arXiv:1508.00493v3 [] 12 Jan 2017

On subgroups of R. Thompson’s group F
Gili Golan, Mark Sapir∗
January 13, 2017

Abstract
We provide two ways to show that the R. Thompson group F has maximal subgroups of
infinite index which do not fix any number in the unit interval under the natural action of F
on (0, 1), thus solving a problem by D. Savchuk. The first way employs Jones’ subgroup of
the R. Thompson group F and leads to an explicit finitely generated example. The second
way employs directed 2-complexes and 2-dimensional analogs of Stallings’ core graphs, and
gives many implicit examples. We also show that F has a decreasing sequence of finitely
generated subgroups F > H1 > H2 > ... such that ∩Hi = {1} and for every i there exist
only finitely many subgroups of F containing Hi .

1
1.1

Introduction
The stories about F

Recall that the R. Thompson group F consists of all piecewise linear increasing homeomorphisms of the unit interval [0, 1] with a finite number of linear segments with slopes of the
form 2n , n ∈ Z, and endpoints of the form 2an , a, n ∈ N∪{0}. The group F has a presentation
with two generators and two defining relations [6, 17] (see below).
The group F is one of the most mysterious objects in group theory. For example, almost
every 6 months a new “proof” of amenability or non-amenability of F appears. The reason
why all these proofs are wrong is that F is very counter-intuitive. Statements which are
“obviously true” turn out to be wrong.
This can also be illustrated by the story of finding the Dehn function of F . First it
was declared exponential because a sequence of loops in the Cayley graph of F was found
with exponential fillings, and it was “obviously impossible” to find fillings of these loops with
smaller area. Then it was proved to be subexponential and the most probable conjecture (for
2
the same reason as before) was that it is something like 2log n . Then a polynomial of degree
5 upper bound was found and it was conjectured that this bound is optimal. Then a few
lower bounds were found until Guba proved that F has quadratic Dehn function, the smallest
∗
The research was partially supported by the NSF grant DMS-1500180. The paper was written while the
second author was visiting the Max Planck Institute for Mathematics in Bonn.

1

possible Dehn function of a non-hyperbolic finitely presented group (see the references in [17,
Chapter 5]).

1.2

Subgroups of F

Subgroups of F have been extensively studied (see, for instance, [5, 6, 13, 3, 1, 7, 18, 19, 2,
10, 22]). For example, it is known that the derived subgroup [F, F ] of F is simple, and every
finite index subgroup of F contains [F, F ] [6]. Finite index subgroups of F are described in
[2] and all solvable subgroups of F are described in [1]. It is also known [13] that the wreath
products F ≀ Z embeds into F (even without distortion [22]). One of the most interesting
and counter-intuitive results about subgroups of F is that in a certain natural probabilistic
model on the set of all finitely generated subgroups of F , every finitely generated nontrivial
subgroup appears with a positive probability [7]. The survey in this paragraph is far from
complete. We just wanted to give a rough description of the area.
In [18, 19] Dmytro Savchuk studied subgroups HU of the group F which are the stabilizers
of finite sets of real numbers U ⊂ (0, 1). He proved that if U consists of one number, then
HU is a maximal subgroup of F . He also showed that the Schreier graphs of the subgroups
HU are amenable. He asked [19, Problem 1.5] whether every maximal subgroup of infinite
index in F is of the form H{α} , that is, fixes a number from (0, 1).1
In this paper, we are going to answer that question by establishing a maximal subgroup
of infinite index in F that does not fix any real number in (0, 1) (see Section 3). In fact we
give two solutions of this problem. One, explicit solution, gives an example as the preimage
→
−
of the subgroup F , first described by Vaughan Jones [15] and then studied by us in [10],
under a certain injective endomorphism of F (Theorem 3.15). As a by-product of the proof,
→
−
we give a positive answer to a question asked by Saul Schleimer: we prove that F is maximal
in a certain subgroup of index 2 of F and, moreover, there are exactly three subgroups of F
→
−
containing F (that fact, Corollary 3.14, is also very counter-intuitive).
Another, implicit, solution gives an example (in fact many examples) of maximal subgroups containing certain proper finitely generated subgroups H of F which do not fix any
point in (0, 1) and satisfy H[F, F ] = F . The most difficult part of that solution is to prove
that H is indeed a proper subgroup of F . Note that the solvability of the membership problem for subgroups in F is a very interesting open problem (it is mentioned in [13]) and the
generation problem is an important open case of the membership problem (we need to check
if the generators of F are in the subgroup). To prove that H is proper we employ diagram
groups of directed 2-complexes [14] and a 2-dimensional analog of Stallings foldings.
We also establish a nice property of Savchuk’s subgroups HU .
Let us call a subgroup H of a group G of quasi-finite index if the interval [H, G] in the
lattice of subgroups of G is finite, that is there are only finitely many subgroups of G that are
bigger than H. For example, every maximal subgroup and every subgroup of finite index are
of quasi-finite index. We say that a finitely generated group G is quasi-residually finite if it
contains a decreasing sequence of finitely generated subgroups: G > G1 > G2 > ... such that
each Gi is of quasi-finite index in G and ∩Gi = {1}. Clearly every residually finite group is
1

In the formulation of [19, Problem 1.5] the words “of infinite index” are missing. D. Savchuk informed us that
this is a misprint. Obviously none of the finite index maximal subgroups of F fixes a point in (0, 1).

2

quasi-residually finite. Since the derived subgroup of the R. Thompson group F is infinite
and simple, F is not residually finite. Nevertheless, we prove that if U is a finite set of finite
binary fractions, then HU is of quasi-finite index in F (this is a natural generalization of a
result from [18]), and hence F is quasi-residually finite (Theorem 4.1).

2

Preliminaries on F

2.1

Generators and normal forms

The group F is generated by two functions x0 and x1 defined as follows [6]


t
if 0 ≤ t ≤ 12

1



2t
if
0
≤
t
≤

2t − 1 if 1 ≤ t ≤ 5
4
2
2
8
x0 (t) = t + 41
x1 (t) =
if 41 ≤ t ≤ 21
5
3
1


if
≤
t
≤
t
+
t

1
1
8
8
4

+
if
≤
t
≤
1

1
t
2
2
2
if 34 ≤ t ≤ 1
2 + 2

The composition in F is from left to right.
Every element of F is determined by its action on the set of finite binary fractions which
can be represented as .α where α is an infinite word in {0, 1} where all but finitely many
letters are 0. For each element g ∈ F there exists a finite collection of pairs of words (ui , vi )
in the alphabet {0, 1} such that every infinite word in {0, 1} starts with exactly one of the
ui ’s. The action of F on a number .α is the following: if α starts with ui , we replace ui by
vi . For example, x0 and x1 are the following functions:


.0α
if t = .0α




if t = .00α
.10α
.0α
if t = .100α
x1 (t) =
x0 (t) = .10α if t = .01α


.110α
if t = .101α



.11α if t = .1α

.111α if t = .11α

For the generators x0 , x1 defined above, the group F has the following finite presentation

[6]
x2

x0
−1
0
F = hx0 , x1 | [x0 x−1
1 , x1 ] = 1, [x0 x1 , x1 ] = 1i.

Often, it is more convenient to consider an infinite presentation of F . For i ≥ 1, let
xi
xi+1 = x1 0 where ab denotes b−1 ab. In these generators, the group F has the following
presentation [6]
x
hxi , i ≥ 0 | xi j = xi+1 for every j < ii.
There is a natural notion of normal forms of elements of F related to this infinite presentation (see [6]). A word w in the alphabet {x0 , x1 , x2 , . . . }±1 is said to be a normal form
if
m −tn
1
w = xsi11 . . . xsim
xjn . . . xj−t
,
(2.1)
1
where
(1) i1 ≤ · · · ≤ im 6= jn ≥ · · · ≥ j1 ; s1 , . . . , sm , t1 , . . . tn > 0, and

3

(2) if xi and x−1
occur in (2.1) for some i ≥ 0 then either xi+1 or x−1
i
i+1 also occurs in (2.1).
Every element of F is uniquely represented by a normal form w. Moreover, the normal
form of an element w ∈ F is the shortest possible representation of the element as a word
in {x0 , x1 , x2 , . . . }±1 . A word w of the form (2.1) which satisfies condition (1) but not
necessarily condition (2) above, is called a semi-normal form.
Given a word w in the alphabet {x0 , x1 , x2 , . . . }±1 , there is a simple algorithm for finding
the corresponding normal form of w (see [20]). First, one uses the following rewriting system
Σ.
(σ1 ) xi xj → xj xi+1 for i > j.
−1
(σ2 ) x−1
i xj → xj xi+1 for i > j.
−1
(σ3 ) x−1
for j > i.
i xj → xj+1 xi
−1
−1 −1
(σ4 ) x−1
for j > i.
i xj → xj+1 xi

(σ5 ) x−1
i xi → 1.
(σ6 ) xi x−1
→ 1.
i
The rewriting system is terminating and confluent (for the terminology see [17]), hence every
word w can be transformed by applying a sequence of rewriting rules to a unique word w′
for which no rewriting rule can be applied. That word w′ does not contain subwords that
are equal to the left hand sides of the rewriting rules, hence it is a semi-normal form (it is
easy to verify). It is obvious that the word w′ is equal to w in F (see [20, Lemma 1]).
The method of getting a normal form from a semi-normal form will not be important to
us. Suffice it to know, that if w′ is a semi-normal form which does not satisfy condition (2)
above, then the length of the corresponding normal form will be at most |w′ | − 2 (see [20,
Lemma 5]). Below, we will not distinguish between an element of F and its normal form.
Every normal form w is a product pq −1 where p, q are positive words. The word p is
called the positive part of w and q −1 is the negative part of w. Every element whose normal
form is a positive word (i.e., its negative part is empty), is called a positive element. The set
of positive elements of F forms a submonoid of F because the rewriting rules do not increase
the number of negative letters in a word.

2.2

Jones’ subgroup

Vaughan Jones [15] showed that, similar to braids, elements of F can be used to construct all
→
−
links. He also showed that a certain subgroup F of F can be used to construct all oriented
→
−
links. In [10] we answered several questions of Jones about F . In particular we proved that
→
−
F is isomorphic to the group F3 introduced by K. Brown [4]: it is a version of F where all
slopes of the piecewise-linear functions are powers of 3 and break points of the derivative are
→
−
finite 3-adic fractions. We also showed that F is generated in F by elements x0 x1 , x1 x2 , x2 x3 .
→
−
By conjugating x1 x2 and x2 x3 by (x0 x1 )n one gets that for all n ≥ 0, xn xn+1 ∈ F .

4

2.3

Directed complexes and diagram groups

The material from this subsection will be used only in Sections 3.2 and 5.3. Here we basically
follow [14].
Definition 2.1. For every directed graph Γ let P be the set of all (directed) paths in Γ,
including the empty paths. A directed 2-complex is a directed graph Γ equipped with a set
F (called the set of 2-cells), and three maps top· : F → P, bot· : F → P, and −1 : F → F
called top, bottom, and inverse such that
• for every f ∈ F , the paths top(f ) and bot(f ) are non-empty and have common initial
vertices and common terminal vertices,
•

−1

is an involution without fixed points, and top(f −1 ) = bot(f ), bot(f −1 ) = top(f )
for every f ∈ F .

We shall often need an orientation on F , that is, a subset F + ⊆ F of positive 2-cells,
such that F is the disjoint union of F + and the set F − = (F + )−1 (the latter is called the
set of negative 2-cells).
If K is a directed 2-complex, then paths on K are called 1-paths (we are going to have
2-paths later). The initial and terminal vertex of a 1-path p are denoted by ι(p) and τ (p),
respectively. For every 2-cell f ∈ F , the vertices ι(top(f )) = ι(bot(f )) and τ (top(f )) =
τ (bot(f )) are denoted ι(f ) and τ (f ), respectively.
We shall denote each cell F by top(f ) → bot(f ). And we can denote a directed 2complex K similar to a semigroup presentaton hE | top(f ) → bot(f ), f ∈ F i where E is the
set of all edges of K (note that we ignore the vertices of K).
For example, the complex hx | x → x2 i is the Dunce hat obtained by identifying all edges
in the triangle (Figure 2.1) according to their directions. It has one vertex, one edge, and
one positive 2-cell.

x

✒❅
❅
❅x
❅
❅
✲
❘
❅

x
Figure 2.1: Dunce hat
With the directed 2-complex K, one can associate a category Π(K) whose objects are
directed 1-paths, and morphisms are 2-paths, i. e. sequences of replacements of top(f ) by
bot(f ) in 1-paths, f ∈ F . More precisely, an atomic 2-path (an elementary homotopy) is a
triple (p, f, q), where p, q are 1-paths in K, and f ∈ F such that τ (p) = ι(f ), τ (f ) = ι(q). If δ
is the atomic 2-path (p, f, q), then ptop(f )q is denoted by top(δ), and pbot(f )q is denoted
by bot(δ); these are called the top and the bottom 1-paths of the atomic 2-path. Every
nontrivial 2-path δ on K is a sequence of atomic paths δ1 , . . . , δn , where bot(δi ) = top(δi+1 )
for every 1 ≤ i < n. In this case n is called the length of the 2-path δ. The top and the

5

bottom 1-paths of δ, denoted by top(δ) and bot(δ), are top(δ1 ) and bot(δn ), respectively.
Every 1-path p is considered as a trivial 2-path with top(p) = bot(p) = p. These are the
identity morphisms in the category Π(K). The composition of 2-paths δ and δ ′ is called
concatenation and is denoted δ ◦ δ ′ .
With every atomic 2-path δ = (p, f, q), where top(f ) = u, bot(f ) = v we associate the
labeled plane graph ∆ in Figure 2.2. An arc labeled by a word w is subdivided into |w|
edges. All edges are oriented from left to right. The label of each oriented edge of the graph
is a symbol from the alphabet E, the set of edges in K. As a plane graph, it has only one
bounded face; we label it by the corresponding cell f of K. This plane graph ∆ is called the
diagram of δ. Such diagrams are called atomic. The leftmost and rightmost vertices of ∆ are
denoted by ι(∆) and τ (∆), respectively. The diagram ∆ has two distinguished paths from
ι(∆) to τ (∆) that correspond to the top and bottom paths of δ. Their labels are puq and
pvq, respectively. These are called the top and the bottom paths of ∆ denoted by top(∆)
and bot(∆).

u
p
s

q
s

f

s

s

v
Figure 2.2: An atomic diagram
The diagram corresponding to the trivial 2-path p is just an arc labeled by p; it is called
a trivial diagram and it is denoted by ε(p).
Let δ = δ1 ◦ δ2 ◦ · · · ◦ δn be a 2-path in K, where δ1 , . . . , δn are atomic 2-paths. Let ∆i
be the atomic diagram corresponding to δi . Then the bottom path of ∆i has the same label
as the top path of ∆i+1 (1 ≤ i < n). Hence we can identify the bottom path of ∆i with the
top path of ∆i+1 for all 1 ≤ i < n, and obtain a plane graph ∆, which is called the diagram
of the 2-path δ.
Two diagrams are considered equal if they are isotopic as plane graphs. The isotopy must
take vertices to vertices, edges to edges, it must also preserve labels of edges and cells. Two
2-paths are called isotopic if the corresponding diagrams are equal.
Concatenation of 2-paths corresponds to the concatenation of diagrams: if the bottom
path of ∆1 and the top path of ∆2 have the same labels, we can identify them and obtain a
new diagram ∆1 ◦ ∆2 .
Note that for any atomic 2-path δ = (p, f, q) in K one can naturally define its inverse
2-path δ −1 = (p, f −1 , q). The inverses of all 2-paths and diagrams are defined naturally. The
inverse diagram ∆−1 of ∆ is obtained by taking the mirror image of ∆ with respect to a
horizontal line, and replacing labels of cells by their inverses.
Let us identify in the category Π(K) all isotopic 2-paths and also identify each 2-path of
the form δ ′ δδ −1 δ ′′ with δ ′ δ ′′ . The quotient category is obviously a groupoid (i. e. a category
with invertible morphisms). It is denoted by D(K) and is called the diagram groupoid of

6

K. Two 2-paths are called homotopic if they correspond to the same morphism in D(K).
For each 1-path p of K, the local group of D(K) at p (i.e., the group of homotopy classes
of 2-paths connecting p with itself) is called the diagram group of the directed 2-complex K
with base p and is denoted by DG(K, p).
The following theorem is proved in [12] (see also [14]).
Theorem 2.2. If K is the Dunce hat (see Figure 2.1) and p is the edge of it, then DG(K, p) is
isomorphic to the R. Thompson group F . The generators x0 , x1 of F are depicted in Figure
2.3 (all edges in the diagrams are labeled by x and oriented from left to right).

✉

✉

✉

✉

✉

x0

✉

✉

✉

✉

x1

Figure 2.3: Generators of the R. Thompson group F
Diagrams over K corresponding to homotopic 2-paths are called equivalent . The equivalence relation on the set of diagrams (and the homotopy relation on the set of 2-paths of
K) can be defined very easily as follows. We say that two cells π1 and π2 in a diagram ∆
over K form a dipole if bot(π1 ) coincides with top(π2 ) and the labels of the cells π1 and π2
are mutually inverse. Clearly, if π1 and π2 form a dipole, then one can remove the two cells
from the diagram and identify top(π1 ) with bot(π2 ). The result will be some diagram ∆′ .
As in [12], it is easy to prove that if δ is a 2-path corresponding to ∆, then the diagram ∆′
corresponds to a 2-path δ ′ , which is homotopic to δ. We call a diagram reduced if it does not
contain dipoles. A 2-path δ in K is called reduced if the corresponding diagram is reduced.
Thus one can define morphisms in the diagram groupoid D(K) as reduced diagrams over
K with operation “concatenation + reduction" (that is, the product of two reduced diagrams
∆ and ∆′ is the result of removing all dipoles from ∆ ◦ ∆′ step by step; that process is
confluent and terminating, so the result is uniquely determined [12, Lemma 3.10]).

3
3.1

Savchuk’s problem
An explicit example

Let G be the subgroup of F generated by the elements y0 = x0 x2 and y1 = x1 x2 .
Lemma 3.1. The subgroup G is isomorphic to F and the map xi 7→ yi , i = 0, 1 extends to
an isomorphism F → G.

7

Proof. The generators y0 , y1 satisfy the defining relations of F and do not commute. Since all
proper homomorphic images of F are Abelian [4, Theorem 4.13], the defined homomorphism
from F onto G is an isomorphism.
Proposition 3.2. The group G is composed of all elements w ∈ F whose normal form is of
even length. In particular, G has index 2 inside F .
Proof. Clearly, the normal form of every element in G is of even length. For the other
±1
direction, we will prove in several steps that for every i, j ≥ 0, the element x±1
belongs
i xj
to G.
Step 1: For every n ≥ 0 the element x2n belongs to G.
Indeed,
x20 = (x0 x2 )(x1 x2 )−1 (x0 x2 ) ∈ G.
−1 2
−1
2
Since x0 x2 ∈ G we have x−1
∈ G, hence x3 x−1
2 x0 x0 = x0 x3
0 x0 = x3 x0 = x0 x4 ∈ G.
−1 −1 2
−1
Furthermore x4 x0 x0 = x0 x5 ∈ G.
−1
Since x20 ∈ G and x0 x2 ∈ G, we have that (x20 )x0 x2 = x20 x2 x−1
5 ∈ G, hence x2 x5 ∈ G. Thus,
−1
−1
−1
−1
−1
we have that x0 x5 x5 x2 = x0 x2 ∈ G, so x2 x0 = x0 x1 ∈ G and x0 x1 ∈ G.
2
Since, x0 x2 = x1 x0 ∈ G we have x1 x0 x−2
0 x0 x1 = x1 ∈ G. Thus, for every n ≥ 0, we have
2n

(x21 )x0 = x21+2n ∈ G.
2
Similarly, x0 x−1
3 , x3 ∈ G imply that x0 x3 = x2 x0 ∈ G. Thus

x2 x0 x0−2 x0 x2 = x22 ∈ G.
2
2
Conjugating by x2n
0 shows that for any n ≥ 0, x2+2n ∈ G. So finally we have that xn ∈ G
for all n ≥ 0.
Step 2: The element xi xi+1 belongs to G for all i ≥ 0.
Indeed, we already have that x0 x1 , x1 x2 ∈ G. Now

x2 x3 = (x2 x0 )(x−2
0 )(x0 x3 ) ∈ G.
Conjugating x1 x2 and x2 x3 by x2n
0 for every n, gives that for every n ≥ 0, the element
xn xn+1 ∈ G.
Step 3: The element xi xj belongs to G for every i ≤ j.
The proof is by induction on the difference j − i. For j − i ∈ {0, 1}, the statement follows
from Steps 1 and 2. If j − i > 1, then
−1
xi xj = (xi xi+1 )(xi+1
xj ).
−2
From Step 2, xi xi+1 ∈ G. By the induction hypothesis and Step 1, x−1
i+1 xj = xi+1 (xi+1 xj ) ∈
G.
±1
Step 4: The element x±1
belongs to G for all i, j ≥ 0.
i xj
±1
Indeed, by Step 1, it’s enough to consider words of the form xi xj since x±1
=
i xj
−2ǫ
−2δ
(xi )(xi xj )(xj ) where ǫ, δ ∈ {0, 1}. Thus, Step 3 and the relation xi xj = xj xi+1 for
i > j complete the proof.

8

→
−
From Proposition 3.2, it follows that G = hx0 x2 , x0 x1 , x1 x2 , x2 x3 i = h F , x0 x2 i. We will
→
−
→
−
prove below, that for every w ∈
/ F , we have h F , wi ⊇ G. We start with the following special
case.
→
−
Lemma 3.3. We have: h F , x20 i = G.
→
−
Proof. In the proof of [10, Lemma 4.4], we proved that x20 x1 (x0 x1 x2 )−1 ∈ F . Thus,
→
−
2
−1 2
x0 x2 = x−2
)x0 (x2 x3 ) ∈ h F , x20 i,
0 (x0 x1 (x0 x1 x2 )
→
−
so h F , x20 i contains x0 x2 and thus, contains G. The opposite inclusion follows immediately
from Proposition 3.2.
→
−
Lemma 3.4. Let w = xi xj for j ≥ i ≥ 0 such that j 6= i + 1. Then h F , wi = G.
→
−
Proof. Proposition 3.2 implies that the subgroup h F , wi is contained in G. We prove the
other direction in several steps.
Step 1: The lemma holds for i = 0 and any j 6= 1.
Indeed, for x0 x2 , it follows from the definition of G. The case j = 0 is Lemma 3.3. For
j = 3,
→
−
x20 = (x0 x1 )(x0 x3 )(x2 x3 )−1 ∈ h F , x0 x3 i,
and we are done by the previous case. For j > 3,
→
−
x0 xj−2 = (xj−3 xj−2 )(x0 xj )(xj−1 xj )−1 ∈ h F , x0 xj i,
→
−
→
−
so h F , x0 xj i contains h F , x0 xj−2 i and we are done by induction.
Step 2: The lemma holds for every i, j ≥ 0 such that j > i + 1.
Indeed, for i = 0 this is true by Step 1.
We prove the statement for every even i by induction. For i = 2 and j > 3,
→
−
x0 xj−2 = (x0 x1 )(x2 xj )(x1 x2 )−1 ∈ h F , x2 xj i.
→
−
Thus, x0 xj−2 ∈ h F , x2 xj i and by Step 1 we are done. For any even i > 2 and j > i + 1,
xi−2 xj−2 = (xi xj )(x0 x1 )

−1

→
−
∈ h F , xi xj i.

Thus, by the induction hypothesis for i − 2 we are done.
Now, suppose i, j ≥ 0, i is odd and j > i + 1. Then
→
−
xi+1 xj+2 = (xi xj )−1 (xi xi+1 )(xj+1 xj+2 ) ∈ h F , xi xj i.
Since i + 1 is even, we are done by the previous case.
Step 3: The lemma holds if i = j.
Indeed, for i = j = 0, this is proved in Step 1. For any i > 0,
→ 2
−
xi−1 xi+1 = (xi−1 xi )x−2
i (xi xi+1 ) ∈ h F , xi i
and thus we are done by Step 2.

9

In order to extend Lemma 3.4 to arbitrary words w, we shall need new definitions and a
few auxiliary lemmas.
Let w be a positive normal form. To find the normal form of the product xi w for some
i ≥ 0, one uses the rewriting rule (σ1 ) from Σ, as long as possible. That is, if w = xi1 . . . xin
and i1 < i, then xi moves over xi1 and becomes xi+1 . Then if for the second letter i2 < i + 1,
then xi+1 continues moving and skips over xi2 .
Definition 3.5. Let w = xi1 . . . xin be a positive normal form. We say that the letter xi ,
i ≥ 0 skips over w if xi w = wxi+n in F .
Lemma 3.6. Let w = xi1 . . . xin be a positive normal form and xi , i ≥ 0 a letter. The
following assertions hold.
(1) The letter xi skips over w if and only if for all j = 1, . . . , n we have ij < i + j − 1.
(2) If xi skips over w, then for all k > i the letter xk also skips over w.
Proof. Part (2) follows immediately from Part (1). Indeed, if the inequalities in (1) hold
for i then they hold for all k > i. Let us prove Part (1). If for all j = 1, . . . , n we have
ij < i + j − 1, then xi clearly skips over w. For the other direction, assume that xi w = wxi+n
in F but there exists j = 1, . . . , n such that ij ≥ i + j − 1. Let r be the minimal such j
and w2 the suffix of w starting from xir . Let w1 be the prefix of w of length r − 1, so that
w = w1 w2 . Applying the rewriting rule (σ1 ) to xi w as long as possible, we get the normal
form w1 xi+r−1 w2 .
We claim that there must exist a sequence of applications of rules from Σ (a derivation)
turning wxi+n into w1 xi+r−1 w2 . Indeed, since Σ is confluent and terminating there is a
unique semi-normal form which one can get by applying a sequence of applications of rules
from Σ to the word wxi+n . The resulting semi-normal form is necessarily positive, and as
such, it is a normal form. By uniqueness, it must be equal to w1 xi+r−1 w2 .
Since the word wxi+n is positive, a derivation applied to it must consist entirely of
applications of (σ1 ). Each application of (σ1 ) increases the sum of indices of letters in the
word. In contradiction to the sum of indices of letters in wxi+n = w1 w2 xi+n being strictly
greater than the corresponding sum for the normal form w1 xi+r−1 w2 .
Definition 3.7. Let w = xi1 . . . xin be a positive normal form. We say that w is a block if
xi1 6= xin and for every j = 1, . . . , n we have ij < i1 + j.
Remark 3.8. Let w = xi1 . . . xin be a positive normal form. It follows form Lemma 3.6
that w is a block if and only if
(1) w contains at least two distinct letters; and
(2) xi1 +1 skips over w; equivalently, xk skips over w for all k > i1 .
Since the equalities xi w = wxi+n and xi−1 w = wx−1
i+n in F are equivalent, one could replace
the second condition of Remark 3.8 by the condition: xk±1 w = wx±1
k+n for all k > i1 .
Intuitively, a block is a positive normal form with at least two distinct letters, such that
for all i, if the letter xi skips over the first letter of w, then it skips over the entire w.
Examples of blocks: x0 x1 , x22 x34 x7 , etc.
Lemma 3.9. Let B = xi1 . . . xin be a block. Then

10

(1) If B ′ is a translation of B, i.e., B ′ = xi1 +k . . . xin +k for some k ≥ 0, then B ′ is also a
block.
′
(2) For every j 6= i1 we have xj−1 B = B ′ x−1
r , where B is a translation of B and r ≥ 0.

(3) If B1 = xi2 . . . xin is not a block (in which case, we say that B is a minimal block),
then there exists j ∈ {2, . . . , n} such that ij = i1 + j − 1.
Proof. (1) Follows easily from the definition.
±1
(2) If j > i1 , then by Remark 3.8 xj±1 B = Bxj+n
. Otherwise, the rewriting rule x−1
j xi →
−1
−1
′ −1
xi+1 xj for i > j, shows that xj B = B xj , where B ′ is a translation of B (with the
translation number k = 1).
(3) If xi2 = xi1 +1 , then B satisfies (3) for j = 2. Otherwise, B being a block, implies
that i2 = i1 and thus the length |B| = n > 2. It follows that the first and last letters of
B1 are distinct. Thus, if B1 is not a block, then there exists some j = 2, . . . , n such that
ij ≥ i2 + j − 1 = i1 + j − 1. Now the fact that B is a block guarantees that ij < i1 + j. Thus,
ij = i1 + j − 1.
→ −
−
→ →
−
Lemma 3.10. Every double coset C = F a F of F contains a positive element w such that
for every w′ ∈ C, we have |w| ≤ |w′ |.
Proof. Let w ∈ C be an element of minimal length. If w is positive, we are done. Otherwise,
it is possible to replace w by a positive word, without increasing its length, as follows. Let
x−1
be the last letter in the normal form of w. Replacing w by w(xi xi+1 ) yields a word
i
in C of the same length, with the negative part of the normal form being shorter. Thus,
proceeding by induction, gives the result.
→
−
Lemma 3.11. Let C be a double coset of F and let w be a positive normal form of minimal
length in C. Then w does not contain any block.
Proof. We begin by making a general observation. Let W be a normal form which contains
a block B. Thus, we have W = w1 Bw2 for some possibly empty w1 and w2 . It follows easily
from Lemma 3.9(2) that if one multiplies W from the left by x−1
for some j, then either
j
−1
−1
′ ′ ′
xj W is shorter than W , or xj W is of the form w1 B w2 for some w1′ , B ′ , w2′ such that
|w1′ | = |w1 |, |w2′ | = |w2 | + 1 and such that B ′ is a translation of B.
→ −
−
→
→
−
Now, let C = F w F be a double coset of F and use Lemma 3.10 to assume that w is a
positive normal form of minimal length in C. We claim that w does not contain any block B.
Indeed, assume by contradiction that w is of the form w = w1 Bw2 for some minimal block
B. It is possible to replace w by an element w′ ∈ C of the same length, such that w′ = B ′ w2′
where B ′ is a translation of B, as follows. If w1 is empty, we take w′ = w. Otherwise, let
−1
w1 = xj w1′′ . Then w′′ := (xj xj+1 )−1 w = xj+1
w1′′ Bw2 ∈ C. By the above observation and
′′
′ ′ ′
the minimality of w we have that w = w1 B w2 , when |w1′ | = |w1′′ | = |w1 | − 1, |w2′ | = |w2 | + 1
and B ′ is a translation of B. Proceeding by induction on the length of the prefix w1 , we get
the result.
Thus, we shall assume that w = Bw2 . Let B = xi1 . . . xin . Then
x−1 Bw2 = x−1
t = (xi1 xi1 +1 )−1 w = xi−1
i1 +1 xi2 . . . xin w2 ∈ C.
1 +1 i1

11

By lemma 3.9(3), the minimality of the block B implies that there exists some j = 2, . . . n
such that ij = i1 + j − 1. If j is the minimal such index, then the letter xij cancels in t with
x−1
i1 +1 , in contradiction with w having minimal length in C (by Lemma 3.10).
→
−
→
−
Theorem 3.12. Let w ∈ F \ F . Then the group h F , wi contains G. If the normal form of
→
−
→
−
w is of even length, then h F , wi = G. Otherwise, h F , wi = F .
→
−
Proof. We denote H = h F , wi. It suffices to prove that H ⊇ G, since the rest of the theorem
follows easily from Proposition 3.2. Clearly, we can replace w by any element of the double
→ −
−
→
coset C = F w F . Thus, by Lemmas 3.10 and 3.11, we can assume that w is positive and
does not contain any block. If w = xi for some i ≥ 0, then H = F . If w is of length 2, then
w = xi xj such that j 6= i + 1 (indeed, w is not a block). Thus, we are done by Lemma 3.4.
If |w| > 2, we write w = w′ xnj where j ≥ 0, n ≥ 1 and w′ is either empty, or its last letter is
not xj . If w′ is empty, then
n

xj xj+1+n = (xj xj+1 )xj ∈ H
and we are done by Lemma 3.4. Assume that w′ is not empty and let k = |w′ |. If
(∗) j ≥ k and xj−k w′ = w′ xj ,
then xj−k skips over w′ and so xj−k+1 also skips over w′ (Lemma 3.6(2)). As such we have
that
(xj−k xj−k+1 )w = (w′ xnj )−1 (xj−k xj−k+1 )w′ xnj = (w′ xnj )−1 w′ (xj xj+1 )xnj
= (w′ xnj )−1 w′ xn+1
xj+n+1 = xj xj+n+1 ∈ H
j
and we are done by Lemma 3.4.
Thus it suffices to prove that (∗) must hold. Assume by contradiction that it doesn’t.
Note, that in that case, for every i ≥ 0, if xi skips over w′ then xi skips over the entire
w. That is, if for some i, we have xi w′ = w′ xi+k , then xi w′ xnj = w′ xnj xi+k+n . Indeed,
xi w′ = w′ xi+k implies that xi w′ xnj = (w′ xi+k )xnj . If i + k > j, the result is clear. If
d = j − (i + k) ≥ 0, then since xi skips over w′ and i + d ≥ i, by Lemma 3.6(2) we have
xi+d w′ = w′ xj , in contradiction to (∗) not holding (note that i + d = j − k ≥ 0).
Let i be the minimal index such that xi w′ = w′ xi+k . Let w′ = xi1 . . . xik . We claim that
there is some index ir such that ir = i + r − 2. Indeed, xi skipping over w′ , implies that for
all r = 1, . . . k, we have ir < i + r − 1 (Lemma 3.6(1)). If for all r, we have that ir < i + r − 2,
then one can replace xi with xi−1 , by contradiction to the minimality of i (note that i ≥ 1).
Let B be the suffix of w starting from xr for r as above. That is, B = xir . . . xik xnj Let
A be the prefix of w of length r − 1 so that w = AB. Then from the above, xi w = xi AB =
Axi+r−1 B = ABxi+k+n . In particular, xi+r−1 skips over B. Since the first letter of B is
xir = xi+r−2 and the first and last letters of B are distinct, B is a block inside w (by Remark
3.8), in contradiction to the choice of w.

12

Remark 3.13. There are several descriptions of the subgroup G of F . Vaughan Jones in
→
−
[15] mentioned that F is contained in the subgroup G′ of F consisting of functions whose
slope at 1 is of the form 22n , n ∈ Z. In [2] that subgroup is denoted by K(1,2) (it is proved
there that this subgroup is isomorphic to F ). Clearly Theorem 3.12 shows that G′ = G.
Saul Schleimer asked (private communication) whether there are subgroups strictly be→
−
tween F and G. Theorem 3.12 answers that question:
→
−
Corollary 3.14. F is a maximal subgroup of infinite index inside G.
The next theorem gives a counterexample to Savchuk’s problem [19, Problem 1.5].
Theorem 3.15. Let Ψ : F → G be the isomorphism taking xi to yi , i = 0, 1 (see Lemma
→
−
3.1). Then H = Ψ−1 ( F ) is a maximal subgroup of infinite index in F and H does not
stabilize any x ∈ (0, 1).
Proof. It follows from Corollary 3.14 that H is a maximal subgroup of F of infinite index.
Recall that y0 = x0 x2 and y1 = x1 x2 . One can check that
x0 x1 = y0 y1 y0−1 y1−1 y0 .
Thus,

Ψ−1 (x0 x1 ) = x0 x1 x0−1 x1−1 x0 = x0 x1 x−1
2 ∈ H.

On binary fractions, g = x0 x1 x2−1 acts as follows:


.0α
if





if
.10α
g(t) = .1100α if



.1101α if



.111α
if

t = .00α
t = .010α
t = .011α
t = .10α
t = .11α

It is clear from the binary description, that x0 x1 x−1
2 does not stabilize any x ∈ (0, 1). Indeed,
Ψ changes a prefix of every binary fraction other than 0 = .0N and 1 = .1N .
→
−
Since F is isomorphic to the Thompson-Brown group F3 [10], we have
Corollary 3.16. The R. Thompson group F has a maximal subgroup which is isomorphic
to F3 .

3.2

Implicit examples

Here is another idea how to prove existence of maximal subgroups of infinite index in F
which do not fix a point in (0, 1). We will show below that this idea actually works.
Let g, h be two elements of F such that M0 = hg, hi does not fix a point in (0, 1).
Suppose that the images of g, h in the free Abelian group F/[F, F ] generate the whole group
F/[F, F ]. Then M0 cannot be contained in any non-trivial subgroup of finite index in F
because subgroups of finite index all contain the derived subgroup [F, F ]. If M0 6= F ,

13

then M0 is contained in a maximal subgroup M of F (by Zorn’s lemma since F is finitely
generated), and M is of infinite index in F and does not fix any point in (0, 1). The problem
is how to prove that M0 6= F . As we have mentioned in the introduction there is no known
algorithm to decide when two (or any finite number > 1) of elements of F do not generate
the whole F . But here is a partial algorithm which works quite often even when the images
of these elements in F/[F, F ] generate the whole F/[F, F ].
Recall the procedure (first discovered by Stallings [21]) of checking if an element g of a
free group Fn belongs to the subgroup H generated by elements h1 , ..., hk . Take paths labeled
by h1 , ..., hk . Identify the initial and terminal vertices of these paths to obtain a bouquet
of circles K ′ with a distinguished vertex o. Then fold edges as follows: if there exists a
vertex with two outgoing edges of the same label, we identify the edges. As a result of all
the foldings (and removing the hanging trees), we obtain the Stallings core of the subgroup
H which is a finite automaton A(H) with o as its input/output vertex. Then g ∈ H if and
only if A(H) accepts the reduced form of g. It is well known that the Stallings core does not
depend on the generating set of the subgroup H.
In the case of diagram groups we will use the same strategy but instead of automata we
will have directed complexes and instead of words - diagrams. We give the construction for
general diagram groups and then apply it to Thompson group F .
Definition 3.17. Let K = hEK | FK i be a directed 2-complex. A 2-automaton over K is a
directed 2-complex L = hEL |FL i with two distinguished 1-paths pL and qL (the input and
output 1-paths), together with a map φ from L to K which takes vertices to vertices, edges
to edges and cells to cells, which is a homomorphism of directed graphs and commutes with
the maps top, bot and −1 . We shall call φ an immersion.
For example, every diagram ∆ over K is a 2-automaton with a natural immersion φ∆
and the distinguished paths top(∆) and bot(∆).
Definition 3.18. Let L, L′ be two 2-automata over K. A map ψ from L′ to L which takes
vertices to vertices, edges to edges and cells to cells, which is a homomorphism of directed
graphs and commutes with the maps top, bot,−1 and the immersions is called a morphism
from L′ to L provided ψ(pL′ ) = pL , ψ(qL′ ) = qL .
Definition 3.19. We say that a 2-automaton L over K accepts a diagram ∆ over K if there
is a morphism ψ from the 2-automaton ∆ to the 2-automaton L.
Let ∆ be a diagram over K with top and bottom paths having the same labels (i.e., a
spherical diagram in terminology of [12]). Let us identify top(∆) with bot(∆). We can
view the result as a 2-automaton L′ over K drawn on a sphere with the distinguished paths
p = q = top(∆) = bot(∆). The 2-automaton L′ clearly accepts ∆, and any diagram of the
form ∆ ◦ ∆ ◦ ....
Suppose we want a 2-automaton that accepts all reduced diagrams that are equal to ∆n
for some n. Then we should do an analog of the Stallings foldings. Namely, let L′ be the
2-automaton as above. Now every time we see two cells that have the same image under
the immersion of L′ and share the top (resp. bottom) 1-path, then we identify their bottom
(resp. top) 1-paths and identify the cells too. This operation is called folding of cells (see
[14, Remark 8.8]). Clearly the result is a directed 2-complex and the immersion of L′ induces
an immersion of the new directed 2-complex. Thus we again get a 2-automaton. Let L be

14

the 2-automaton obtained after all possible foldings in L′ . Then ∆ ◦ ∆ ◦ ... is still accepted
by L. But also the reduced diagrams ∆n are accepted by L for every n.
More generally, let ∆1 , ..., ∆n be (reduced) diagrams from the diagram group DG(K, u)
i.e., diagrams over K with the same label u of their top and bottom paths. Then we can
identify all top(∆i ) with all bot(∆i ) and obtain a 2-automaton L′ over K with the distinguished paths p = q = top(∆i ) = bot(∆i ). We can view L′ as a “bouquet of spheres”. That
automaton accepts any concatenation of diagrams ∆i and their inverses. Let us do all the
foldings in L′ . We obtain a 2-automaton L.
The proof of the following lemma is quite similar to the proof of [12, Lemma 3.10] and is
left to the reader as an exercise.
Lemma 3.20. The rewriting system where objects are finite 2-automata over a directed
2-complex K and moves are foldings, is confluent and terminating.
By Lemma 3.20 and [17, Lemma 1.7.8] the 2-automaton L is uniquely determined by L′ .
Remark 3.21. Since folding is a “local” operation, Lemma 3.20 implies that even if we start
with an infinite directed 2-automaton over a directed 2-complex K, the end result of all the
foldings does not depend on the order in which the foldings are performed.
Lemma 3.22. The 2-automaton L accepts all reduced diagrams from the subgroup of the
diagram group DG(K, u) generated by ∆1 , . . . , ∆n .
Proof. Suppose that cells π1 and π2−1 form a dipole in a diagram ∆ accepted by L, p1 =
top(π1 ), q1 = bot(π1 ) = bot(π2 ) , p2 = top(π2 ), and ∆′ is obtained from ∆ by removing
this dipole. We claim that then ∆′ is also accepted by L. This will immediately imply the
statement of the lemma. To prove the claim let φ be a morphism from ∆ to L which takes
both top(∆) and bot(∆) to the input/output 1-path of L. Then the images of the cells
π1 , π2 are cells in L that have the same images under the immersion of L into K and share
a common top or bottom path. Hence these two cells must be the same in L. Therefore
φ(p1 ) = φ(p2 ). Hence φ induces a morphism φ̄ from ∆′ into L. The morphism φ̄ coincides
with φ on top(∆) and bot(∆). Hence L accepts ∆′ .
Remark 3.23. Suppose ∆′1 , ..., ∆′m is another generating set of H = h∆1 , ..., ∆n i and all
diagrams ∆′i are reduced. Let L′ be the 2-automaton corresponding to ∆′1 , ..., ∆′m . It is easy
to see that since ∆i is accepted by L′ and ∆′j is accepted by L for every i, j, we have L′ = L.
Thus L does not depend on the choice of generating set of the subgroup H (we can even
use any infinite generating set, the resulting 2-automaton will be the same). Thus L can be
called the Stallings 2-core of the subgroup H.
Remark 3.24. In general the 2-automaton L constructed above can also accept diagrams
not from the subgroup H = h∆1 , . . . , ∆n i but in many cases it accepts only the diagrams
from H. In that case we call H closed (see Section 5).
Lemma 3.22 can be used to construct many proper subgroups of F that 1) do not fix
a point in (0, 1) and 2) whose image in F/[F, F ] is the whole F/[F, F ] (and hence prove
existence of examples solving Savchuk’s problem). We illustrate this by the following
Lemma 3.25. The subgroup H = hx0 , x1 x2 x−1
1 i is a proper subgroup of F and satisfies the
two properties 1) and 2) above.

15

Proof. It is easy to check that x0 does not fix a point in (0, 1) which implies 1). The image
of x1 x2 x−1
1 in F/[F, F ] coincides with the image of x1 . Hence H[F, F ] = F which implies 2).
It remains to prove that H is a proper subgroup of F . We shall prove that x1 6∈ H.
Let us denote the positive cell of the Dunce hat hx | x → x2 i by π. The diagrams for x0
and x1 x2 x−1
1 viewed as 2-automata are in Figure 3.4 below (the immersion to the Dunce hat
maps all positive cells to π, and all edges to the only edge of the Dunce hat).

e8
γ5
e1
γ1

e2
r

e3

γ2
r

e9

e4

γ7 e13
r

e5

er 10 r e12 r e14
γ10
e17

r

γ3

γ4

γ6

γ8

e11

r e15

r

e16

r

γ9
e18
γ11
e19

e6
e7

γ12

e20

Figure 3.4: The diagrams for x0 and x1 x2 x−1
1
Together the two diagrams have 20 edges (labeled by e1 , . . . , e20 ) and 12 cells (labeled
γ1 , . . . , γ12 ). To construct the 2-automaton L for these two diagrams, we first need to identify
the top and the bottom paths of both diagrams. So we set e1 = e7 = e8 = e20 . Now the
positive cells γ1 , γ4 , γ5 , γ12 need to be folded because these cells share the top 1-path e1 . So
we need to identify γ1 = γ4 = γ5 = γ12 and the edges e2 = e3 = e10 and e5 = e6 = e9 = e19 .
Now the cells γ3 , γ6 , γ11 have common top edge e5 . So we need to fold these three cells.
Thus γ3 = γ6 = γ11 , e4 = e11 = e17 , e5 = e16 = e18 . Then the cells γ7 and γ10 share the
top edge e4 . So we set γ7 = γ10 , e13 = e14 . Furthermore γ9 and γ3 now share the top edge
e5 . So we need to set e4 = e15 . No more foldings are needed, and the 2-automaton L is
presented in Figure 3.5 (there the cells and edges are supposed to be identified according to
their labels: all e1 edges are the same, all γ7 -cells are the same, etc.).
Now suppose that x1 is accepted by L. The diagram ∆ for x1 with labels of edges and
cells is in Figure 3.6. We should have a morphism ψ from ∆ to L sending f1 and f10 to e1 .
Then ψ(δ1 ) = ψ(δ6 ) = γ1 since L has only one cell with top edge e1 . This forces ψ(f2 ) = e2 ,
ψ(f3 ) = ψ(f9 ) = e5 . Since L has only one positive cell with top edge e5 , we should have
ψ(δ2 ) = γ3 . That means ψ(f4 ) = e4 , ψ(f5 ) = e5 . Again L has only one positive cell with
top edge e4 . Therefore ψ(δ3 ) = γ7 , hence ψ(f6 ) = e12 , ψ(f7 ) = e13 . Now ψ must map the
positive cell δ4 to a cell with bottom edges ψ(f7 ) = e13 and ψ(f5 ) = e5 . But L does not

16

e1
γ1
e1

e2

γ3
γ7 e13 e4

γ1

e2
r

e5

γ2

r e5

r e4

γ8 e
re2 re12 r e13
r 4

r

γ7
e4

γ3

γ1

r e5

r

γ3
e5
γ3
e5

e5
e1

γ1

e1

Figure 3.5: The 2-core for hx0 , x1 x2 x−1
1 i

f1
f3
δ1

δ2
f6

s f2 s

f4

δ3

f7

s

δ4

δ5
δ6

s f5

s

f8

f9
f10

Figure 3.6: The 2-automaton for x1
have such a cell, a contradiction. This contradiction shows, by Lemma 3.22, that H does
not contain x1 and hence is a proper subgroup of F .
Remark 3.26. One more way to show that H = hx0 , x1 x2 x−1
1 i is a proper subgroup of F is
to prove that H does not act transitively on the set of finite binary fractions from [0, 1]. That

17

can be also done using the 2-core of H (see [8]). One can ask whether a maximal subgroup
of F of infinite index can act transitively. The answer is “yes", see [8]. That gives a strong
negative answer to Savchuk’s question.

4

F is quasi-residually finite

Let G be a finitely generated group. Recall that a subgroup H < G is of quasi-finite index
if the interval [H, G] in the lattice of subgroups of G is finite, and G is quasi-residually finite
if it has a decreasing sequence of finitely generated subgroups G > H1 > H2 ... of quasi-finite
index in G with ∩Hi = {1}. In this section we prove
Theorem 4.1. The R. Thompson group F is quasi-residually finite.
Proof. Let U be a finite set of finite binary fractions in (0, 1) and HU be the stabilizer of this
set in F . It is clear that if U1 ⊂ U2 ⊂ U3 ... and ∪Ui is the set of all finite binary fractions in
(0, 1), then F > HU1 > HU2 ... and ∩HUi = {1}. It is also clear that each HU is isomorphic
to the direct product of |U | + 1 copies of F , hence is finitely generated. It remains to show
that HU is of quasi-finite index in F . In fact we will prove more:
Claim 1. The lattice L1 of subgroups of F containing HU is anti-isomorphic to the
Boolean lattice L2 of subsets of U , and the map V 7→ HV for V ⊆ U is an anti-isomorphism
from L2 to L1 .
We shall prove
Claim 2. Let g ∈ F \ HU . Then the subgroup hHU , gi contains one of the subgroups
HV where V is a proper subset of U .
Note that Claim 2 implies Claim 1. Indeed, every subgroup G of F containing HU is
generated by HU and a sequence of elements g1 , g2 , . . . such that g1 6∈ HU and for every i,
gi+1 6∈ hHU , g1 , . . . , gi i. By Claim 2, G is equal to hHV , g1 , g2 , . . .i where V is a proper subset
of U and by induction on the size |U | the subgroup G is equal to HW for some W ⊆ U . Now
it is obvious that if V 6= W , then HV 6= HW and that V ⊆ W if and only if HW ≤ HV .
Hence the map V 7→ HV from the lattice of subsets of U to the lattice of subgroups of F
containing HU is a lattice anti-isomorphism.
Let us prove Claim 2. Let u1 , ..., un be the elements of U in increasing order. Let us
denote u0 = 0. Since g 6∈ HU , it does not fix one of the numbers uj , j ≥ 1. Let ui be
the smallest number in U not fixed by g. Let V = U \ {ui }. It is enough to show that
HV ⊆ hHU , gi. To do so, we will prove first that hHU , gi contains some element of HV \ HU .
Let Ug ⊆ U be the subset of elements of U which are not fixed by g. In particular ui ∈ Ug .
Since for any u ∈ Ug , the elements in the orbit g n (u) for n ∈ N are all distinct, there is a
power k ∈ N for which g k (Ug ) ∩ U = ∅. By [6, Lemma 4.2] there is a function h ∈ F which
fixes all the points in U ∪ g k (Ug \ {ui }) and does not fix g k (ui ). In particular h ∈ HU . Let
g1 = g k hg −k . Then g1 fixes all u ∈ U \ {ui } (recall that composition in F is from left to
right). By construction, g1 does not fix ui . Therefore g1 ∈ hHU , gi ∩ (HV \ HU ).
To finish the proof, we will show that hHU , g1 i = HV . Since HU and g1 are contained
in HV one inclusion is trivial. For the other direction, let f ∈ HV . We can assume that f
does not belong to HU , so in particular f (ui ) 6= ui . Since f is increasing, either f (ui ) < ui
or f −1 (ui ) < ui . Without loss of generality let f (ui ) < ui . Similarly we can assume that

18

g1 (ui ) < ui . Lemma 4.2 of [6] implies that the group HU acts transitively on the set of finite
binary fractions from the interval (ui−1 , ui ). Therefore there exists an element h1 ∈ HU such
that f h1 (ui ) = g1 (ui ). Therefore p = f h1 g1−1 fixes ui . Since p also belongs to HV , we get
that p ∈ HU . Therefore f = pg1 h1−1 ∈ hHU , g1 i.
Remark 4.2. It is easy to see that the proof of Theorem 4.1 can be adapted for every group
G of order preserving maps Q → Q where Q is a dense subset of R provided G is locally
transitive meaning that for every open interval (a, b) in R the subgroup of all elements of G
fixing Q \ (a, b) pointwise acts transitively on (a, b) ∩ Q. Thus every such group G is quasiresidually finite. That includes the group Homeo+ (R) of all increasing homeomorphisms of
R.
Certainly not every finitely generated group is quasi-residually finite. The simplest (to
explain) example is a torsion Tarski monster where every proper subgroup is finite and cyclic
[16]. More generally every infinite group with the descending chain condition on subgroups
is not quasi-residually finite.
For every non-residually finite group, it is interesting to know if it is quasi-residually
finite. The next problem lists some of the most famous non-residually finite groups.
Problem 4.3. Is any of these groups quasi-residually finite:
• The free Burnside group Bm,n for all sufficiently large exponents n and rank m ≥ 2,
• The Higman group H = ha, b, c, d | b−1 ab = a2 , c−1 bc = b2 , d−1 cd = c2 , a−1 da = d2 i,
• The Baumslag-Solitar group BS(2, 3) = ha, b | a3 = b−1 a2 bi,
• The Baumslag group B = ha, b, c | c−1 ac = b, b−1 ab = a2 i,
• The R. Thompson groups T and V ?
Remark 4.4. Recently Jordan Nikkel showed (unpublished) that the R. Thompson groups
T and V are quasi-residually finite.
→
−
By Theorems 3.12 and Claim 1 in the proof of Theorem 4.1, if H is Jones’ subgroup F or
one of Savchuk’s subgroups HU for a finite set U of binary fractions, then the interval [H, F ]
in the lattice of subgroups of F is modular (in fact, distributive): the three-element chain in
the first case and a Boolean lattice in the second case. If H is a finite index subgroup of F ,
then it is normal in F and F/H is Abelian because H contains the derived subgroup of F .
Hence the lattice [H, F ] is also modular.
Problem 4.5. Is there a subgroup H of F of quasi-finite index such that the interval [H, F ]
is not a modular lattice?
Finally notice that there are finitely generated and infinitely generated maximal subgroups of F (say, H{α} is finitely generated if α is a finite binary fraction, and infinitely
generated if α is irrational [19]). Since F is finitely generated, every proper subgroup of F
is contained in a maximal subgroup of F .
Problem 4.6. Is it true that every finitely generated proper subgroup of F is contained in
a finitely generated maximal subgroup of F ?
We cannot finish the paper about F without asking an amenability question.

19

→
−
Problem 4.7. Is the Schreier graph of Jones’ subgroup F amenable?
Note that if the answer is “no”, then F itself is not amenable.

5 Some further results and open problems about subgroups of F
Since the paper was submitted several new results about subgroups of the Thompson group
F have been obtained by the authors. Here we list some of these results.

5.1

Stabilizers of finite sets of points

As noted above, if a finite set U ⊂ (0, 1) consists of finite binary fractions, then the subgroup
HU , the stabilizer of U in F , is isomorphic to the direct product of |U | + 1 copies of F .
It is mentioned in [18], that if U = {α} where α is irrational, then HU is not finitely
generated. In [11] we clarify the algebraic structure of FU for arbitrary finite U . For every
such U = {α1 , α2 , ..., αn } where α1 < α2 < ... < αn we define its type τ (U ), as the word of
length n in the alphabet {1, 2, 3} as follows: for every i, the ith letter in τ (U ) is 1 if αi is a
finite binary fraction, 2 if αi is a rational but not a finite binary fraction, 3 if αi is irrational.
Theorem 5.1. (1) If τ (U ) = τ (V ) then HU and HV are isomorphic.
(2) HU is finitely generated if and only if U does not contain irrational numbers. In that
case the minimal number of generators of HU is 2k + m + 2 where k is the number of 1s in
τ (U ) and m is the number of 2s in τ (U ).
In each case, we completely describe HU as an iterated HNN extension of a direct product
of several copies of F , several copies of the derived subgroup [F, F ] of F and several copies of
the normal subgroup L of F of all functions with slope 1 at 0. In particular, if α is a rational
number which is not a finite binary fraction, then H{α} is isomorphic to the ascending HNN
extension of F × F corresponding to the endomorphism φ defined as follows (it does not
depend on α by Theorem 5.1):
φ:

(x0 , 1) → (x0 ⊕ 1, 1),
(x1 , 1) → (x1 ⊕ 1, 1),
(1, x0 ) → (1, x1 ),
(1, x1 ) → (1, x2 )

where 1 is the identity function and the sum f ⊕ g of two functions f, g in F is defined as
the function from F which on [0, 21 ] coincides with the shrunk by the factor of 2 copy of f ,
i.e., 12 f (2t), and on [ 21 , 1] coincides with the shrunk by the factor of 2 copy of g shifted by 12 ,
i.e., 21 g(2t − 1) + 12 .

5.2

Amenable maximal subgroups

Problem 5.2. Does F contain an elementary amenable maximal subgroup?
The problem is open, but the following result was recently obtained by the first author.

20

Theorem 5.3 (Golan [8]). There is a sequence of finitely generated subgroups B < K < F
such that B is elementary amenable and maximal in K, K is normal in F and F/K is
infinite cyclic.
Note that the subgroup B is isomorphic to one of Brin’s subgroups of F from [3]. Also
note that K contains the derived subgroup of F , hence it contains a copy of F . Thus K is
amenable if and only if F is.

5.3

The closure operation on the lattice of subgroups of F

The notion of 2-core of a subgroup of F gives rise to the following natural notion.
Definition 5.4. The closure Cl(H) of a subgroup H of F is the subgroup of F consisting
of all diagrams that are accepted by the 2-core L(H) of H.
It is clear that all usual conditions of the closure operation are satisfied, that is, H ≤
Cl(H) (Lemma 3.22), Cl(Cl(H)) = Cl(H) and if H1 ≤ H2 , then Cl(H1 ) ≤ Cl(H2 ). If H
is finitely generated, then the 2-automaton L(H) is finite, and so the membership problem
in Cl(H) is decidable, hence the distortion function of Cl(H) is recursive. Recall that it is
still not known [13] whether the distortion function of every finitely generated subgroup of
F is recursive, and in fact no finitely generated subgroup with super-polynomial distortion
function is known.
Problem 5.5. Is every closed subgroup of F quasi-isometrically embedded, i.e., its distortion
function is linear?
The closure operation preserves several properties of a subgroup. For example, it is not
hard to see (and follows from Theorem 5.9 below) that the closure of every non-trivial cyclic
subgroup generated by a function that has no fixed points except 0 and 1 is cyclic. Moreover,
using the description of solvable subgroups from [2] the first author proved
Theorem 5.6 (Golan, [8]). If H is finitely generated and solvable of class k, then Cl(H) is
also finitely generated and solvable of class k.
Still the following problem is open.
Problem 5.7. Is it true that if H is finitely generated, then Cl(H) is finitely generated?
Note that in Theorem 5.6, one cannot replace “solvable” by “elementary amenable”. Indeed, if B is the group from Theorem 5.3, then there is a copy of B in F whose closure
contains a subgroup isomorphic to F .
Definition 5.8. Let h ∈ F . Then components of h are all elements of F that coincide with
h on a closed interval [a, b] with finite binary a, b and are identity outside [a, b] (in that case
h necessarily fixes a and b).
For example, if h = f ⊕ g, then f ⊕ 1 and 1 ⊕ g are components of h. Clearly components
of h pairwise commute and h is a product of its “minimal” components, i.e., components
with connected support (the set of all non-fixed points of [0, 1]).

21

It is easy to prove that if h is in a subgroup H of F , then the components of h are in the
closure of H. The converse was conjectured by Guba and the second author about 20 years
ago, and was recently proved by the first author of this paper.
Theorem 5.9 (Golan, [8]). The subgroup Cl(H) coincides with the subgroup of H generated
by all components of elements of H. Moreover Cl(H) coincides with the subgroup of F
consisting of piece-wise linear functions f (with finitely many pieces) such that every linear
piece of f is a restriction of a linear piece of some function from H.
This theorem immediately implies the following
Corollary 5.10. Let X be any subset of the unit interval [0, 1]. Then the stabilizer of X
in F is a closed subgroup of F , that is Cl(H) = H. Thus all Savchuk’s subgroups HU and
→
−
Jones’ subgroup F are closed.
Problem 5.11. Is it true that every maximal subgroup of F of infinite index is closed?
Every closed subgroup H of F is a diagram group. Indeed, it is a diagram group of
the 2-automaton L(H) viewed as a directed 2-complex. Moreover closed subgroups can be
characterized “abstractly” as precisely the diagram groups DG(K, a) where a is an edge and
K is a directed 2-complex where every positive cell has the form t → lr (t, l, r are edges) and
no foldings can be performed (which means that for every t there is only one cell of the form
t → lr, and for every pair (l, r) there is only one cell of the form t → lr). Indeed, if K is
such a directed 2-complex, then mapping every edge to x and every cell t → lr to x → x2
gives a morphism from K to the Dunce hat (Figure 2.1) and it is not difficult to prove that
the corresponding homomorphism from DG(K, a) to F is injective and the image is closed
(that follows from Theorem 5.9).
Problem 5.12. Suppose that Cl(H) = F and H is not inside a proper subgroup of finite
index of F . Is it true that H = F ?
The positive answer to Problem 5.12 would give an easy algorithm to decide whether a
finite set of elements of F generates the whole F , i.e., solves the generation problem for F .
Indeed first we decide whether these elements are not inside any finite index subgroup of F .
That can be done by looking at their images in F/[F, F ]. Then construct the 2-core L(H)
and check if it coincides with the 2-core L(F ).
Note that in [9], the first author proved that the generation problem for F is in fact
decidable (the notion of 2-core is also used in the algorithm from [9]).
Note also that a positive answer to Problem 5.12 would imply a positive answer to
Problem 5.11. Indeed, if H ≤ F is a non-closed maximal subgroup of infinite index in F
then Cl(H) = F and H is not contained in any proper finite index subgroup of F .

References
[1] Collin Bleak, A geometric classification of some solvable groups of homeomorphisms.
J. Lond. Math. Soc. (2) 78 (2008), no. 2, 352–372.

22

[2] Collin Bleak, Bronlyn Wassink, Finite index subgroups of R. Thompson’s group F .
arXiv:0711.1014 .
[3] Matthew G. Brin, Elementary amenable subgroups of R. Thompson’s group F . Internat. J. Algebra Comput. 15 (2005), no. 4, 619–642.
[4] Kenneth S. Brown, Finiteness properties of groups. Proceedings of the Northwestern
conference on cohomology of groups (Evanston, Ill., 1985). J. Pure Appl. Algebra 44
(1987), no. 1–3, 45–75.
[5] José Burillo, Quasi-isometrically embedded subgroups of Thompson’s group F . J. Algebra 212 (1999), no. 1, 65–78.
[6] J. Cannon, W. Floyd, and W. Parry, Introductory notes on Richard Thompson’s
groups. L’Enseignement Mathematique, 42 (1996), 215–256.
[7] Sean Cleary, Murray Elder, Andrew Rechnitzer, Jennifer Taback, Random subgroups
of Thompson’s group F . Groups Geom. Dyn. 4 (2010), no. 1, 91–126.
[8] Gili Golan, Closed subgroups of the R. Thompson group F . preprint, 2016.
[9] Gili Golan. The generation problem for the R. Thompson group F . preprint, 2016.
[10] Gili Golan, Mark Sapir, On Jones’ subgroup of R. Thompson group F , accepted
in Advanced Studies in Pure Mathematics. Proceedings of the MSJ-SI conference
"Hyperbolic Geometry and Geometric Group Theory", arXiv: 1501.00724.
[11] Gili Golan, Mark Sapir, Stabilizers of finite sets of points in R. Thompson’s group
F . preprint, 2016.
[12] Victor Guba, Mark Sapir. Diagram groups. Memoirs of the Amer. Math. Soc. 130,
no. 620 (1997), 1–117.
[13] Victor Guba, Mark Sapir, On subgroups of the R. Thompson group F and other
diagram groups. (Russian) Mat. Sb. 190 (1999), no. 8, 3–60; translation in Sb. Math.
190 (1999), no. 7-8, 1077–1130.
[14] Victor Guba, Mark Sapir, Diagram groups and directed 2-complexes: homotopy and
homology. Journal of Pure Appl. Algebra 205 (2006), no. 1, 1–47.
[15] Vaughan Jones, Some unitary representations of Thompson’s groups F and T , arXiv:
1412.7740.
[16] A.Yu. Olshanskii, An infinite group with subgroups of prime orders. Izv. Akad. Nauk
SSSR Ser. Mat. 44 (1980), no. 2, 309–321.
[17] Mark Sapir, Combinatorial algebra: syntax and semantics, Springer Monographs in
Mathematics, 2014.
[18] Dmytro Savchuk, Some graphs related to Thompson’s group F . Combinatorial and
geometric group theory, 279—296, Trends Math., Birkhäuser/Springer Basel AG,
Basel, 2010.
[19] Dmytro Savchuk, Schreier graphs of actions of Thompson’s group F on the unit
interval and on the Cantor set. Geom. Dedicata 175 (2015), 355–372.
[20] Vladimir Shpilrain and Alexander Ushakov, Thompson’s group and public key cryptography Lecture Notes Comp. Sc. 3531 (2005), 151–164.

23

[21] John R. Stallings, Foldings of G-trees. Arboreal group theory (Berkeley, CA, 1988),
355–368, Math. Sci. Res. Inst. Publ., 19, Springer, New York, 1991.
[22] Yan Wu, Xiaoman Chen, Distortion of wreath products in Thompson’s group F. Chin.
Ann. Math. Ser. B 35 (2014), no. 5, 801–816.

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