Dehn Functions of Metabelian Groups

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Notation and Conventions

We establish a commutative algebra approach to estimate upper bounds for the Dehn function of a given finite presented metabelian group. It first gives a uniform upper bound for Dehn functions for all finitely presented metabelic groups.

The Word Problem

The Dehn function

A finitely presented group has a decidable word problem if and only if its Dehn function is bounded above by a recursive function [24]. A finite generated group is hyperbolic if and only if it has sub-quadratic Dehn function.

Metabelian Groups

Despite the dependence of the Dehn function on finite representations of a group, all Dehn functions in the same finitely presented group are equivalent under≈[17], i.e. given a finitely presented group Gwith finite presentationsPand P′, one can show thatδP≈δP′. Thus, we define the Dehn function of a finitely presented group G,δG as the Dehn function of any of its finite presentations.

Main Results

The metabelized Baumslag-Solitar groupBS(n,m) =˜ ha,t|(an)t=amiS2,m>2,m=n+1 has at most a quadratic relative Dehn function. For every l∈N there exists a finitely generated metabelian group such that its relative Dehn function is asymptotically greater than or equal to nl.

Properties of the Dehn Function

Van Kampen Diagram

We say a Van Kampen diagram is minimal, it has the minimum number of cells across all such diagrams of the same word. For a word w=G1, the area of ​​is the same as the number of cells of a minimal van Kampen diagram.

Estimate the Upper Bound

Under a series of such operations, we can convert a word into another word that is equal to the original word inG. By counting the number of relators we use, we can estimate the upper limit of the cost of this conversion. To demonstrate this idea, we calculate the upper bound of the Dehn function of a finitely generated abelian group.

Estimate the Lower Bound

Combining the result of Section II.3, we obtain that finitely generated abelian groups with torsion-free rank greater than or equal to 2 have a quadratic Dehn function. A Dehn function of a finitely generated abelian group G is quadratic if the torsion-free rank of G is greater than one, and linear otherwise.

Figure II.3: the [a, b] cell

Properties of Finitely Generated Metabelian Groups

Examples of Metabelian Groups and their Dehn Functions

Note that all known examples of Dehn functions of finitely represented metatable groups above (up to equivalence) are bounded by an exponential function. Is the Dehn function of the finite metabelian group presented above (up to equivalence) bounded by an exponential function.

Finitely Presented Metabelian Groups

Therefore, the statement in Theorem III.3.3 is true for Gif and only if it is true for G1. G∞ is finitely presented and the defining relations are given by (III.3)-(III.6) for any fixed positive real number>R.

Preliminaries on Module Theory

The membership problem Task IV.1.1 can be considered the word problem for the quotient M/S.

A Well-order on a Polynomial Ring

We also denote supp(f) to be the set of modulus monomials with nonzero coefficients. Recall that the abinary relation on a set X is a subset of X×X, that is, it is a set of ordered pairs(x1,x2) x1,x2∈X. Next, we define the leading coefficient off to be the leading monomial coefficient, denoted by LC(f).

One remark is that the Noetherian on U as well as on X is the set of modular monomials in M, i.e. there is no infinite descending chain of modular monomials.

Gr¨obner Basis

First, notice that we change the coefficient of M(g0)ofh to a positive number after a reduction. There are then only finitely many possible reductions that can be applied to the term containing the monomialM(g0). This provides our motivation for defining the Gröbner basis: a generating set such that each element in it has a unique reduced form moduloF. The following theorem shows that the Gr¨obner basis always exists, while not all submodules have a basis.

According to our definition, the Gr¨obner basis is not unique since adding any element ∈N to a Gr¨obner basis results in another Gr¨obner basis.

Division Algorithm

Let P be the set of all such hu that generateSoverZwheneverhu can be defined (since Sum can be empty), and let L be the set of leading terms of elements of P. We take a finite set ¯FofP such that the set of leading terms of elements in ¯F is F. It follows , that there exists an element f ∈F¯ such that cuu=LT(x f) for somex∈X and h is reducible to f, which contradicts the irreducibility of h.

When we have Gr¨obner bases in hand, we can calculate R(g) sideng∈Sif and only ifR(g) =0.

Reduction Step

So from now on, we will assume that T is a free abelian group of rank kandG an extension of a softened module T from the free abelian group T. Then there exists a finite subset Λ⊂C(A)∪ C(A∗) such that for every characterχ:T→R, there exists λ∈Λ such thatχ(λ)>0. Then we are able to define a sequence of groupsGr as we did in Section III.3, but for our purpose, we will need a larger R.

Ifk>0, the result follows directly from Proposition V.1.2 by passing the problem to a subset of finite index.

The Ordered Form of Elements

To show the existence, let us construct an explicit algorithm that rewrites the word w∈ hhAiiH∞ into a word of ordered form. Furthermore, by combining factau=au¯ from Proposition III.3.4 (a), we can write an ordered form of the following type. The uniqueness of the ordered form can be justified by the fact that the set of words in the ordered form is isomorphic to the free T-moduleM.

Note that depending on how we define ordered from, the ordered form of each word is unique.

Main Lemmas

In particular, for w∈Fsuch thatπ(w) =1, it costs at most stn2relations in H to convert it to a product of conjugates of elements in A. Furthermore, Lemma V.3.2 allows us to estimate the cost of commuting two conjugates of elements in A. Since the normal closure of A inHis abelian, this lemma provides a tool to estimate the cost of converting words inhhAiiH, especially inG.

In particular, by combining the three lemmas presented in this section (Lemma V.3.1, Lemma V.3.2, and Lemma V.3.3), we are able to convert any word in hhAiiH to its ordered form.

The T -module in Metabelian Groups

This is the basis for transforming the word problem in groupG into a submodule membership problem in the freeT-module generated by A. We now estimate the cost of relations in group H to perform each of the above module operations. For c∈Z, the cost of relations in H of the transformation fcto∏mi=1acλi i is at most (|c| −1)(m)2P2K2Q, where the right-hand side is written in ordered form.

Therefore, the cost of (V.7) is bounded by mP2K2Q. By repeating previous process times, we obtain an upper bound m2P2K2Q. c) Conjugating each term of at1′,t′∈T takes zero relations.

Proof of Proposition V.1.2

The equality holds in the free group generated by A ∪T, thus the cost of relations that convertwtow1 is 0. Therefore, the total number of relations we need to convert each fiαito its order fi′ is bounded by. In general, the cost of converting all fiαi's to their order forms is limited by.

Summarizing all the costs fromw1tow4and with factsC>K, we conclude that the area is bounded above by.

The Cost of Metabelianness

The claim can be proved by always choosing to move the leftmost pair everywhere∈T ∪T−1,a∈A ∪A−1. Similar to Lemma V.3.3, there exists a constantC2 such that the cost of the second equality in terms of relations is boundedC2·2|u|. Consequently, [[x,y],[z,w]] can be written as a product of at most 8n2 conjugates of elements in A with a cost of at most 8n2C22n.

The rest of the proof is the same as when T is quite abelian.

The Relative Dehn Functions of Metabelian Groups

Therefore, it is worthwhile to denote the relative Dehn function of a finitely generated metabelian group Gby ˜δG. Then its finitely generated and metabelian function and the relative Dehn of GandHare is equal to the equivalence. But it can be proved that the relative function Dehn iBS(1,2) is not instead of the usual function Dehn 2n[15].

In general, it is difficult to calculate the relative Dehn function of a finitely generated metabolic group.

Connections Between Dehn Functions and Relative Dehn Functions

In the next section we will estimate how much cost we save by introducing the relative presentation. Now in this thesis we have three different types of Dehn functions: the Dehn function, the relative Dehn function and the Dehn function of a module. So in a relative sense we save a lot of costs because we assume that metabelianity is free.

Then we have a relation between the relative Dehn function G and the Dehn function of the submodule S.

Estimate the Relative Dehn Function

According to the proof of Theorem V.1.2, adding the left-hand side of (VI.1) costs at most max{δˆA3(2n3),2n} to equivalence. In this case, that is, when the module is over the Laurent polynomial of one variable and is generated by one variable, the Dehn function of the module is well studied. The metatabulated Baumslag-Solitar group ˜BS(n,m) =ha,t|(an)t=amiS2 has at most cubic relative Dehn function whenn6=mand at most quartic relative Dehn function whenn=m.

The metabelized Baumslag-Solitar group BS(n,˜m) =ha,t|(an)t=amiS2,m>2,m=n+1 has at most quadratic relative Dehn function.

Relative Dehn Function of the Lamplighter Group L 2

Therefore, we can improve the result in [15, Theorem 6.1] by the following corollary of Statement VI.4.4. Thus, the total cost of canceling C of the empty graph is just the sum of the weights of the edges in C. Because each time we cancel pairs of numbers based on the order of the original sequencejam1,m2,.

It follows that the cost of cancellation is limited by the total weight of p.

Figure VI.1: the corresponding graph of the sequence 2,3,5,3,5,8,2,8

Relative Dehn Functions and Subgroup Distortions

And it is not difficult to check that infinite subgroups of finitely generated abelian groups are undistorted. For every finitely generated infinite subgroup H6A≀Z, there exists l∈N such that the distortion of H is in A≀Z. If, moreover, G is finitely represented, the Dehn function G above is asymptotically bounded by an exponential function.

IfGi is represented finitely, by theorem VI.3.1, if the relative Dehn function is polynomially bounded, δG(n) is bounded above by the exponential function.

Motivations

We answer this question by computing Dehn functions for the wilder class of finitely represented metatable groups constructed by Baumslag. Any crown product of a free abelian group of finite rank with a finitely generated abelian group can be embedded in a metabelian group with an exponential Dehn function. A free metatable group of finite rank is embedded in a finite represented metatable group with an exponential Dehn function.

The question of whether such groups can be embedded in a finitely represented metabelian group with Dehn polynomial function remains, even in the case of eZ≀Z.

Embeddings of Wreath Product of Abelian Groups

In particular ifr=k=land we let fi=1+tifor alli,WF contain a copy of the free metabelian group rankr. Assuming that the aui band intersects one of the u1 bands, again by Lemma VII.2.4 (ii), (iii), it cannot cross the au1 band twice, nor intersect itself. Then, if there exists a cell with tsfors>1 in the upper band, then according to Lemma VII.2.4 (i) γ1top is a word ah11ah22.

This leads to a contradiction, since the image of the left side inU×T is not trivial.

Figure VII.1: an example of a u 1 -bands

Tight upper bound for Dehn functions of metabelian groups

Yves Cornulier suggests that the technique of [11] can be used to show that the Dehn function of split finite metabelian groups is polynomially or exponentially bounded. If this is true, combining the results from [6] and [35], each finitely represented metatable group has a polynomial or exponential Dehn function. But so far their technique cannot be applied to the Baumslag group Γ, although it has been proved that Γ has an exponential Dehn function [21].

Embeddings of finitely generated metabelian groups

The van Kampen diagram of w n