . } enumerate the non-trivial elements of H. If

Si := {H/〈〈η1〉〉, . . . , H/〈〈ηi〉〉}

then there is shortest hi that is injective on {η1, . . . , ηi}. A sub-

sequence of {hi} converges to a faithful tree contradicting The-

orem(Shortening).

Corollary. We may assume each Hs is a limit group.

• The sets that appear in the theorem are calledfactor sets.There aren’t many explicit examples.

• If H = H1 ∗ H2, then S(H) = S(H1) ∗ S(H2) := {s1 ∗ s2 | si ∈S(Hi)}.

• S(Zn) = {s : Zn → Z}

• If H is a closed, orientable, genus g surface group and if s :H → Fg represents the standard retraction of the surface onto arank g graph, then S(H) = {s}. (Zieschang, Stallings)

• The non-orientable version is due to Grigorchuk and Kurchanovand one map does not suffice. Note: for n = 1, 2, or 3, nP isnot a limit group.

5. Examples of factor sets

• We need to better understand real trees. We will use foliated

2-complexes to visualize real trees.

Example (Triangles).

L}}L

Each colored band is decomposed into parallel line segments with

a transverse Lebesgues measure.