An Irrational-slope Thompson's Group

An Irrational-slope

Thompson’sGroup

Josep BurilloU. Politècnicade Catalunya

(Joint withB.Nucinkis

and L.Reeves)

Introduction

DefinitionDefinition

Subdivisions

Binary Trees

PresentationGenerators

Relations

Presentation

Normal Form

CommutatorandAbelianisationSimple Commutator

Abelianisation

MetricPropertiesMetric Estimates

Distortion

Future

An Irrational-slope Thompson’s Group

Josep BurilloU. Politècnica de Catalunya

(Joint with B.Nucinkis and L.Reeves)

Western Sydney University Algebra Seminar20 August 2020

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


Subdivisions

Binary Trees


Relations

Presentation

Normal Form


Abelianisation


Distortion

Future

Introduction

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


Subdivisions

Binary Trees


Relations

Presentation

Normal Form


Abelianisation


Distortion

Future

History

In 1969, in a context of logic, Richard J. Thompsonintroduced a family of groups (called since Thompson’sgroups) which captured properties of commutativity andassociativity.

Higman realised these groups were important from thegroup-theoretic point of view. In particular, the grouporiginally called Thompson’s group (now called V ) was thefirst known example of a finitely presented, infinite, simplegroup.

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


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Abelianisation


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History

An interesting subgroup of V , called F , was described byGeoghegan in 1979, and he conjectured that:

1 F has infinite cohomological dimension.

2 F is simply connected at infinity.

3 F has no nonabelian free subgroups.

4 F is not amenable.

The first three properties were readily proved by severalpeople (Brown, Geoghegan, Brin, Squier), and property 4 isfamously still open.

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


Subdivisions

Binary Trees


Relations

Presentation

Normal Form


Abelianisation


Distortion

Future

Thompson’s group F

Thompson’s group F is the group of homeomorphisms of[0,1] such that:

• are piecewise linear and orientation-preserving,• have breakpoints in Z[ 1

2 ],• slopes are powers of 2.

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


Subdivisions

Binary Trees


Relations

Presentation

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Abelianisation


Distortion

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Thompson’s group F

Elements can be obtained with subdivisions of the interval,which can be encoded with carets:

1

3/4

1/2

1/4 1/2 1

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


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Abelianisation


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Generalisations

Several authors generalised this group to other breaks andslopes, for instance Z[ 1

n ] and powers of n, or allowingpowers of 2 and 3 and breaks of the type a/2n3m.

Bieri and Strebel wrote a wonderful memoir about groups ofPL maps. Given A ⊂ R a subring, and given Λ ⊂ A∗ asubgroup of units of A, they define the group G(I,A,Λ),which is the group of orientation-preserving,piecewise-linear homeomorphisms of I with breaks in A andslopes in Λ.

Hence F is the group G([0,1],Z[ 12 ], 〈2〉).

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


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Abelianisation


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Definition

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


Subdivisions

Binary Trees


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Abelianisation


Distortion

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Definition

Let τ =√

5−12 = 0.6180339887..., which is a zero of the

polynomial X 2 + X − 1.

Consider the ring Z[τ ] of elements of the type a + bτ , wherea and b are integers.

Then the group Fτ is the group G([0,1],Z[τ ], 〈τ〉) inBieri-Strebel notation, that is, the group of piecewise-linearmaps of the interval [0,1] with breakpoints in Z[τ ] andslopes powers of τ .

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


Subdivisions

Binary Trees


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Abelianisation


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Subdivisions

Since 1 = τ + τ 2, the unit interval can be split into twointervals of lengths τ and τ 2, which can be done in twoways:

[0,1] = [0, τ ] ∪ [τ,1] [0,1] = [0, τ 2] ∪ [τ2,1].

And any smaller interval can be subdivided further usingτ k = τ k+1 + τ k+2 (for all k = 0,1,2, . . .).

Then, the interval can be subdivided into n intervals whoselengths are all powers of τ .

An Irrational-slope

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Subdivisions

Given two such subdivisions in n intervals, an element of Fτ

can be obtained by mapping the intervals linearly inorder-preserving fashion.

The group Fτ was introduced by Cleary in 2000, where heproved that the group is of type FP∞. In that paper it is alsoproved that any element of Fτ can be obtained in this waywith subdivisions of the interval.

An Irrational-slope

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Binary Trees

Hence, this opens the possibility to describe elements of Fτ

using binary trees, in a similar fashion as it is done for F . Asubdivision will be represented by a caret, but we have todistinguish the two types of subdivisions available. This isdone using unbalanced carets.

00 1 1τ τ2

The caret on the left is called a y-caret, while theright-hand-side one is an x-caret.

An Irrational-slope

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Binary Trees

An example of an element:

τ

τ2

An Irrational-slope

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Binary Trees

And a tree looks like . .. .... ....

1

0

2

6

3

4

5

. .

τ ττ τ τ τ33 4 456τ5

where the level of a vertex marks the length of the interval(e.g. a dot at level 3 is an interval of length τ 3).

An Irrational-slope

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and L.Reeves)

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Elements

These subdivisions have a particularity which is importantfor this group. There exist two trees which represent thesame subdivision:

An Irrational-slope

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Elements

Hence in a tree we can perform what we call a basic move.

An Irrational-slope

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Multiplication

Multiplication is a bit more complicated than in F. Sincemultiplication is a composition, one needs to match thetarget tree of the first element with the source tree of thesecond. But the carets may not match, so the processrequires finding subdivisions and using basic moves.

Imagine we need to multiply two copies of the element y −10 :

An Irrational-slope

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and L.Reeves)

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MultiplicationHere is the process:

An Irrational-slope

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Multiplication

This is guaranteed by the following lemma.

LemmaGiven two trees, we can always find a common subdivision,i.e., a tree which is a subdivision of both.

Obviously this will involve some basic moves. Here is anexample of two trees and how to find a common subdivision.

An Irrational-slope

Thompson’sGroup



and L.Reeves)

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Abelianisation


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Multiplication1)

3)

2)

4)

5) 6)

TT’

T"

An Irrational-slope

Thompson’sGroup



and L.Reeves)

Introduction


Subdivisions

Binary Trees


Relations

Presentation

Normal Form


Abelianisation


Distortion

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Presentation

An Irrational-slope

Thompson’sGroup



and L.Reeves)

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Generators for F

The set of generators for F is the family of elements xn

given by

......n+1 carets n+1 carets

An Irrational-slope

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and L.Reeves)

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Generators

We copy these generators for F to obtain those for Fτ . Wefirst define a spine, formed by x-carets:

An Irrational-slope

Thompson’sGroup



and L.Reeves)

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Generators

We observe that the process we used to switch caret typesto perform multiplications allows us to change carets on theright hand side of the trees, so that we always have a spine.

Hence instead of mixing types of carets, we only need touse the spine made of x-carets only, and attach an x-caretor a y-caret to obtain a generating set.

An Irrational-slope

Thompson’sGroup



and L.Reeves)

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x-generators

This is the generator xn:

n ncarets carets.

..

..

.

An Irrational-slope

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y -generators

And then the second series with a y-caret, creating thegenerators yn:

n n caretscarets...

...

An Irrational-slope

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Relations

These generators satisfy the standard sets of relations ofthe Thompson type, but with both types of generators:

• xjxi = xixj+1

• xjyi = yixj+1

• yjxi = xiyj+1

• yjyi = yiyj+1

The combinatorics of the trees are the same, this involvesonly how to attach two carets to a spine. Hence we onlyneed to consider all possibilities for the two carets added.

An Irrational-slope

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Relations

But they also satisfy a new relation, which is specific for Fτ ,representing the basic move, and given by:

y2n = xnxn+1

An Irrational-slope

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Relations

See, for instance, why the relation y 21 = x1x2 is true. The

two elements are the same but because of the basic movethey admit two expressions in the generators:

An Irrational-slope

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and L.Reeves)

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Presentation

Hence we have the presentation:

Theorem (B, Nucinkis, Reeves) A presentation for Fτ isgiven by the generators xi , yi , with the relations

1 xjxi = xixj+1

2 xjyi = yixj+1

3 yjxi = xiyj+1

4 yjyi = yiyj+1

5 y2i = xixi+1

for 0 ≤ i < j .

An Irrational-slope

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and L.Reeves)

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Presentation

To see this is a presentation, we start with a word on thegenerators which represents the identity. Performing themultiplications we obtain a tree diagram for it, where the twotrees represent the same subdivision (since the element isthe identity).

We need to show that this element is consequence of therelations. This is shown by the following lemma

LemmaGiven two trees which represent the same subdivision, wecan go from one to the other by using only basic moves andwithout adding any carets.

An Irrational-slope

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Presentation

Hence each basic move is represented by a relation of type(5), so these relations are sufficient to represent thiselement. The Thompson relations are necessary to movecarets around and perform the basic move relations in theright place of the word.

The proof of this lemma involves using geometric powerseries based on τ .

An Irrational-slope

Thompson’sGroup



and L.Reeves)

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PresentationImagine we have two trees representing the samesubdivision, but the root carets are of different types.

τ ττ 542

then for the first tree to have a break in the correspondingpoint given by the second tree, somewhere below it theremust be two consecutive carets of the same type so a basicmove can be performed.

An Irrational-slope

Thompson’sGroup



and L.Reeves)

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Binary Trees


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Abelianisation


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Presentation

same type caretbreak appearsbasic move can be done

opposite caretbreak does not appear

An Irrational-slope

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and L.Reeves)

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Presentation

break can never appear

no matter how deep the tree is

with alternating−type carets

An Irrational-slope

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Presentation

This process will never end if we do not have the repeatedcaret type (and hence we can have a basic move). This isbecause when alternating caret types, we have the followingstrict inequalities

n∑

k=1

τ2k < τ < 1 −n∑

k=1

τ2k+1

and an infinite tree would be required, according to thepower series

∞∑

k=1

τ2k = τ = 1 −∞∑

k=1

τ2k+1

An Irrational-slope

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Presentation

Since the trees are finite, if the break must appear in bothtrees, in one of them there must be two consecutive treeswhere a basic move can be done.

Hence we only need relations of type (5) and the Thompsonrelations to transform one tree to the other and then theserelations are enough for a presentation.

An Irrational-slope

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and L.Reeves)

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Normal Form

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and L.Reeves)

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ConstructionThe relation y2

i = xixi+1 allows for the construction of aneasy normal form.

First of all use the Thompson relators to reorder thegenerators in the standard fashion. Any element of Fτ

admits an expression of the type

ui1ui2 . . . uinv−1jm

. . . v−1j2

v−1j1

where

1 the letters u and v represent either x or y ,

2 i1 ≤ i2 ≤ . . . ≤ in and j1 ≤ j2 ≤ . . . ≤ jm.

If a higher index appears before a smaller one, use theThompson relator to switch them. This is the same as in F,but here we get x- and y-generators mixed up.

An Irrational-slope

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The special relation

Once all generators with the same index are together, nowthe special relation comes to our aid. Since we have therelation y2

i = xixi+1, we have that xixi+1 commutes with yi :

yixixi+1 = y3i = xixi+1yi

and this means that

yixi = xixi+1yix−1i+1 = xiyixi+2x−1

i+1

This amounts to yixi being replaced by xiyi followed byterms with higher index that are moved farther down theword and are dealt with at a later time.

An Irrational-slope

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Only one y

So, in a given subindex, we can accumulate thex-generators first and the y-generators later. Even more,since y2

i can be replaced by xixi+1, we can make it so wehave at most only one y-generator per index:

xa00 yε0

0 xa11 yε1

1 . . . xann yεn

n y−δmm x−bm

m . . . y−δ11 x−b1

1 y−δ00 x−b0

0

where ai ,bi ≥ 0 and εi , δi ∈ {0,1}.

An Irrational-slope

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The special relation

This commuting relation between xixi+1 and yi has an easyexpression in carets:

An Irrational-slope

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Only ys on the left

The equivalent of the algebraic method above means thatall y-carets can be pushed down to the bottom of the tree,i.e. each y-caret has no left children (since these would bex-gens with the same index.

But if a y-caret is at the bottom of the tree with no leftchildren, we can attach another y-caret and convert them tox-generators. Of course y-carets need to be added in bothtrees. But with this process, we can have the target treecompletely free of y-carets.

An Irrational-slope

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Only ys on the leftHere is an example:

An Irrational-slope

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Seminormal Form

Hence, every element admits an expression in seminormalform given by

xa00 yε0

0 xa11 yε1

1 . . . xann yεn

n x−bmm x−bm−1

m−1 . . . x−b11 x−b0

0

where ai ,bi ≥ 0 and εi ∈ {0,1}. Observe that the form hasonly x-generators except for maybe one y-generator perindex in the positive part.

An Irrational-slope

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Normal Form

As it happens with F , this seminormal form is not unique. Atrue normal form needs to satisfy the two extra conditions:

• If ai and bi are both nonzero, then at least one ofai+1,bi+1, εi , εi+1 is nonzero.

• If w contains a subword of the form xiyixi+2ux−1i+1x−1

i ,then u contains a generator with index either i + 1 ori + 2.

The first condition is standard in Thompson, it correspondsto the tree-pair diagram being reduced. The secondcondition is specific to Fτ , and corresponds to hiddencancellations.

An Irrational-slope

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Hidden CancellationsA hidden cancellation is a cancellation that only appearsafter a basic move is performed. See an example:

=

=

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Normal Form

Theorem (B, Nucinkis, Reeves) Every element of Fτ admitsa unique normal form.

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Commutator and Abelianisation

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Simple Commutator Subgroup

As it happens with F , Fτ satisfies this property.

Theorem (B, Nucinkis, Reeves) The commutator subgroupF ′τ is a simple group.

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Simple Commutator Subgroup

The proof of this fact is done the standard way: using atheorem by Higman which constructs simple secondcommutator subgroup due to the high transitivity of thegroup. Then prove that the first and second commutatorsare equal, hence showing the commutator is simple.

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Abelianisation

Another interesting feature of Fτ is that, due to the specialrelation y2

i = xixi+1, the abelianisation has torsion.

Theorem (B, Nucinkis, Reeves) The abelianisation of Fτ isisomorphic to Z

2 ⊕ Z/

2Z .

This is the first known example of group of the Thompsonfamily that has torsion in the abelianisation.

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Metric Properties

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Metric Estimates

Also similarly to F , the group Fτ admits an estimate of theword metric by carets.

Theorem (B, Nucinkis, Reeves) The word metric can beestimated by the number of carets of the reduced diagram,up to a multiplicative constant.

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Distortion

We can consider different subgroups of Fτ which areisomorphic to F . For instance, if we restrict the diagram tohave only one type of carets, we obtain the subgroups Fx

(whose elements have only x-carets in their diagram) andFy (with only y-carets). Since the metric on these copiescan also be estimated by the number of carets, weimmediately obtain the following result:

Theorem (B, Nucinkis, Reeves) The subgroups Fx and Fy

are undistorted in Fτ .

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Future research

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Future research: T and V

This work opens the doors to several lines of futureresearch. For instance:

The construction can clearly be extended to the T and Vversions of Fτ , obtaining groups Tτ and Vτ .

The presentations are similar, just adding the correspondingtorsion elements. In particular the relations y 2

i = xixi+1 arestill valid.

Then these two groups are not simple anymore, but theirabelianisation is Z

/2Z and they both admit index-two

subgroups which are now simple.

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Future research: otherpolynomials

Similar groups can be obtained using different polynomials.Jason Brown (L. Reeves’s student) has extended theseconstructions to the groups obtained by considering zerosof the polynomials

1 = x2 + nx

where carets have n short legs and one long one. Relationsare very similar and everything works in similar fashion.

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For polynomials1 = mx2 + nx

things are more complicated. This is the contents ofNucinkis’ student Nick Winstone’s thesis. In particular, forthese groups not every element is obtained withsubdivisions of the intervals. Winstone has acharacterisation for those.

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Even for those where every element is obtained withsubdivisions, things are different here.

The reason is that for polynomials such as 1 = 2x 2 + 2x ,the four subintervals are two of each type, and the intervalcan be split in the middle and this involves rationalbreakpoints (unlike our case, where they are all irrational). Aparticularity is that the group of units now has rank two,because both 2 and the irrational root are units.

So this is work in progress.

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Thank you!

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An Irrational-slope Thompson's Group

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