Dehn functions of finitely presented metabelian groups

The structure of this paper

We will state some preliminary concepts and lemmas in Section 2. In Section 3, we prove Corollary 3.4 that allows us to solve the membership problem for some special free modules in a reasonable time. Then we shall briefly revisit the proof of the “if” part of a result of Bieri and Strebel [6] in Section 4, which characterizes finitely presented metabelian groups. In Section 5, we provide the proof of Theorem 1.3. We will talk about relative Dehn functions and the connections between them and Dehn functions in Section 6. In Section 7, we prove Theorem 1.5 and discuss the obstructions for finding metabelian groups with Dehn function greater than the exponential function.

2.1 Notation

We denote the set of rational integers by ℤ and the set of real numbers by ℝ. The set of natural numbers is indicated by ℕ, where our convention is that 0 ∉ N . In addition, we let R + = { x ∈ R ∣ x > 0 } .

If 𝐺 is a group, we will denote by G ′ = [ G , G ] the derived (commutator) subgroup and by G ab ≅ G / G ′ the abelianization. For elements x , y ∈ G , n ∈ N , our conventions are x n ⁢ y = y - 1 ⁢ x n ⁢ y , [ x , y ] = x - 1 ⁢ y - 1 ⁢ x ⁢ y . We use a double bracket ⟨ ⟨ ⋅ ⟩ ⟩ G to denote the normal closure of a set in the group 𝐺. Sometimes, we omit the subscript when there is no misunderstanding in the context. For a set 𝒳, we denote the free group generated by the set 𝒳 as F ⁢ ( X ) . We also use F ⁢ ( X ) to represent the set of reduced words in the alphabet X ∪ X - 1 .

In addition, for a group 𝐺 and a commutative ring 𝐾 with 1 ≠ 0 , we let K ⁢ G be the group ring of 𝐺 over 𝐾. An element λ ∈ K ⁢ G is usually denoted as

where all but finitely many α g are 0. We also regard 𝜆 as a function λ : G → K with finite support, where λ ⁢ ( g ) = α g .

We say a group 𝐺 is an extension of a group 𝐴 by a group 𝑇 if 𝐴 is a normal subgroup of 𝐺 and T ≅ G / T . If 𝐴 is abelian, then 𝐴 is a module over Z ⁢ T and the action of 𝑇 on 𝐴 is given by conjugation. In this case, we also say that 𝐺 is an extension of a 𝑇-module 𝐴 by 𝑇.

2.2 Dehn function

Let 𝐺 be a finitely presented group in the category of all groups, i.e. G = ⟨ X ∣ R ⟩ , where | X | , | R | < ∞ . We denote this presentation by 𝒫. The length of a word w ∈ G is the length of the corresponding reduced word in F ⁢ ( X ) . A word 𝑤 in the alphabet X ∪ X - 1 is equal to 1 if and only if it lies in the normal closure of 𝑅, i.e.

The smallest possible 𝑘 is denoted by Area P ⁡ ( w ) . In addition, if w 1 = G w 2 , the cost of converting w 1 to w 2 in 𝐺 with respect to the presentation 𝒫 is defined to be Area P ⁡ ( w 1 ⁢ w 2 - 1 ) . The Dehn function of 𝐺 with respect to the presentation 𝒫 is defined as

It is convenient for us to talk about functions up to the equivalence relation ≈ which we defined in the introduction, because the Dehn function is independent of finite presentations of a group in terms of ≈.

We shall then denote the Dehn function of a finitely presented group 𝐺 by δ G ⁢ ( n ) . Moreover, the Dehn function is a quasi-isometric invariant up to ≈.

Proposition 2.1 follows immediately from the fact that Cayley graphs of the same group over different generating sets are quasi-isometric.

Here is another useful consequence of Theorem 2.2.

2.3 Van Kampen diagrams

One way to visualize the area of a given word is to consider what is called a van Kampen diagram. Let G = ⟨ X ∣ R ⟩ be a finitely presented group and 𝑤 a reduced word which is equal to 1. Then, by the previous discussion, 𝑤 has a decomposition as follows:

For every decomposition (2.1), we can draw a diagram which consists of a bouquet of “lollipops”. Each “lollipop” corresponds to a factor r i f i , the stem of which is a path labeled by f i and the candy of which is a cycle path labeled by r i . Going counterclockwise around the “lollipop” starting and ending at the tip of the stem, we read f i - 1 ⁢ r i ⁢ f i . Thus the boundary of the bouquet of “lollipops” is labeled by the word which is the right-hand side of (2.1).

Note that we obtained 𝑤 from the right-hand side of (2.1) by canceling all consecutive pairs of x ⁢ x - 1 or x - 1 ⁢ x , x ∈ X on the boundary and removing subgraphs whose boundaries labeled by x ⁢ x - 1 or x - 1 ⁢ x , x ∈ X (which is a “dipole” or a sphere). In the diagram, the corresponding process is identifying two consecutive edges with the same label but different orientation on the boundary. After finitely many such reductions, we will obtain a diagram whose boundary is labeled by 𝑤.

Figure 1

A bouquet of “lollipops” and its corresponding van Kampen diagram

The resulting diagram is the van Kampen diagram of 𝑤. The edges are labeled by elements in 𝑋, and cells (i.e. the closure of a bounded connected component of the plane minus the graph) are labeled by words from R ∪ R - 1 .

For example, in the group ⟨ a , b ∣ [ a , b ] = 1 ⟩ , the van Kampen diagram of

is depicted in Figure 2.

Figure 2

Obtain the van Kampen diagram of [ a 2 , b ]

The following is called the van Kampen lemma.

We say a van Kampen diagram is minimal if it has the minimal number of cells over all such diagrams of the same word. For a word w = G 1 , the area of 𝑤 is the same as the number of cells of a minimal van Kampen diagram. This fact is useful to estimate the lower bound of the Dehn function. We will use the van Kampen diagram to compute Dehn functions of a class of examples in the last section.

Other applications of the van Kampen diagram can be found in many books. For example, in the book [21], one can find the study of van Kampen diagrams used to construct groups with extreme properties such as infinite bounded torsion group, Tarski monsters, etc.

3.1 A well-order on the free module

Let R := Z ⁢ [ x 1 , … , x k ] be a polynomial ring over ℤ. Consider a free 𝑅-module 𝑀 with basis elements e 1 , … , e m . A term in 𝑀 is a product of an integer, a monomial in 𝑅 and an element from the basis. A typical term looks like a ⁢ μ ⁢ e i , where a ∈ Z , 𝜇 is a monomial in 𝑅. Let 𝒯 be the set of all terms in 𝑀. In addition, we will call μ ⁢ e i a monomial in 𝑀 and denote by 𝒰 the set of monomials of 𝑀. The set of monomials in the polynomial ring 𝑅 in the usual sense will be denoted by 𝒳. For a term g ∈ T , we denote by C ⁢ ( g ) and M ⁢ ( g ) the coefficient and monomial part of 𝑔, respectively. An element in 𝑀 is a finite sum of terms. From now on, we only consider reduced elements in 𝑀, in a sense that no terms share the same monomial. We also denote by supp ⁡ ( f ) the set of monomials with non-zero coefficients.

Our first goal is to put a well-order on 𝒯. To construct such an order, we put well-orders on ℤ, 𝒳 and { e 1 , … , e m } separately. Then we extend this to 𝑀.

On ℤ, we define an order ≺ as follows:

Under this order, all negative numbers are larger than any positive number. Let a , b ∈ Z , where a ≺ b ; then there exist unique 𝑞 and 𝑟 such that a = q ⁢ b + r , 0 < r < | b | . Note that r ≺ a , depending on whether 𝑎 is positive or negative; thus we can reduce any number to its remainder in this sense. The remainder is always positive, which is important to us. It is not hard to see that ≺ on ℤ is a well-order.

For monomials in 𝑅, we use the degree lexicographical order (also called shortlex or graded lexicographical order) ≺ which is defined with respect to

i.e. for μ 1 = x 1 n 1 ⁢ x 2 n 2 ⁢ … ⁢ x k n k , μ 2 = x 1 m 1 ⁢ x 2 m 2 ⁢ … ⁢ x k m k ,

where ≺ lex is the usual lexicographical order which in defined in the following way:

In fact, ≺ on 𝒳 is a well-order, while ≺ lex may not be. (See [2].)

Finally, we fix an order e 1 ≻ e 2 ≻ ⋯ ≻ e s . We now set ≺ on 𝑇 to be the lexicographical order based on X ≻ { e 1 , … , e m } ≻ Z . For instance,

Since ≺ on each of 𝒳, { e 1 , … , e m } , ℤ is a well-order, we obtain a well-order ≺ on 𝒯.

With the well-order ≺ , we are able to compare any two terms. Consequently, we can define the leading monomial LM ⁡ ( f ) of 𝑓 to be the largest monomial among supp ⁡ ( f ) . For example,

Next, we define the leading coefficient of 𝑓 to be the coefficient of the leading monomial, denoted by LC ⁡ ( f ) . Then the leading term of 𝑓 can be defined as

We then extend ≺ to 𝑀. For g , f ∈ M , we define g ≺ f inductively as follows:

Since ≺ on 𝑇 is a well-order, then so is ≺ on 𝑀.

Note that ≺ is compatible with multiplication by elements from 𝒳, i.e. if g ≺ h , then μ ⁢ g ≺ μ ⁢ f , μ ∈ X .

One remark on ≺ is that it is Noetherian on 𝒰 as well as 𝒳, the set of monomials in 𝑀, i.e. there is no infinite descending chain of monomials. However, the corresponding statement is not true for ≺ on 𝒯 because we have an infinite descending chain for negative numbers. This issue can be avoided by what we will introduce in the next subsection: polynomial reduction.

3.2 Polynomial reduction and Gröbner bases

Now let us define the key ingredient for the application of Gröbner bases: polynomial reduction.

For two monomials μ ⁢ e and μ ′ ⁢ e ′ from 𝒰, we say μ ⁢ e ∣ μ ′ ⁢ e ′ if μ ∣ μ ′ and e = e ′ . Let F = { f 1 , … , f l } be a finite subset of 𝑀, and let 𝑆 be the submodule generated by 𝐹. Given g , h ∈ M , we define the polynomial reduction g → F h as follows: if there exists f ∈ F and a term g 0 ∈ T of 𝑔 such that LM ⁡ ( f ) ∣ M ⁢ ( g 0 ) , LC ⁡ ( f ) ≺ C ⁢ ( g 0 ) , then

where C ⁢ ( g 0 ) = q ⁢ LC ⁡ ( f ) + r , q , r are unique integers such that 0 < r < | LC ⁡ ( f ) | .

For g → F h , read “𝑔 reduces to ℎ modulo 𝐹”. If there is no such 𝑓 and g 0 , then we say that 𝑔 is irreducible modulo 𝐹.

Note that we naturally have h ≺ g if g → F h . We claim that → F is Noetherian, i.e. there is no infinite reduction sequence. First, note that we turn the coefficient of M ⁢ ( g 0 ) of ℎ to a positive number after a reduction; then there are only finitely many possible reductions that can be applied to monomial M ⁢ ( g 0 ) . Thus if we assume that g 0 is the largest term that can be reduced in 𝑔 modulo 𝐹, then after finitely many reductions, the monomial of the largest term that can be reduced is strictly less than the original one. Since ≺ is Noetherian for monomials, we only have a reduction of finite length for any given g ∈ M .

Let → F * be the reflexive and transitive closure of → F . Then, for each g ∈ M , there exists h ∈ M such that g → F * h and ℎ is irreducible modulo 𝐹. We call ℎ a reduced form of 𝑔 modulo 𝐹. Unfortunately, the reduced form of an element in 𝑀 may not be unique. In fact, at each step of reduction, we may have multiple choices of f ∈ F that can be applied to this reduction. This yields our motivation for defining a Gröbner basis: a generating set 𝐹 such that every element in 𝑀 has a unique reduced form modulo 𝐹. In theoretical computer science, this property is called the Church–Rosser property (see [7]).

We write g ≡ S h if g - h ∈ S . It is not hard to check that ≡ S defines an equivalence relation on 𝑀. We define the normal form of 𝑔 to be the smallest element in its equivalence class with respect to ≺ . It is well-defined since ≺ is a well-order. We denote the normal form of 𝑔 by NF ⁡ ( g ) .

Let S u = { g ∈ S ∣ LM ⁡ ( g ) = u } , u ∈ U and L u = { LC ⁡ ( g ) ∣ g ∈ S u } . It is not hard to see that L u is an ideal in ℤ. Thus it is generated by the smallest element in this ideal with respect to ≺ . We write h u for an element such that LC ⁡ ( h u ) generates L u since ℤ is a principle ideal domain. Note that the leading coefficient of h u is always positive since, by our definition of ≺ , negative numbers are larger than positive numbers. For our purpose, we denote this by c u ; hence LT ⁡ ( h u ) = c u ⁢ u . Let 𝑃 be the set of all such h u which generate 𝑆 over ℤ whenever h u can be defined (since S u might be empty).

Let 𝐹 be a Gröbner basis for 𝑆. From the proof of [24, Proposition 10.6.3], 𝐹 can be constructed as a finite set such that P = X ⁢ F := { u ⁢ f ∣ u ∈ X , f ∈ F } , which allows us to reduce g ∈ S by deleting the leading terms with elements in 𝐹. In the next subsection, we will use this specific 𝐹 for the Gröbner basis.

3.3 Division algorithm

For an element g ∈ M , 𝑔 can be written as a finite sum of distinct terms, i.e.

where c i ∈ Z , u i ∈ U and u 1 ≻ u 2 ≻ ⋯ ≻ u d . We define the length of 𝑔 to be | g | := ∑ i = 1 d | c i | . And if the leading monomial of 𝑔 is x 1 n 1 ⁢ x 2 n 2 ⁢ … ⁢ x k n k ⁢ e i , then we define deg ⁡ ( g ) = ∑ i = 1 k n i . One immediate observation is that if g ≺ h , then deg ⁡ ( g ) ⩽ deg ⁡ ( h ) .

Let F = { f 1 , … , f l } be a Gröbner basis for a submodule 𝑆 and

such that deg ⁡ ( g ) ⩽ n , | g | ⩽ p . Since g ∈ S , then g → F * 0 . Thus there exists a finite sequence of reductions

At each step, if we always choose to cancel the leading term of g i using the polynomial reduction (this is always possible since 𝑔 can be reduced to 0), we may assume that LM ⁡ ( g 0 ) ≻ LM ⁡ ( g 1 ) ≻ LM ⁡ ( g 2 ) ≻ ⋯ ≻ LM ⁡ ( g r ) = 0 . Thus the number of steps of reduction is bounded by the number of monomials less than or equal to LM ⁡ ( g ) . Recall that 𝑚 is the rank of the free module; then

where G k ⁢ ( n ) is the growth function of a free commutative monoid with a free generating set of size 𝑘 (see [23, Example 3.7.1]). It is well known that G k ⁢ ( n ) is a polynomial of degree 𝑘. In fact,

At the 𝑗-th step of reduction, we have g j = g j - 1 - a j ⁢ μ j ⁢ f i j , where a j ∈ Z , μ j ∈ X , 1 ⩽ i j ⩽ l and LT ⁡ ( g j - 1 ) = LT ⁡ ( a j ⁢ μ i ⁢ f i j ) . Then

Let C = max ⁡ { | f 1 | , | f 2 | , … , | f l | } . We also observe that

Additionally, we have | a 1 | ⩽ | g 0 | = p , and

Combining (3.1) and (3.2) inductively,

Adding all the steps up, we have

Note that

In general, one important consequence of Gröbner bases is the following.

Let 𝑇 be the free abelian group of rank 𝑘 with basis t 1 , … , t k . We can regard the group ring Z ⁢ T as a factor ring of Z ⁢ [ t 1 , t 1 - 1 ⁢ … , t k , t k - 1 ] , i.e.

Then a submodule generated by a finite set 𝐹 over Z ⁢ T can be identified as a submodule generated by F ∪ { t 1 ⁢ t 1 - 1 - 1 , … , t k ⁢ t k - 1 - 1 } over Z ⁢ [ t 1 , t 1 - 1 ⁢ … , t k , t k - 1 ] .

Therefore, we have a similar result for group rings.

4.1 A geometric lemma

Let R n be the Euclidean vector space with the usual inner product ⟨ ⋅ , ⋅ ⟩ . We denote the norm induced by this inner product by ∥ x ∥ = ⟨ x , x ⟩ . If r > 0 , then B r denotes the open ball of radius 𝑟, i.e. B r = { x ∈ R n ∣ ∥ x ∥ < r } .

We consider a finite collection ℱ of finite subsets L ⊂ R n . We say that an element x ∈ R n can be taken from B r if either x ∈ B r or if there exists L ∈ F such that

We omit the proof of this lemma but we will use the explicit choice of r 0 and ε : ( r 0 , ∞ ) → R + in the proof of [6, Lemma 1.1]. Let S n - 1 be the unit sphere in R n , and consider the function f : S n - 1 → R given by

The function 𝑓 is continuous. By the assumption on ℱ, we have f ⁢ ( u ) > 0 for all u ∈ S n - 1 . Since S n - 1 is compact, we can define

Then our choice of r 0 and 𝜀 are

4.2 A theorem by R. Bieri and R. Strebel

Let 𝑇 be a finitely generated abelian group, written multiplicatively. A (real) character of 𝑇 is a homomorphism χ : T → R of 𝑇 into the additive group of the field of real numbers ℝ. Let tor ⁡ T be the torsion subgroup of 𝑇. Then

where 𝑘 is the rank of 𝑇. We fix a homomorphism θ : T → R k . For every character χ : T → R , there is a unique ℝ-linear map χ ¯ : R k → R such that χ = χ ¯ ∘ θ . And by the Riesz representation theorem, there is a unique element x χ ∈ R k such that χ ¯ ⁢ ( y ) = ⟨ x χ , y ⟩ for all y ∈ R k , whence χ ⁢ ( t ) = ⟨ x χ , θ ⁢ ( t ) ⟩ (see [6]). Therefore, each character 𝜒 corresponds a vector x χ in R k . Conversely, given a vector 𝑥 in R k , we can define a corresponding character by χ ⁢ ( t ) = ⟨ x , θ ⁢ ( t ) ⟩ . This will be a useful realization for characters on 𝑇.

Every character χ : T → R can be extended to a “character” of the group ring χ : Z ⁢ T → R ∪ { + ∞ } by putting χ ⁢ ( 0 ) = + ∞ and

One can check that χ ⁢ ( λ ⁢ μ ) ⩾ χ ⁢ ( λ ) + χ ⁢ ( μ ) for all λ , μ ∈ Z ⁢ T . Moreover, if 𝑇 is free abelian, the group ring Z ⁢ T has no zero divisors. In this case, it follows that χ ⁢ ( λ ⁢ μ ) = χ ⁢ ( λ ) + χ ⁢ ( μ ) (see [6]).

For every 𝑇-module 𝐴, the centralizer C ⁢ ( A ) of 𝐴 is defined to be

If 𝐴 is a left (right) 𝑇-module, then we write A * for the right (resp. left) 𝑇-module with 𝑇-action given by a ⁢ t = t - 1 ⁢ a (resp. t ⁢ a = a ⁢ t - 1 ).

We say a 𝑇-module 𝐴 is tame if 𝐴 is finitely generated as a 𝑇-module and there is a finite subset Λ ⊂ C ⁢ ( A ) ∪ C ⁢ ( A * ) such that, for every non-trivial character χ : Z ⁢ T → R , there is λ ∈ Λ with χ ⁢ ( λ ) > 0 .

Robert Bieri and Ralph Strebel proved the following theorem that characterizes finitely presented metabelian groups.

For our purpose, let us sketch the proof of the “if” part of this theorem more precisely.

To prove Theorem 4.3, we have to introduce some preliminary concepts in order to provide a reasonable sketch. We first define ordered and semi-ordered words. Let 𝐹 be the free group freely generated by T = { t 1 , … , t k } . Let F ¯ ⊂ F denote the subset of all ordered words of 𝐹, i.e.

If w ∈ F , we write w ¯ for the unique word from F ¯ representing 𝑤 modulo the derived subgroup F ′ . In addition, a word w ∈ F is said to be semi-ordered if it is of the form

where 𝜎 is a permutation of the symbols { 1 , … , k } .

Let θ : F → R k be the homomorphism given by

for 1 ⩽ i ⩽ k . For every w ∈ F , define the trace Tr ⁡ w ⊂ R n as follows: if

is freely reduced, then

Next, we define a sequence of auxiliary groups. Let 𝒜 be a finite set, and choose an assignment picking an element a i ⁢ j ∈ A for every pair of integers ( i , j ) with 1 ⩽ i < j ⩽ k . For every r ∈ R + ∪ { + ∞ } , let H r be the group generated by the set A ∪ T with the following defining relations:

We have some useful properties for the group H r .

For r = ∞ , H ∞ is metabelian, and it is an extension of a free abelian group by another free abelian group. In fact, by the definition of H ∞ and Proposition 4.4 (a), we have [ a u , b v ] = 1 for any u , v ∈ F , a , b ∈ A , and a u is of infinite order. Then ⟨ ⟨ A ⟩ ⟩ H ∞ is free abelian of infinite rank with basis { a u ∣ a ∈ A , u ∈ F ¯ } . By definition, H ∞ / ⟨ ⟨ A ⟩ ⟩ is generated by 𝒯. It is abelian since we include all commutators [ t i , t j ] in 𝒜. Each t i is of infinite order. It follows that H ∞ / ⟨ ⟨ A ⟩ ⟩ is also free abelian. But let us emphasize that H ∞ is infinitely related.

Now back to the proof of Theorem 4.3. We first claim that the problem can be reduced to the case when 𝑇 is a free abelian group. Let π : G → T be the epimorphism, and let T 1 ⩽ T be a complement of the torsion subgroup of 𝑇. Then G 1 = π - 1 ⁢ ( T 1 ) has finite index in 𝐺, and G 1 is an extension of an abelian group by a finitely generated free abelian group. Note that 𝐺 is finitely presented if and only if G 1 is finitely presented. Moreover, if 𝐴 is a tame 𝑇-module, then 𝐴 is also a tame T 1 -module [6, Proposition 2.5]. Therefore, the statement of Theorem 4.3 is true for 𝐺 if and only if it is true for G 1 .

Now we assume that 𝑇 is a free abelian group of rank 𝑘, 𝐴 is a tame 𝑇-module and 𝐺 is an extension of 𝐴 by 𝑇. Denote by π : G ↠ T the epimorphism such that A ≅ ker ⁡ π .

Let T = { t 1 , … , t k } be a subset of 𝐺 such that { π ⁢ ( t 1 ) , … , π ⁢ ( t k ) } forms a basis of 𝑇, and let 𝒜 be a finite subset of 𝐴 containing all commutators a i ⁢ j = [ t i , t j ] for 1 ⩽ i < j ⩽ k and generating 𝐴 as a 𝑇-module. We write w ^ ∈ T for the image of w ∈ F under 𝜋.

Since 𝐴 is a tame 𝑇-module, there is a finite subset Λ ⊂ C ⁢ ( A ) ∪ C ⁢ ( A * ) with the property that, for every character χ : T → R , there exists λ ∈ Λ such that χ ⁢ ( λ ) > 0 . Recall that F := F ⁢ ( T ) and F ¯ is the set of ordered words of 𝐹. For every r ∈ ( 0 , + ∞ ] ,we define the group G r to be given by generators A ∪ T and defining relations