Subgroups of F P_{2} groups

Ian J. Leary^{∗}

October 19, 2016

Abstract

We show that every countable group embeds in a group of type F P_{2}.

1 Introduction

The Higman embedding theorem is a high-point of combinatorial group theory that makes precise the connection between group presentations and logic: it states that a finitely generated group G embeds in some finitely presented group if and only if G is recursively presented, i.e., there is an algorithm to write down the relations that hold in G [3].

A group G is F P_{2} if its augmentation ideal I_{G} is finitely presented as a module for its group algebra ℤG. Every finitely presented group is F P_{2}, and every F P_{2} group is finitely generated. Bestvina and Brady gave examples of F P_{2} groups that are not finitely presented [1], although these examples arose as subgroups of finitely presented groups. In [5] the author constructed groups of type F P_{2} that do not embed in any finitely presented group. Given these examples it becomes natural to look for an analogue of the Higman embedding theorem for F P_{2} groups. Our main theorem answers this question.

Theorem 1. Every countable group embeds in an F P_{2} group.

Our proof of this theorem is modelled on Valiev’s proof of the Higman embedding theorem as described in [6, Sec. IV.7], which is a simplification of Valiev’s first proof [8]. Our proof is much simpler than these antecedents because we are not obliged to consider recursive sets. We make the following definition, which is an analogue of Higman’s notion of a benign subgroup.

Definition 2. A subgroup H of a finitely generated group G is a homologically benign subgroup if the HNN-extension

G_{H} = ⟨G, t: t^{−1}ht = h h ∈ H⟩

can be embedded in an F P_{2} group.

∗Partially supported by a Research Fellowship from the Leverhulme Trust. This work was done at MSRI, Berkeley, where research is supported by the National Science Foundation under Grant No. DMS-1440140.

Theorem 1 implies that all subgroups of finitely generated groups are homologically benign, however showing that various subgroups are homologically benign plays a major role in the proof of Theorem 1. The result below details what we need from [5]; after the statement we outline how to deduce it from results stated in [5].

Theorem 3. For any fixed l ≥ 4 and any set S of integers with 0 ∈ S, there is an F P_{2} group J = J(S) and a sequence j_{1}, …, j_{l} of elements of J such that j_{1}^{s}j_{2}^{s} ⋯ j_{l}^{s} = 1 if and only if s ∈ S.

Proof. The groups G_{L}(S) that are constructed in [5] depend on a connected flag simplicial complex L and a set S ⊆ ℤ. If L has perfect fundamental group and contains an edge loop of length l that is not homotopic to a constant map, then J = G_{L}(S) has the claimed properties. See [5, section 2] for an explicit example of a suitable L in the case l = 4; examples for larger l can be obtained by taking subdivisions of this L.

We expand a little by giving the precise results within [5] that guarantee the various properties of the group J = J(S). When 0 ∈ S ⊆ ℤ, [5, theorem A] gives a presentation for G_{L}(S) with generators the directed edges of L. By [5, theorem B], the group G_{L}(S) is F P_{2} if and only if the fundamental group of L is perfect. If j_{1}, …, j_{l} is a directed loop in L that does not bound a disk then by [5, Lemma 44], the word j_{1}^{s}j_{2}^{s} ⋯ j_{l}^{s} in the given generators for J is equal to the identity if and only if s ∈ S.

Theorem 3 enables one to encode arbitrary subsets of ℕ in presentations for F P_{2} groups. This theorem replaces those parts of Valiev’s proof that concern Diophantine equations or those parts of Higman’s proof that concern Turing machines, each of which is used to encode recursive subsets of ℕ in finite presentations.

2 The proofs

Since this section is closely modelled on Lyndon and Schupp’s account of the Higman embedding theorem [6, Sec. IV.7], we have tried to stay close to the notation that they use. We also omit arguments that are identical to those in [6].

Since we will be working with presentations, it is convenient to have a characterization of the F P_{2} property in terms of presentations. Recall that the Cayley complex for a presentation of a group G is the universal cover of the presentation 2-complex. The group G acts freely on its Cayley complex, with one orbit of vertices and with orbits of 1- and 2-cells corresponding to the generators and relators respectively in the presentation. We define a partial Cayley complex to be a G-invariant subcomplex of the Cayley complex; partial Cayley complexes are in bijective correspondence with subcomplexes of the presentation complex.

Proposition 4. Let H be given by a presentation with finitely many generators and a countable set of relators r_{1}, r_{2}, . . .. The following are equivalent.

(i) H is F P_{2}.

(ii) There exists m so that for each i > m, the loop defined by r_{i} represents zero in the homology of the partial Cayley complex corresponding to all the generators and the relators r_{1}, …, r_{m}.

(iii) There is a connected free H-CW-complex with finitely many orbits of cells and perfect fundamental group.

Proof. Equivalence of (i) and (ii). Let X be the Cayley complex for H and let X_{m} be the partial Cayley complex containing all 1-cells and only the 2-cells that correspond to the relators r_{1}, …, r_{m}. Let C_{∗}(X) and C_{∗}(X_{m}) denote the cellular chain complexes of X and X_{m}. The image of the map d_{1} : C_{1}(X) → C_{0}(X) is isomorphic to the augmentation ideal I_{H}. Hence H is F P_{2} if and only if the kernel of d_{1} is finitely generated as a ℤH-module. Since H_{1}(X) is trivial, this kernel is equal to the image d_{2}(C_{2}(X)). The stated condition on loops is equivalent to d_{2}(C_{2}(X_{m})) = d_{2}(C_{2}(X)). If this holds then clearly d_{2}(C_{2}(X)) is finitely generated. Conversely, any finite subset of d_{2}(C_{2}(X)) is contained in some d_{2}(C_{2}(X_{m})), so if d_{2}(C_{2}(X)) is finitely generated then there exists m with d_{2}(C_{2}(X_{m})) = d_{2}(C_{2}(X)).

(ii) ⟹ (iii) and (iii) ⟹ (i). Each X_{i} is a connected H-CW-complex with finitely many orbits of cells, and if (ii) holds then H_{1}(X_{m}) ≅ H_{1}(X) is trivial. Given any H-CW-complex Y as in (iii), pick a maximal subtree T in Y/H, let T̃ be the set of lifts of T in Y, and note that T̃ is equivariantly isomorphic to T × H. The cellular chain complex C_{∗}(Y, T̃) gives a finite presentation for the relative homology group H_{1}(Y, T̃) as a ℤH-module. Since H_{1}(Y) = 0, H_{1}(Y, T̃) is isomorphic to I_{H}.

Next we give the homological version of the Higman Rope Trick [6, IV.7.6].

Lemma 5. If R is a homologically benign normal subgroup of a finitely generated group F, then F/R is embeddable in an F P_{2} group.

Proof. Fix R as in the statement, and let H be an F P_{2} group containing the group F_{R} = ⟨F, t: t^{−1}rt = r, r ∈ R⟩. Let L be the subgroup of F_{R} ≤ H generated by F and t^{−1}F t, so that L ≅ F ∗_{R} F. As in [6, IV.7.6] there is a homomorphism ϕ : L → F/R whose restriction to F is equal to the quotient map F → F/R and whose restriction to t^{−1}F t is the trivial homomorphism. Viewing L as a subgroup of H, the map l ↦ (l, ϕ(l)) defines a second copy of L inside H × F/R. Let K be the HNN-extension in which the stable letter conjugates these two copies:

K = ⟨H × F/R, s: s^{−1}(l, 1)s = (l, ϕ(l)), l ∈ L⟩.

The group K is generated by the generators for H, the generators for F/R and the element s. As defining relators we may take the relators for F/R, the relators for H, finitely many relators stating that the generators for H and the generators for F/R commute, and finitely many relators of the form s^{−1}(l, 1)s(l, ϕ(l))^{−1} for l in some generating set for L. As in [6, IV.7.6], the relators that hold between the generators for F/R can be eliminated from this presentation for K, leaving just the relators for H and finitely many other relators.

To see that K is F P_{2}, we use Proposition 4 applied to the presentation 2-complex with the generators and relators described above. The generators and relators for H are contained in those for K, so we may look at the partial Cayley complex for K corresponding to just these generators and relators. This 2-complex is isomorphic to a disjoint union of copies of the Cayley complex for H (one copy for each coset of H in K). Let r_{1}, r_{2} … be an enumeration of the relators for H. Since H is F P_{2}, there exists m so that for i > m, the relator r_{i} represents zero in the homology of the partial Cayley complex for H with just the relators r_{1}, …, r_{m}. It follows that these same loops represent zero in the homology of the partial Cayley complex for K discussed above.

Now consider the partial Cayley complex for K, taking all the generators, the commutation relators between generators for H and F/R, the finitely many relators involving s, and the relators r_{1}, …, r_{m}. For i > m, the loops in this complex defined by r_{i} represent the zero element of homology, since they already represent 0 in the smaller partial Cayley complex consisting of a disjoint union of copies of the Cayley complex for H. Hence this presentation for K satisfies condition (ii) of Proposition 4, and so K is F P_{2}.

Lemma 6. Let G be a finitely generated group which is embeddable in an F P_{2} group.

• Every finitely generated subgroup of G is homologically benign in G.

• If H and K are homologically benign subgroups of G, then so are their intersection and the subgroup that they generate.

Proof. Almost identical to the proof of [6, Lemma IV.7.7], except that it relies on the fact that a free product with amalgamation P = M ∗_{G} N is F P_{2} provided that M and N are F P_{2} and G is finitely generated rather than on a similar statement for finite presentability. This can be proved easily using Proposition 4.

Lemma 7. Fix l ≥ 4, and for any integer s define v_{s} := c_{0}^{s}c_{1}^{s} ⋯ c_{l}^{s}de^{s}, an element of the free group H = ⟨c_{0}, …, c_{l}, d, e_{l}⟩ of rank l + 3. For any S ⊆ ℤ with 0 ∈ S, the subgroup

V_{S} := ⟨v_{s}: s ∈ S⟩ ≤ ⟨c_{0}, …, c_{l}, d, e_{l}⟩

is homologically benign and is freely generated by the given elements.

Proof. If a reduced word in the elements v_{s} is written out in terms of the elements c_{0}, …, c_{l}, d, e, the only cancellation that can take place involves c_{0} and e. Thus the subwords c_{1}^{s}c_{2}^{s} ⋯ c_{l}^{s}d survive uncancelled, which implies that the elements v_{s} are free generators for the subgroup V_{ℤ} of H.

We claim that V_{ℤ} is benign, and hence homologically benign. To see this, define an ascending HNN-extension of the free group H = ⟨c_{0}, …, c_{l}, d, e⟩ by

u^{−1}c_{i}u = c_{0}c_{1} ⋯ c_{i−1}c_{i}c_{i−1}^{−1}c_{i−2}^{−1} ⋯ c_{0}^{−1}, u^{−1}du = c_{0}c_{1} ⋯ c_{l}de, u^{−1}eu = e.

Since u^{−1}v_{s}u = v_{s+1} for all s ∈ ℤ and v_{0} = d it follows that

⟨d, u⟩ ∩ H = V_{ℤ}.

Hence V_{ℤ} is benign in ⟨c_{0}, …, c_{l}, d, e, u⟩ and therefore also in the free group H.

Fix some S ⊆ ℤ, and claim that V_{S} is homologically benign in H. To see this, let J = J(S) and j_{1}, …, j_{l} ∈ J be as in the statement of Theorem 3, and let

K = ⟨c_{0}, d, e⟩ ∗ (⟨c_{1}, c_{2}, …, c_{l}⟩ × J) = H ∗_{⟨c_{1},...,c_{l}⟩} (⟨c_{1}, c_{2}, …, c_{l}⟩ × J).

The group K is F P_{2}, since it has a presentation in which the only relators are the relators of J and finitely many commutation relators between c_{1}, …, c_{l} and the generators of J. Note also that H is a retract of K, since the map ψ : K → H defined by

ψ(c_{i}) = c_{i}, ψ(d) = d, ψ(e) = e, ψ(j) = 1 for j ∈ J

is a group homomorphism.

Now define a homomorphism ϕ : K → K by

ϕ(c_{0}) = c_{0}, ϕ(d) = d, ϕ(e) = e, ϕ(c_{i}) = c_{i}j_{i} for i > 0, ϕ(j) = j for j ∈ J.

It is easily checked that ϕ is an automorphism and that ψ ∘ ϕ = ψ. Since V_{ℤ} is homologically benign in K, so is ϕ(V_{ℤ}). Now consider the intersection V_{ℤ} ∩ ϕ(V_{ℤ}). If w, w′ ∈ V_{ℤ} are elements such that w = ϕ(w′), then w = w′ since w = ψ(w) = ψ(ϕ(w′)) = w′. Since ϕ(v_{s}) = v_{s} if and only if s ∈ S, it follows that V_{ℤ} ∩ ϕ(V_{ℤ}) = V_{S}, and hence that V_{S} is homologically benign.

As in [6, IV.7], let L be the free group L = ⟨a, b⟩, and let F be the free group of rank l + 6 with F = ⟨a, b, c_{0}, …, c_{l}, d, e, h⟩. Define a Gödel numbering γ of all words on the alphabet {a, b, a^{−1}, b^{−1}} by the formula

γ(∅) = 0, γ(a) = 1, γ(b) = 2, γ(a^{−1}) = 3, γ(b^{−1}) = 4,

and extending to longer words by concatenation, viewing a concatenation of digits as a number. Thus γ is a bijection between the words and the subset of ℕ consisting of zero and all integers whose decimal digits lie in the set {1, 2, 3, 4}.

To any word w on {a, b, a^{−1}, b^{−1}}, associate a codeword g_{w} ∈ L defined by

g_{w} := whc_{0}^{γ(w)}c_{1}^{γ(w)} ⋯ c_{l}^{γ(w)}de^{γ(w)}.

The subgroup G of F generated by all the elements g_{w} is freely generated by them.

Lemma 8. The subgroup G is benign in F.

Proof. Almost identical to the argument in [6, IV.7]. Make a group F^{∗} defined as the fundamental group of a graph of groups with one vertex group F, and four edges corresponding to stable letters u_{λ} for λ ∈ {a, b, a^{−1}, b^{−1}}, each of which defines an ascending HNN-extension of F with relations

u_{λ}^{−1}hu_{λ} = λh, u_{λ}^{−1}au = a, u_{λ}^{−1}bu = b, u_{λ}^{−1}c_{i}u = c_{0}^{γ(λ)}c_{1}^{γ(λ)} ⋯ c_{i−1}^{γ(λ)}c_{i}^{10}c_{i−1}^{−γ(λ)} ⋯ c_{0}^{−γ(λ)},

u_{λ}^{−1}du = c_{0}^{γ(λ)}c_{1}^{γ(λ)} ⋯ c_{l}^{γ(λ)}dc_{l}^{−γ(λ)} ⋯ c_{1}^{−γ(λ)}c_{0}^{−γ(λ)}, u_{λ}^{−1}eu_{λ} = e^{10}.

In F^{∗}, we have that for any word w = λ_{1} ⋯ λ_{n},

u_{λ_{1}}^{−1} ⋯ u_{λ_{n}}^{−1}g_{∅}u_{λ_{n}} ⋯ u_{λ_{1}} = u_{λ_{1}}^{−1} ⋯ u_{λ_{n}}^{−1}hdu_{λ_{n}} ⋯ u_{λ_{1}} = g_{w},

and if w = uλ then u_{λ}g_{w}u_{λ}^{−1} = g_{u}.

To show that G is benign in F, it suffices to show that in F^{∗},

G = F ∩ ⟨g_{∅}, u_{a}, u_{b}, u_{a^{−1}}, u_{b^{−1}}⟩.

From the equations given above, it is clear that the left-hand side is contained in the righthand side. As in [6, IV.7], to prove the converse it suffices to show that whenever z ∈ G and λ ∈ {a, b, a^{−1}, b^{−1}} are such that u_{λ}zu_{λ}^{−1} ∈ F, then in fact u_{λ}zu_{λ}^{−1} ∈ G, or equivalently z ∈ u_{λ}^{−1}Gu_{λ}. For this, write z = g_{w_{1}}^{ϵ_{1}} ⋯ g_{w_{n}}^{ϵ_{n}} as a reduced word in the elements g_{w}, with ϵ_{i} = ±1. When this expression for z is rewritten in terms of the generators for F and reduced, each subword of the form (c_{1}^{γ(w_{i})}c_{2}^{γ(w_{i})} ⋯ c_{l}^{γ(w_{i})}d)^{ϵ_{i}} survives uncancelled, and any two such subwords are separated by a non-trivial word in the other generators a, b, c_{0}, e, h. Each of the natural free generators for u_{λ}^{−1}F u_{λ} except u_{λ}^{−1}du_{λ} = c_{0}^{γ(λ)}c_{1}^{γ(λ)} ⋯ c_{l}^{γ(λ)}de^{γ(λ)} has total exponent of each c_{i} divisible by 10. From this it follows that each γ(w_{i}) is congruent to γ(λ) modulo 10, and hence that w_{i} = x_{i}λ for some shorter word x_{i}, so that z ∈ u_{λ}^{−1}Gu_{λ} as required.

Corollary 9. Every subgroup of the free group L = ⟨a, b⟩ is homologically benign.

Proof. Let N be a subgroup of L, and define a subset S = S(N) ⊆ ℕ as the set of Gödel codes for words w on {a, b, a^{−1}, b^{−1}} that are equal (as elements of L) to an element of N:

S = {γ(w): w ∈_{L} N}.

Now let Y_{S} be the free product ⟨a, b, h⟩ ∗ V_{S} ≤ F, where V_{S} is as defined in the statement of Lemma 7. By that lemma, V_{S} is homologically benign, and hence Y_{S} is homologically benign in F. Since Y_{S} is freely generated by {a, b, h, v_{s}: s ∈ S}, it is easy to see that G ∩ Y_{S} is freely generated by {g_{w}: w ∈ N}. (Recall that v_{s} = c_{0}^{s}c_{1}^{s} ⋯ c_{l}^{s}de^{s}.) Hence G ∩ Y_{S} is homologically benign. The subgroup generated by G ∩ Y_{S} and the finite set {c_{0}, …, c_{l}, d, e, h}, which is equal to N ∗⟨c_{0}, …, c_{l}, d, e, h⟩, is therefore also homologically benign and the intersection of this group with L is equal to N.

We are now ready to complete the proof of Theorem 1. By the Higman-Neumann-Neumann embedding theorem [4, 6], any countable group can be embedded in a 2generator group. This 2-generator group is isomorphic to L/N for some normal subgroup N. By Corollary 9, N is homologically benign, and so by Lemma 5, L/N can be embedded in an F P_{2} group.

3 Closing remarks

An opinion attributed to Gromov [2, Ch. 1] is that any statement that is valid for every countable group should be trivial. With this in mind, is there an easier, more direct proof of Theorem 1? Is there one that is not modelled on a proof of the Higman embedding theorem and that does not does not rely on Theorem 3, or other results from [1, 5]?

The groups J = J(S) in Theorem 3 may be chosen to have cohomological dimension cd J = 2 in addition to the stated properties. By keeping track of the cohomological dimension at each stage of the argument one obtains the following strengthened version of Corollary 9, and hence a strengthened version of Theorem 1:

Corollary 10. For every subgroup N of the free group L = ⟨a, b⟩, the HNN-extension ⟨L, t: t^{−1}nt = n n ∈ N⟩ embeds in an F P_{2} group of cohomological dimension five.

Theorem 11. Every countable group G embeds in an F P_{2} group G^{∗}, with cd G^{∗} ≤ cd G + 5. Every torsion element in G^{∗} is conjugate to an element of G.

The proof of the Higman embedding theorem in [6, IV.7] shows that every recursively presented group G of finite cohomological dimension embeds in a finitely presented group G^{∗} of finite cohomological dimension. However, cd G^{∗} increases with the complexity of the Diophantine equation used to encode the relators in G. Applying Sapir’s aspherical version of the Higman embedding theorem [7] gives the following.

Theorem 12. For every recursive subgroup N of the free group L = ⟨a, b⟩, the HNNextension ⟨L, t: t^{−1}nt = n n ∈ N⟩ embeds in a finitely presented group of cohomological dimension two.

Combining this with the Higman rope trick [6, IV.7.6] gives a version of the Higman embedding theorem which is an analogue of Theorem 11, but with a better bound on cd G^{∗}.

Theorem 13. Every recursively presented group G embeds in a finitely presented group G^{∗} with cd G^{∗} ≤ cd G + 2. Every torsion element in G^{∗} is conjugate to an element of G.

References

[1] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445–470.

[2] E. Ghys and P. de la Harpe, Panorama, in the book Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser (1990) 1–25.

[3] G. Higman, Subgroups of finitely presented groups, Proc. Roy. Soc. Ser. A 262 (1961) 455–475.

[4] G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949) 247–254.

[5] I. J. Leary, Uncountably many groups of type F P, arXiv:1512.06609v2, 35 pages.

[6] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory (reprint of the 1977 edition), Classics in Mathematics. Springer-Verlag (2001).

[7] M. Sapir, A Higman embedding preserving asphericity, J. Amer Math. Soc. 27 (2014) 1–42.

[8] M. K. Valiev, One theorem of G. Higman, Algebra and Logic 7 (1968) 135–143.

Author’s address:

   i.j.leary@soton.ac.uk

School of Mathematical Sciences,

University of Southampton,

Southampton,

SO17 1BJ