Solvability in Groups of Piecewise-linear
Homeomorphisms of the Unit Interval
Collin Bleak
November 28, 2005
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PLo(I) and F
Definition 1. PLo (I) is the group of piecewise-linear, orientation-preserving
homeomorphisms of the unit interval which allow only finitely many breaks in
slope, under the operation of composition.
α
PSfrag replacements
In this talk, we will classify the solvable subgroups of PLo (I), as well as
those in R. Thompson’s group F (a subgroup of PLo (I)), and we will produce
a structure theorem for the non-solvable subgroups of PLo (I) and F. The following question is therefore only partially answered.
Question 1. Can we provide structure theorems for subgroups of PL o (I) describing the nature of groups in each of the classes Solve ⊂ EG ⊂ AG ⊂ NF?
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1.1 The classes Solve, EG, AG, and NF
1. Solve
A group G is solvable if there exists a finite series of groups
1 = G 0 / G1 / . . . / G n = G
so that for each i ∈ {0, 1, . . . , n − 1} the group Gi+1 /Gi is abelian.
2. EG
A group G is elementary amenable if it is in the class EG of groups. The
class EG is the smallest class of groups which contains the finite groups
and the abelian groups, and which is closed under the operations of
(a) taking subgroups,
(b) taking quotients,
(c) taking extensions, and
(d) taking direct limits.
We will see several examples of non-solvable, elementary amenable groups
in this talk.
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3. AG
A group is in the class AG if it is amenable. A discrete group is amenable
if it admits a measure – a function that assigns to each subset of G a
number from 0 to 1 so that
(a) the measure of the whole group is 1,
(b) the measure is finitely additive, and
(c) the measure is right-invariant.
A question of current interest is whether F is non-amenable. Initially, this
was to provide a first finitely-presented counterexample to von Neumann’s
conjecture. A separate counter-example has just been found by Olshanskii
and Sapir (Pubs. of IHES #96, 2002). Nonetheless, interest in the possible
non-amenability of F persists.
4. NF
A group G is in the class NF if G fails to contain a copy of the free group on
two generators.
It is known that Solve ⊂ EG ⊂ AG ⊂ NF, and that there are groups in each
set difference. It is not known whether, for instance, PLo (I) has any groups
which are in the class AG\EG.
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2
Orbits and Orbitals
Definition 2. Suppose G ≤ PLo (I), the support of G is the subset of [0, 1] which
is moved by the action of G.
Definition 3. Suppose G ≤ PLo (I). We define an orbital of G to be a component
of the support of the action of G.
Note that if A is an orbital of G, then A is an open interval which is the
convex hull of the orbit of any point in the interval under the action of H.
Suppose g ∈ PLo (I), we call an interval A = (a, b) an orbital of g if A is an
orbital of hgi. We note that on A, g either moves all points right or all points
left.
PSfrag replacements
0 a
α
b
c
1
A = (a, b) and B = (c, 1) are the orbitals of α.
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3
The Dynamics of Conjugation
Lemma 3.1. Suppose g, h ∈ PLo (I), and gh = h−1 gh.
1. The orbitals of g and gh are in one-to-one correspondence.
2. If A is an orbital of g, then Ah is an orbital of gh , furthermore, gh moves
points left on Ah if and only if g moves points left on A.
The following diagram demonstrates a convenient “sketch-hand” of these
notions; given α ∈ PLo (I) we sketch a smooth approximation of the set
{(x, xα − x) | x ∈ Supp(α)} .
For multiple elements, we offset their sketches vertically as is convenient. We
will use this method of sketching elements of PLo (I) in the remainder of the
talk.
h
g
gh
PSfrag replacements
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4
Signed Orbitals and Towers
Definition 4. A signed orbital for a group G ≤ PLo (I) is a pair (A, g) so that A
is an orbital of g which is an element of G.
Note: The set of signed orbitals of PLo (I) forms a poset.
Definition 5. Given a group G ≤ PLo (I) and a set T of signed orbitals of G,
we will say T is a tower of G if T satisfies the following properties:
1. T is a chain in the partial order on the signed orbitals of G.
2. Given any interval A ⊂ I, T has at most one element of the form (A, g).
PSfrag replacements
(A, f )
(B, g)
(C, h)
f
g
h
Here we can easily spot a tower T = {(A, f ), (B, g), (C, h)} for the group
h f , g, hi.
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5
Geometric Classification of Solvable subgroups of PLo(I)
Definition 6. The depth of a group H ≤ PLo (I) is the supremum of the cardinalities of the towers of the group.
This definition allows us to geometrically characterize the solvable subgroups of PLo (I).
Theorem 5.1. Let n ∈ N and H ≤ PLo (I). H has derived length n if and only
if the depth of H is n.
PSfrag replacements
f
g
gf
g−1
[g, f ] = g−1 · g f
h
To explain the theorem, consider that if A is a “top-level” element orbital
for a group G, then no element in G0 will have orbital A, so no tower of G0 can
have an element of the form (A, ∗).
Corollary 5.2. A subgroup H ≤ PLo (I) is non-solvable if and only if H admits
towers of arbitrary height.
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Transition Chains
Definition 7. Given a subgroup G of PLo (I) and a set C = {(Ai , gi ) | i ∈ I } of
signed orbitals of G for some index set I . If C satisfies the following properties:
1. given any x < y with x ∈ Ai and y ∈ A j for some i, j ∈ I , we have (x, y) ⊂
PSfrag replacements
∪k∈I Ak , and
2. given any i ∈ I , there is a point pi ∈ Ai , which is not in A j for any index
j 6= i,
then we will call C a transition chain for the group G.
g
δ
γ
0 a
b
c
f
e
d
From the diagram above, if we set
A1 = (a, c),
p1 =
A2 = (b, e),
p2 =
A3 = (d, f ), p3 =
a+b
2
c+d
2
e+ f
2
then the set
{(A1 , δ), (A2 , γ), (A3 , δ)}
forms a transition chain of length three for the group hδ, γi.
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1
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A Route to Thompson’s group F
Theorem 7.1. (Brin) Suppose G ≤ PLo (I) and that G has an element α and an
orbital A = (a, b) so that α has an orbital contained in A which shares exactly
one end of A, while α has no orbital in A sharing the other end of A, then G
contains a subgroup isomorphic with Thompson’s group F.
α
PSfrag replacements
a
b
A
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Suppose g and h are one-bump functions as below, where the orbital of g is
larger than a fundamental domain of h. Let us consider gh .
h
PSfrag replacements
g
gh
g and gh have orbitals that allow us to create a transition chain of length
two. While that does not always force a copy of Thompson’s group F to exist,
it does force the existence of infinite towers.
We now conjugate g by k = gh to get f = gk .
PSfrag replacements
k
g
f
A
The group h f , gi must contain a copy of Thompson’s group!
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8
Interesting operations, and a classification theorem
Given two subgroups G, and H of PLo (I), we can realize G
L
H by contracting
the action of G into the interval [0, 21 ] and contracting the action of H into the
interval [ 12 , 1], and then taking the union of the resulting pl functions.
PSfrag replacements
G = ha, b, ci
a
c
b
H = hd, ei
d
e
PSfrag replacements
becomes
G
a
b
0
L
H
d
c
e
1
2
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1
Another operation that is easy to accomplish in PLo (I) is taking a restricted
wreath product with Z, eg A 7→ A o Z.
You might ask what a restricted wreath product is, so here is a description.
GoH ∼
=(
M
G) o H
h∈H
The group action is by (right) multiplication on the index in the sum by the
element of H.
Given a subgroup A ≤ PLo (I), we realize A o Z easily in PLo (I) by squeezing
A horizontally into a single fundamental domain of a single bump function. See
a realization of Z o Z below.
h
PSfrag replacements
g
h−2
g
h−1
g
gh
13
gh
2
Noticing that the direct sum of two solvable groups is solvable, and that taking a restricted wreath product with Z increases derived length by one, we see
that either of these operations applied finitely many times with solvable group
inputs will produce a solvable group output. Further, solvability is also preserved by passing to subgroups, so we are encouraged to define the following
class of groups.
Definition 8. Let R be the smallest non-empty class of groups which is closed
under the following three operations:
1. taking subgroups,
2. taking wreath products with the integers (H 7→ H o Z), and
3. taking bounded direct sums.
(Here bounded direct sums are countable sums where all the summands are
solvable with a universal finite bound on their derived lengths.)
And now, a magic result!
Theorem 8.1. A subgroup H of PLo (I) is solvable if and only if H is isomorphic
to a group in R .
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9
Algebraic classification of the solvable subgroups of PLo(I)
and F
Define G0 = 1, the trivial group, and for all positive integers n define
Gn =
M
(Gn−1 o Z).
k∈Z
Definition 9. Let M = {Gn | n ∈ N}
PSfrag replacements
G2
Theorem 9.1. H is a group in the class R if and only if H is isomorphic to a
subgroup of a group in M .
Theorem 9.2. Let n ∈ N and H be a subgroup of PLo (I). H is a solvable with
derived length n if and only if H is isomorphic to a subgroup of G n , but not to
a subgroup of Gn−1 .
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10
Some Important Groups
Wn = ((. . . ((Z o Z) o Z) o . . . o Z) o Z) o Z
|
{z
}
n-Z’s
PSfrag replacements
a
b
c
d
W=
W4 ∼
= ha, b, c, di
M
Wn
n∈N
W1
W3
W2
PSfrag replacements
W=
L
i∈N Wi
Theorem 10.1. A subgroup H of PLo (I) is non-solvable if and only if H contains an isomorphic copy of W .
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(Zo)∞ = lim Wn
jn
(oZ)∞ = lim Wn
in
Lemma 10.2. Neither (Zo)∞ nor (oZ)∞ embed in W .
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Another group of recent acclaim is the group generated by the elements g
and h below. Guba and Sapir have taken to calling it the “Brin group” as it
feautures in a paper of Brin’s on elementary amenable classes in PL o (I).
g
h
PSfrag replacements
B = hg, hi
Theorem 10.3. Suppose H ≤ PLo (I) admits a transition chain of length two,
then H contains a copy of the group B sketched above.
Here is something else that is known. . .
Lemma 10.4. Every finitely-generated non-solvable subgroup of PL o (I) contains a copy of (oZ)∞ or of (Zo)∞ .
But no one is certain about the answer to this. . .
Question 2. Must a finitely generated non-solvable subgroup of PL o (I) contain
a copy of B?
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11 Wi vs Gi , and a cute fact
Remark 11.1. The following are true:
1. Wi embeds in Gi
2. Gi embeds in W3i−1
Theorem 11.2. Every non-solvable subgroup of PLo (I) contains a copy of every solvable subgroup of PLo (I).
pf:
Suppose H ≤ PLo (I) is solvable with derived length n for some n ∈ N, and
G ≤ PLo (I) is non-solvable. Then H embeds in Gn , which embeds in W3n−1 ,
which embeds in W , which embeds in G.
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