DEHN FUNCTIONS FOR MAPPING TORI OF AMALGAM OF FREE GROUPS QIANWEN SUN_2

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DEHN FUNCTIONS FOR MAPPING TORI OF AMALGAM OF

FREE GROUPS

QIANWEN SUN

Abstract: For an amalgam of two free groups and a particular kind of auto-

morphism, we show that the Dehn function of the corresponding mapping torus is

quadratic.

1. Introduction

Dehn functions have attracted a lot of attention in recent years. Named after

Max Dehn, a Dehn function is an optimal function that bounds the area of an

identity in terms of the deﬁning relators in a ﬁnitely presented group. It is also

closely connected with algorithmic complexity of the word problem, which is the

problem of deciding whether a given word equals to 1. For a ﬁnitely presented

group, it has solvable word problem if and only if it has recursive Dehn function

[6].

A milestone that related Dehn functions and hyperbolic groups, proved by Gro-

mov [7], is that a ﬁnitely presented group is hyperbolic if and only if its Dehn

function is linear, or equivalently, it satisﬁes a linear isoperimetric inequality.

In the same paper, Gromov also pointed out that there exists an isoperimetric

gap: if a ﬁnitely presented group satisﬁes a subquadratic isoperimetric inequality,

then its Dehn function is linear. In particular, no group has a Dehn function

equivalent to ndwith d∈(1,2).

Brinkmann [4] classiﬁed all hyperbolic mapping tori of free groups. In 2000,

Macura [10] proved that the mapping torus of a polynomially growing automor-

phism of ﬁnitely generated free group satisﬁes a quadratic isoperimetric inequality.

Bridson and Groves [1] generalized it to an arbitrary automorphism in 2010.

In this paper, following Bridson and Groves, we consider an amalgam of two free

groups but only with a certain kind of automorphism.

Let Gbe an amalgam of two ﬁnitely generated free groups F1and F2amal-

gamated along ﬁnitely generated subgroups Ai∈Fi, where A1is a malnormal

subgroup. Let ψbe an automorphism of F2that ﬁxes A2pointwise. Then there

exists an induced automorphism ϕof Gsuch that it restricts to identity on F1, and

ψon F2.

Theorem 1.1. The Dehn function of the mapping torus

Mϕ={G, t;t−1at =ϕ(a), a ∈G}

is quadratic.

Note that when A2is generated by one element uand ψ=Adu, the induced map

ϕis the Dehn twist along the edge.

The group Gin Theorem 1.1 is hyperbolic. This is a special case of a more

general result, see [8], where they show that amalgam of two hyperbolic groups are

arXiv:2210.14172v1 [math.GR] 25 Oct 2022

2 QIANWEN SUN

hyperbolic if at least one of the amalgamated subgroup is malnormal, by proving

such an amalgam satisﬁes a linear isoperimetric inequality. For our work, we show

a stronger result: Single Bounded Theorem.

Theorem 1.2. Let Gbe an amalgam of two free groups F1and F2amalgamated

along ﬁnitely generated subgroups A1∈F1and A2∈F2. If A1is a malnormal

subgroup, then there exists a presentation of Gand a constant Bsuch that for

any word z=X1Y1...XnYn=G1,Xi∈F1,Yi∈F2,Area(z)≤B∑n

i=1∣Xi∣in this

presentation.

Here, ∣Xi∣denotes the length of the reduced word in F1representing Xi. We

make this convention throughout the rest of this paper.

As a corollary, we recover the above mentioned fact that Gis hyperbolic.

Corollary 1.3. Let Gbe an amalgam of two free groups F1and F2amalgamated

along ﬁnitely generated subgroups A1∈F1and A2∈F2. If one of Aiis a malnormal

subgroup, then Gis hyperbolic.

Mapping tori have been powerful in low-dimensional topology. Since any outer

automorphism [¯

f]of a surface group is realized by a homeomorphism fof the

underlying surface. The mapping tori is fundamental group of a 3-manifold.

A deep result of Thurston [12] states that in this case the 3-manifold Mfis

hyperbolic if and only if fis a pseudo-Anosov homeomorphism of S. In all other

cases, π1(Mf)has quadratic Dehn functions, for example, because it is automatic

[5].

This paper is organized as follows: ﬁrst in Section 2, we recall some preliminaries.

Then in Section 3, we introduce some combinatorial methods and use them to prove

the Single Bounded Theorem. Finally in Section 4, we introduce some geometric

methods and use them to prove Theorem 1.1.

2. Preliminaries

In this section, we will recall some preliminary knowledge. Some standard ref-

erences on these topics are [2], [9].

2.1. N-reduced set. An N-reduced set is an important concept in the study of

subgroups of a free group. We recall the deﬁnition and some properties here, see

[9].

Consider a set of reduced words W={w1, w2,...}in a free group Fwith a ﬁxed

basis. We call WN-reduced if for all triples v1, v2, v3of the form w±1

i, the following

conditions hold:

(N0) v1≠1;

(N1) v1v2≠1 implies ∣v1v2∣≥max{∣v1∣,∣v2∣};

(N2) v1v2≠1 and v2v3≠1 implies ∣v1v2v3∣>∣v1∣−∣v2∣+∣v3∣.

By establishing the following proposition, Nielsen [11] shows that every ﬁnite

generated subgroup of a free group is free.

Proposition 2.1. If W={w1,...,wm}is a ﬁnite set in a free group F, then W

can be carried by Nielsen transformations into some W′such that W′is N-reduced

and ⟨W⟩=⟨W′⟩

There is a slightly stronger result than Nielsen’s original work due to Zieschang

[13].

DEHN FUNCTIONS FOR MAPPING TORI OF AMALGAM OF FREE GROUPS 3

Proposition 2.2. If W={w1, w2,...,wm}is N-reduced, then one may asso-

ciate with each vin W±1words a(v)and m(v), with m(v)≠1, such that v=

a(v)m(v)a(v−1)−1reduced, and such that if u=v1v2...vt,vi∈W±1and all

vivi+1≠1, then m(v1),...,m(vt)remain uncanceled in the reduced form of u.

2.2. Amalgams. Let G1and G2be groups. Let i1∶H↪G1and i2∶H↪G2be

embeddings. Then the amalgamated free product of G1and G2over His the group

G1∗HG2∶=⟨G1, G2;i1(a)=i2(a), a ∈H⟩.

There are extensive studies on the structure of amalgams. One fact is that the

two natural inclusions G1→G1∗HG2and G2→G1∗HG2are both embeddings.

So G1and G2can be identiﬁed as subgroups of G1∗HG2.

Another classical result is the following lemma.

Lemma 2.3. [[9], IV. Theorem 2.6] Let G=G1∗HG2. Let X1,......,Xn∈G1

and Y1,......,Yn∈G2be elements such that Xi∉Hif i>1and Yi∉Hif i<n

where n>1. Then X1Y1......XnYn≠1in G.

As a corollary,

Corollary 2.4. Let G=G1∗HG2. Let X1,......,Xn∈G1and Y1,......,Yn∈G2

such that z=X1Y1......XnYnrepresents 1in G, then there exists ksuch that

Xk∈Hor Yk∈H. Furthermore, if n>1and all Xi’s and Yi’s are nontrivial except

X1and Yn, then there exists nontrivial Xkor Ykthat is in H.

2.3. Dehn functions. For a ﬁnitely presented group G=⟨X;R⟩, a word w∈F(X)

represents 1 in G if and only if it can be represented as ﬁnite product of conjugates

of elements in R. Among all such representations, there is a smallest integer nsuch

that wcan be represented by a product of conjugates of nelements in R. We call

nthe area of w.

Van Kampen diagrams are a standard tool for the study of isoperimetric func-

tions in groups, see [3] for details. The idea of a van Kampen Diagram is that for

a word w=G1, we have a simply connected planer 2-complex, whose 2-cells are

labeled by relators and whose boundary is labeled by w.

We call the number of 2-cells in a van Kampen diagram its area. The area of

wdeﬁned above is the minimum area of a van Kampen diagram with boundary

labeled by w.

For a ﬁnitely presented group G=⟨X;R⟩, the Dehn function is deﬁned as

D(n)=max{Area(w)∣w=G1,∣w∣≤n}.

Although it depends on the particular presentation, diﬀerent presentations of the

same group give equivalent Dehn functions, see [3]. Here, two functions fand g

are equivalent if f⪯gand g⪯f, where ⪯means that there exists a constant C>0

such that

f(n)≤Cg(Cn +C)+Cn +C.

The Dehn function of a group is deﬁned as the equivalence class of functions.

2.4. Mapping torus. Let us recall the deﬁnition of the mapping torus. For any

automorphism ϕof a group G, the algebraic mapping torus is deﬁned by

Mϕ=⟨G, t;t−1at =ϕ(a), a ∈G⟩.

Note that Gand ⟨t⟩are embedded subgroups in the mapping torus.

4 QIANWEN SUN

Figure 1.

In 2010, Bridson and Groves [1] proved the following theorem about mapping

tori of free groups.

Theorem 2.5. If Fis a ﬁnitely generated free group and ϕis an automorphism

of F, then the mapping torus Mϕsatisﬁes a quadratic isoperimetric inequality.

3. Combinatorial methods

3.1. Relating Pairs. An N-reduced set W={w1,...,wm}in a free group Fforms

a basis for the subgroup ⟨w1,...,wm⟩. We assume all wiare reduced words in this

section.

By a combination we mean a word in ⟨W⟩that is represented by a product

of elements in W±1. If no two adjacent elements are inverse of each other, the

combination is said to be reduced, in which case it is the unique representation in

terms of this basis.

For a reduced combination v=v1...vtwhere vi∈W±1, there may be cancellation

between viand vi+1. By Proposition 2.2, there is a unique partition for each vi:

vi=a−1

i−1siai, where a0and atare trivial, siis nontrivial and contains m(vi), and

v1...vt=s1...stwith no cancellation between siand si+1. We call each siastem,

and each aiatwig in this combination, where ai−1and aiare left and right twig of

virespectively.

If a word is a stem or a twig in some combination, we call it a stem or a twig in

A reduced combination, when expressed as a product of elements in W±1, can be

represented by a path in the Cayley graph of F. After Stalling foldings, the folded

path with labels is called the graph of the combination. We make the convention

that all twigs are placed vertically, and all stems horizontally in the graph.

Figure 1 is an example of such a graph in a free group generated by a, b, c, where

v1=abab,v2=b−1ca2c−1,v3=ca−1bc2. Note that the horizontal line represents the

reduced form of v1v2v3.

For v∈W±1, if it has a partition asb in some combination, we call it a partition

of vin W±1, and sa stem of v.

Deﬁnition 3.1. Arelating pair is a pair of elements in W±1:[v1, v2], equipped

with the following information:

(1) a partition in W±1for each vi:vi=aisibi;

(2) a nontrivial word s, which is called the relating segment, that is an initial

subword of s1, and is also a subword of s2. That is, s1=ss′and s2=lsm

as strings of letters. Note that s′, l, m may be trivial.

A relating pair is trivial if v1=v2, their partitions are the same, and s=s1=s2.

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DEHNFUNCTIONSFORMAPPINGTORIOFAMALGAMOFFREEGROUPSQIANWENSUNAbstract:Foranamalgamoftwofreegroupsandaparticularkindofauto-morphism,weshowthattheDehnfunctionofthecorrespondingmappingtorusisquadratic.1.IntroductionDehnfunctionshaveattractedalotofattentioninrecentyears.NamedafterMaxDehn,aDehnfunctionisano...

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DEHN FUNCTIONS FOR MAPPING TORI OF AMALGAM OF FREE GROUPS QIANWEN SUN_2

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