Torsion, torsion length and finitely presented groups

1 Introduction

It is well known that the set of torsion elements in a group G, Tor⁢(G), is not necessarily a subgroup. One can, of course, consider the subgroup Tor1⁢(G)generated by the set of torsion elements in G: this subgroup is always normal in G.

The subgroup Tor1⁢(G) has been studied in the literature, with a particular focus on its structure in the context of 1-relator groups. For example, suppose G is presented by a 1-relator presentation P with cyclically reduced relator Rk where R is not a proper power, and let r denote the image of R in G. Karrass, Magnus, and Solitar proved ([17, Theorem 3]) that r has order k and that every torsion element in G is a conjugate of some power of r; a more general statement can be found in [22, Theorem 6]. As immediate corollaries, we see that Tor1⁢(G) is precisely the normal closure of r, and that G/Tor1⁢(G) is torsion-free.

More generally, the manner in which Tor1⁢(G) is impacted by the deficiencydef⁡(G) of a finitely presented group G has also been investigated. The deficiency of G is the maximum value of m-n, where m and n are the number of generators and relators respectively as we range over all finite presentations of G. In [4, Corollary 3.6], Berrick and Hillman proved that if G is a finitely presentable group with def⁡(G)>0, and Tor1⁢(G) is either finitely generated or locally finite, then Tor1⁢(G) is actually finite; again, in this situation G/Tor1⁢(G) is torsion-free. They claim that the question of whether Tor1⁢(G) is necessarily trivial under these hypotheses is open; using a result of Karras and Solitar [18, Main Theorem] one immediately sees that this triviality is indeed the case when G is presented by a 1-relator presentation.

In both cases described above, the quotient G/Tor1⁢(G) is torsion-free. Unfortunately, this is not always the case. Consider, for example, the group C presented by the following presentation: 〈x,y,z∣x3=e,y3=e,x⁢y=z3〉; it can be shown that C is a finitely presented word-hyperbolic group ([10, Proposition 3.5]), but that C/Tor1⁢(C)≅ℤ/3⁢ℤ ([10, Proposition 3.1]).

We can, however, iterate the process used to construct Tor1⁢(G) to produce an ascending chain of normal subgroups Tor1⁢(G)≤Tor2⁢(G)≤⋯ of G. For finite n∈ℕ, we define Torn+1⁢(G) via Torn+1⁢(G)/Torn⁢(G)=Tor1⁢(G/Torn⁢(G)); we define Torω⁢(G):=⋃n∈ℕTorn⁢(G). The ordinal for which this chain stabilises is called the torsion length of G and denoted by Tor⁢Len⁡(G). It turns out that G/Torω⁢(G) is the universal torsion-free quotient of G: it is torsion-free, and all other torsion-free quotients uniquely factor through it (see [9, Corollary 3.4]). Thus Tor⁢Len⁡(G) is always bounded above by ω; this bound is attained when the chain mentioned above does not stabilise at any finite stage. Intuitively, Tor⁢Len⁡(G) is the minimal number of times we need to ‘kill off’ torsion to get a torsion-free quotient of G.

The notion of torsion length first appeared, independently, in both [11] and our earlier work [10]. In [11], Cirio, D’Andrea, Pinzari and Rossi defined the torsion degree of a quantum group (here, quantum groups are C*-algebras equipped with a suitable comultiplication). The definition of torsion length aligns with torsion degree when a group is viewed as a quantum group via its associated C*-algebra. Further, they defined the notion of the “connected component at the identity” Q∘ of a quantum group Q and remarked that for an ordinary group G (again viewed as a quantum group via its associated C*-algebra) this object corresponds to G/Torω⁢(G) ([11, Example 3.17]). They also constructed a descending ordinal indexed family of quantum subgroups Gα “converging” to G∘; again, in the classical situation these objects correspond to the quotients G/Torα⁢(G).

The quotient G/Torω⁢(G) was first studied in [6], where Brodsky and Howie investigated this object (they use the notation G^) for various families of groups. A group is locally indicable if every non-trivial finitely generated subgroup admits a surjection onto ℤ: Brodsky and Howie showed that if a group has deficiency def⁡(G)>0, then G/Torω⁢(G) is locally indicable [6, Theorem 3.7]. They also showed that G/Torω⁢(G) is locally indicable when G is 1-relator, or 2-relator with one relator having length ≤4, or 2-relator with one relator having length 5 and the other having length ≤8, or at most 5-relator with each relator having length ≤3. These results appear as [6, Theorems 1.1–1.4].

In [10] we began a preliminary investigation of torsion length. One of the main results of that work, which we generalise here in Theorem 6.13, was the following theorem:

We then showed that a construction used to prove a classic embedding theorem of Higman, Neumann and Neumann (every countable group embeds into a 2-generator group) preserved torsion length. This fact, used with the theorem mentioned above, allowed us to arrive at the following result:

This paper aims to extend [10, Theorem 3.10]. In Theorem 5.7, we prove the following:

We do this by showing that a particular construction used in a proof of the Higman Embedding Theorem preserves this invariant.

Let us be more precise. The Higman Embedding Theorem [16] states that a finitely generated, recursively presented group embeds into a finitely presented group. There are many proofs of this result, but these arguments share a common theme: they are all constructive. One must begin with a finite generating set for the group, and an algorithm that computes its relations, and then explicitly build a finitely presented group from this data. In this paper we pick a particular construction, due to Aanderaa and Cohen [1, 2] and presented in [12], examine it in detail, and conclude that the torsion length of the finitely presented group so constructed is the same as that of the recursively presented group that we started with.

The existence of a finitely presented group with infinite torsion length is then an immediate consequence of [10, Theorem 3.10]: take the recursively presented group constructed in [10, Theorem 3.10] and apply the Aanderaa–Cohen construction.

Section 6 of this paper is concerned with improving Theorem 5.7. The following result appears as Theorem 6.10.

This is done using small cancellation theory. Of particular importance is the following theorem, whose content is contained in Proposition 6.4 and Theorem 6.7; this result is also of independent interest. Combined with Theorem 5.7, it proves Theorem 6.10.

A part of the above theorem appeared in the work [7] of Bumagin and Wise; see Remark 6.11. In Section 7 we finish with a discussion of some open problems relating to torsion length and torsion subgroups.

2 Torn⁢(G), HNN extensions and Britton’s lemma

Observe that Tori+1⁢(G)/Tori⁢(G)=Tor1⁢(G/Tori⁢(G)).

HNN extensions play a critical role in this paper; we briefly introduce them here.

Given a group G, we write 〈G;X∣R〉 to denote (G*FX)/〈〈R〉〉G*FX, where R is any subset of G*FX.

3 Good subgroups of HNN extensions

The notion of a good subgroup was introduced in [12, Proposition 1.34], and named so in [23, Definition 2].

We now study good subgroups which are normal.

4 The Higman embedding construction

The Higman Embedding Theorem states that a finitely generated, recursively presented group can be embedded in a finitely presented group. In this section we provide an overview of a proof of this result, introducing notation and constructions that will be used later in this paper.

5 Properties of the embedding construction

In this section, the groups C,H1,H2 and H3 will be as in Section 4.3.

Proof.

We claim that the following is true:

〈〈X〉〉H1∩〈{tα∣α∈I}〉={e},

〈〈X〉〉H1∩〈{tα⁢wα⁢(b)⁢d∣α∈I}〉={e}.

To see this, consider the map λ:H1→H1 induced by the identity maps on Kℳ, 〈b1,…,bn∣-〉, and 〈d∣-〉, and the trivial map on C.

The map λ, restricted to KM*({e}×〈b1,…,bn∣-〉)*〈d∣-〉, is injective, and thus injective on both 〈{tα∣α∈I}〉 and 〈{tα⁢wα⁢(b)⁢d∣α∈I}〉. However, 〈〈X〉〉H1 is contained in ker⁡(λ).

This proves the first part of the lemma; we now move to the second.

Take the map λ defined above. It is clear that λ extends to a map λ¯:H2→H2, sending p↦p. Again, 〈〈X〉〉H2≤ker⁡(λ¯). As before, we see that the restriction of λ¯ to 〈KM*({e}×〈b1,…,bn∣-〉)*〈d∣-〉,p〉 is injective. It follows that

〈〈X〉〉H2∩A=〈〈X〉〉H2∩Ai={e}

for all 1≤i≤2⁢n.

Finally, consider the inclusions

ι-:A-→H2,ι+:A+→H2.

Step (20) of Section 4.3 tells us that the restriction of λ to A- is injective with image A+, and thus induces an isomorphism λ′:A-→A+; λ′ is inverse to the map ψ+ defined in (15) of Section 4.3. We see that λ¯∘ι+∘λ′=λ¯∘ι-:A-→H2:

It is clear that λ¯∘ι+ is injective, and thus that λ¯∘ι- is as well. Since 〈〈X〉〉H2 is contained in ker⁡(λ¯), we see that 〈〈X〉〉H2∩A-=〈〈X〉〉H2∩A+={e}. This proves the last part of the lemma. ∎

Before we proceed, we need the following observation. It is proved in the same way that [10, Corollary 2.9] is, by using the torsion theorem for HNN extensions.