THOMPSONS GROUP FIS ALMOST3 2-GENERATED GILI GOLAN POLAK

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THOMPSON’S GROUP FIS ALMOST 3

2-GENERATED

GILI GOLAN POLAK

Abstract. Recall that a group Gis said to be 3

2-generated if every non-trivial element of

Gbelongs to a generating pair of G. Thompson’s group Vwas proved to be 3

2-generated by

Donoven and Harper in 2019. It was the ﬁrst example of an inﬁnite ﬁnitely presented non-

cyclic 3

2-generated group. Recently, Bleak, Harper and Skipper proved that Thompson’s

group Tis also 3

2-generated. In this paper, we prove that Thompson’s group Fis “almost”

2-generated in the sense that every element of Fwhose image in the abelianization forms

part of a generating pair of Z2is part of a generating pair of F. We also prove that

for every non-trivial element f∈Fthere is an element g∈Fsuch that the subgroup

hf, gicontains the derived subgroup of F. Moreover, if fdoes not belong to the derived

subgroup of F, then there is an element g∈Fsuch that hf, gihas ﬁnite index in F.

1. Introduction

A group Gis said to be 3

2-generated if every non-trivial element of Gis part of a generating

pair of G. In 2000, settling a problem of Steinberg from 1962 [21], Guralnick and Kantor

proved that all ﬁnite simple groups are 3

2-generated [18]. In 2008, Breuer, Guralnick and

Kantor [4] observed that if a group Gis 3

2-generated then every proper quotient of it must

be cyclic. They conjectured that for ﬁnite groups this is also a suﬃcient condition. The

conjecture was proved in 2021 by Burness, Guralnick and Harper [7].

Note that for an inﬁnite group Gevery proper quotient being cyclic is not a suﬃcient

condition for the group being 3

2-generated. Indeed, the inﬁnite alternating group A∞is

simple but not ﬁnitely generated. Moreover, there are ﬁnitely generated inﬁnite simple

groups which are not 2-generated (see [17]) and in particular, are not 3

2-generated. Recently,

Cox [10] constructed an example of an inﬁnite 2-generated group Gsuch that every proper

quotient of Gis cyclic and yet Gis not 3

2-generated.

Obvious examples of inﬁnite 3

2-generated groups are the Tarski monsters constructed

by Olshanskii [20]. Recall that Tarski monsters are inﬁnite ﬁnitely generated non-cyclic

groups where every proper subgroup is cyclic1. In particular, if Tis a Tarski monster, then

Tis generated by any pair of non-commuting elements of T. Since the center of Tis trivial,

every non-trivial element of Tis part of a generating pair of T.

In 2019, Donoven and Harper gave the ﬁrst examples of inﬁnite non-cyclic 3

2-generated

groups, other than Tarski monsters. Indeed, they proved that Thompson’s group Vis

The research was supported by ISF grant 2322/19.

1There are two types of Tarski monsters. One where every proper subgroup is inﬁnite cyclic and one

where every proper subgroup is cyclic of order pfor some ﬁxed prime p.

arXiv:2210.03564v1 [math.GR] 7 Oct 2022

2 GILI GOLAN POLAK

2-generated. More generally, they proved that all Higman–Thompson groups Vn(see [19])

and all Brin–Thompson groups nV (see [5]) are 3

2-generated. In 2020, Cox constructed

two more examples of inﬁnite 3

2-generated groups with some special properties (see [10]).

Quite recently (in 2022), Bleak, Harper and Skipper proved that Thompson’s group Tis

also 3

2-generated [2].

In this paper, we study Thompson’s group F. Recall that Thompson’s group Fis

the group of all piecewise-linear homeomorphisms of the interval [0,1] with ﬁnitely many

breakpoints where all breakpoints are dyadic fractions (i.e., numbers from Z[1

2]∩(0,1))

and all slopes are integer powers of 2. Thompson’s group Fis 2-generated. The derived

subgroup of Fis inﬁnite and simple and can be characterized as the subgroup of Fof all

functions fwith slope 1 both at 0+and at 1−(see [9]). The abelianization F/[F, F ] is

isomorphic to Z2. The standard abelianization map πab :F→Z2maps every function f∈

Fto (log2(f0(0+)),log2(f0(1−))) (see [9]). Since the abelianization of Fis Z2, Thompson’s

group Fcannot be 3

2-generated. However, we prove that it is “almost” 3

2-generated in the

sense that the following theorem holds.

Theorem 1. Every element of Fwhose image in the abelianization Z2is part of a gener-

ating pair of Z2is part of a generating pair of F.

In fact, we have the following more general theorem.

Theorem 2. Let (a, b),(c, d)∈Z2be such that {a, c} 6={0}and {b, d} 6={0}. Let f∈F

be a non-trivial element such that πab(f)=(a, b), then there is an element g∈Fsuch that

πab(g) = (c, d)and such that

hf, gi=π−1

ab (h(a, b),(c, d)i).

Recall that a subgroup Hof Fis normal if and only if it contains the derived subgroup

of F[9] (in particular, all ﬁnite index subgroups of Fare normal subgroups of F). It

follows that the normal subgroups of Fare exactly the subgroups π−1

ab (h(a, b),(c, d)i) for

(a, b),(c, d)∈Z2. Note that if a=c= 0 or b=d= 0, then π−1

ab (h(a, b),(c, d)i) is not ﬁnitely

generated (see Remark 6below). Theorem 2implies that all other normal subgroups of F

are 2-generated. Hence, we have the following.

Corollary 3. Every ﬁnitely generated normal subgroup of Fis 2-generated.

In particular, every ﬁnite index subgroup of Fis 2-generated. Recall that in [3] (see

also [6]), the ﬁnite index subgroups of Fthat are isomorphic to Fwere characterized. Let

p, q ∈N. We denote by Fp,q the subgroup π−1

ab (pZ×qZ) of F. Then for every p, q ∈Nthe

subgroup Fp,q is isomorphic to Fand these are the only ﬁnite index subgroups of Fthat

are isomorphic to F[3]. In particular, the ﬁnite index subgroups Fp,q of Fare known to

be 2-generated. It was conjectured in [14, Conjecture 12.6] that all ﬁnite index subgroups

of Fare 2-generated.

Note that for every non-trivial (a, b)∈Z2, there exists (c, d)∈Z2such that h(a, b),(c, d)i

is a ﬁnite index subgroup of Z2of the form pZ×qZfor some p, q ∈N(see Lemma 7below).

Hence, Theorem 2implies the following.

THOMPSON’S GROUP FIS ALMOST 3

2-GENERATED 3

Corollary 4. Let f∈Fbe an element whose image in the abelianization of Fis non-

trivial. Then there is an element g∈Fsuch that the subgroup hf, giis isomorphic to F

and has ﬁnite index in F.

Note that Theorem 1shows that in some sense it is “easy” to generate Thompson’s group

F. Several other results demonstrate (in diﬀerent ways) the abundance of generating pairs

of Thompson’s group F. Recall that in [15], we prove that in the two natural probabilistic

models studied in [8], a random pair of elements of Fgenerates Fwith positive probability.

In [12], Gelander, Juschenko and the author proved that Thomspon’s group Fis invariably

generated by 3 elements (i.e., there are 3 elements f1, f2, f3∈Fsuch that regardless of

how each one of them is conjugated, together they generate F.) Using results from [16],

one can show that in fact, there is a pair of elements in Fwhich invariably generates F.

2. Preliminaries

2.1. F as a group of homeomorphisms. Recall that Thompson group Fis the group

of all piecewise linear homeomorphisms of the interval [0,1] with ﬁnitely many breakpoints

where all breakpoints are dyadic fractions and all slopes are integer powers of 2. The group

Fis generated by two functions x0and x1deﬁned as follows [9].

x0(t) = 









2tif 0 ≤t≤1

t+1

4if 1

4≤t≤1

2+1

2if 1

2≤t≤1

x1(t) = 









tif 0 ≤t≤1

2t−1

2if 1

2≤t≤5

t+1

8if 5

8≤t≤3

2+1

2if 3

4≤t≤1

The composition in Fis from left to right.

Every element of Fis completely determined by how it acts on the set Z[1

2]. Every number

in (0,1) can be described as .s where sis an inﬁnite word in {0,1}. For each element g∈F

there exists a ﬁnite collection of pairs of (ﬁnite) 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 Fon a

number .s is the following: if sstarts with ui, we replace uiby vi. For example, x0and x1

are the following functions:

x0(t) = 









.0αif t=.00α

.10αif t=.01α

.11αif t=.1α

x1(t) = 









.0αif t=.0α

.10αif t=.100α

.110αif t=.101α

.111αif t=.11α

where αis any inﬁnite binary word.

The group Fhas the following ﬁnite presentation [9].

F=hx0, x1|[x0x−1

1, xx0

1] = 1,[x0x−1

1, xx2

1]=1i,

where abdenotes b−1ab.

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摘要：

THOMPSON'SGROUPFISALMOST32-GENERATEDGILIGOLANPOLAKAbstract.RecallthatagroupGissaidtobe32-generatedifeverynon-trivialelementofGbelongstoageneratingpairofG.Thompson'sgroupVwasprovedtobe32-generatedbyDonovenandHarperin2019.Itwastherstexampleofaninnitenitelypresentednon-cyclic32-generatedgroup.Recent...

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