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Arbitrary p-Gradient Values

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable groups that are finitely generated, infinite, torsion, non-amenable, and residually-p.


Introduction
Let G be a finitely generated group and d(G) denote the minimum number of generators of G. Recall the Schreier index formula: Let H be a finite index subgroup of a finitely generated group G. The rank gradient of a finitely generated group is, in a sense, a measure of how far the Schreier index formula is from being an equality rather than an inequality. Though this is an interesting question from a group-theoretic standpoint, Mark Lackenby first introduced the rank gradient as a means to study 3-manifold groups [5].
The absolute rank gradient of G is defined by where the infimum is taken over all finite index subgroups H of G.
This paper is to appear in J. Group Theory It will be evident later that the rank gradient of a group is sometimes difficult both to work with and to calculate. It is often more convenient to compute the rank gradient of the pro-p completion G p of the group G for some fixed prime p. When dealing with profinite groups we use the notion of topologically finitely generated instead of (abstractly) finitely generated. The p-gradient of the group G, denoted RG p (G), can be defined as the rank gradient of G p . The p-gradient is also referred to in the literature as the mod-p rank gradient or mod-p homology gradient. A more explicit definition of p-gradient is provided in Section 2.
Since Lackenby first defined rank gradient of a finitely generated group [5], the following conjecture has remained open: Conjecture. For every real number α > 0 there exists a finitely generated group Γ such that RG(Γ) = α.
The aim of this paper is to prove the analogous question for p-gradient: Main Result. For every real number α > 0 and any prime p, there exists a finitely generated group Γ such that RG p (Γ) = α.
Given a prime p and an α > 0 ∈ R, consider a free group F of finite rank greater than α + 1. Take the set of all residually-p groups that are homomorphic images of F that have p-gradient greater than or equal to α and partially order this set by G 1 G 2 if G 1 surjects onto G 2 . Then by a Zorn's Lemma argument this set has a minimal element, Γ. We show RG p (Γ) = α by contradiction by constructing an element which is less than Γ with respect to the partial order. To construct this new smaller element, Theorem 3.2 is used, which was proved using slightly different language and a different method by Barnea and Schlage-Puchta in [3], but is formulated and proved independently here as well.
The methods used to prove this result require and are similar to those used by Schlage-Puchta in his work on p-deficiency and p-gradient [10] and Osin in his work on rank gradient [7]. To prove the above set has a minimal element, we will use direct limits of groups and show the relationship between the p-gradient of each group in the direct limit and the p-gradient of the limit group. This idea (Lemma 3.5) was inspired by Pichot's similar result for L 2 -Betti numbers [8]. It is known that for a finitely generated, residually finite, infinite group the rank gradient is always greater than or equal to the L 2 -Betti number, which provides a useful relationship between these two group invariants [7].
One of the primary goals of Osin's [7] and Schlage-Puchta's [10] papers was to provide a simple construction of non-amenable, torsion, residually finite groups. The construction given in this paper shows that there exist such groups with arbitrary p-gradient (Theorem 3.7). A simple consequence of this result is that there exist uncountably many pairwise noncommensurable groups that are finitely generated, infinite, torsion, nonamenable, and residually-p. The fact that the groups are non-commensurable uses the p-gradient and is almost immediate from the construction, which shows another way in which the p-gradient can be a useful tool.

Rank Gradient and p-Gradient
In this section, some useful results concerning rank gradient and p-gradient are collected, which will be used to prove the main result. If G is finite, then using H = {1} implies RG(G) = −1 |G| . As the following proposition shows, it is not difficult to produce groups with rational rank gradient. Whether an irrational number can be the rank gradient of some finitely generated group remains an open question. We will show later that for every prime p, every positive real number is the p-gradient for some finitely generated group.
Proposition 2.2. Let q > 0 ∈ Q. There exists a finitely presented group G such that RG(G) = q.
Proof. Write q = m n . Let F m+1 be a non-abelian free group of rank m + 1 and let A be any group of order n. Consider G = F m+1 × A. Let ϕ : G → A be the projection onto the second component and let H = ker ϕ. Then As stated earlier, we can define RG p (G) = RG(G p ). However, a more explicit definition of the p-gradient can be stated.
Definition. Let p be a prime. The p-gradient (also called mod-p homology gradient) of G is defined by where d p (G) = d (G/[G, G]G p ) and the infimum is taken over all normal subgroups H such that [G : H] = p k for some k ∈ Z ≥0 .
We will prove that a group and its pro-p completion have the same pgradient, which will then be used to show the p-gradient of a group equals the rank gradient of its pro-p completion. To get this result, some facts about profinite groups must be presented.
Let G be a finitely generated group. The pro-p completion of G for some prime p will be denoted by G p . Let d(G) denote the minimal number of abstract generators of a group G if the group is not profinite and the minimal number of topological generators if the group is profinite. If a group is profinite, the term "finitely generated" will be used to mean topologically finitely generated. The reader is referred to any standard text in profinite groups for the basic results used in this section [4], [11].
When dealing with pro-p completions of a group, it is often convenient to assume that the group is residually-p since the group will imbed in its pro-p completion. To show why this type of assumption will not influence any result about the p-gradient, the following lemma is given.
Definition. Let G be a group and p a prime. Let N , the p-residual of G, be the intersection of all normal subgroups of p-power index in G. The p-residualization of G is the quotient G/N . Note that the p-residualization of G is isomorphic to the image of G in its pro-p completion G p and is residually-p.

Lemma 2.3. Let G be a group and p a prime number. Let
Proof. There is a bijective correspondence between normal subgroups of ppower index in G and normal subgroups of p-power index in G. Let the By the inverse limit definition of pro-p completion and the fact that The following fact is well known. With the following proposition, we will be able to prove that a group and its pro-p completion have the same p-gradient. Parts of this proposition can be found in an exercise in [4].
Proposition 2.5. Let G be a finitely generated group and p a prime. Let ϕ : G → G p be the natural map from G to its pro-p completion. Let H be a normal subgroup of p-power index of G. The following hold: 4) For notational simplicity, assume G is residually-p and thus ϕ is injective. The case of G not residually-p is proved similarly. It is only necessary to show that the pro-p topology on G induces the pro-p topology on the subspace H of G. By Proposition 2.4, subnormal subgroups of p-power index in G form a base for the pro-p topology on G. If K is subnormal of p-power index in H it implies that K is subnormal of p-power index in G. This implies that the subspace topology and the pro-p topology on H are the same. Therefore, H ≃ H p as pro-p groups. (2) and (4)

5) By
It is now possible to prove the relationship between the p-gradient of a group and its pro-p completion.
Theorem 2.7. Let G be a finitely generated group and p a fixed prime. Let G p be the pro-p completion of G. Then RG p (G) = RG p (G p ) = RG(G p ).
Proof. Assume that G is residually-p. By Proposition 2.5 the natural injective map ϕ : G → G p induces an index preserving bijection H → H ≃ H p between the normal subgroups of p-power index in G and the normal subgroups of p-power index in G p . Proposition 2.5 also implies that d p (H) = d p (H p ) for all p-power index normal subgroups H G. It is not difficult to show that d p (K) = d p (K p ) holds for any finitely generated group K. Therefore, RG p (G) = RG p (G p ). If G is not residually-p, let G be the p-residualization of G. By Lemma 2.3, RG p (G) = RG p ( G) and G p ≃ G p . The first equality follows.
The fact that RG p (G) = RG(G p ), where G p is the pro-p completion of G, follows by the above remarks and Theorem 2.6.
Remark. Nikolov and Segal proved Serre's conjecture on finitely generated profinite groups. That is, in a finitely generated profinite group all finite index subgroups are open [6].
The above two theorems provide some useful corollaries.
Proof. If G is finite, then so is G p and thus RG p (G) = RG p (G p ) = RG(G p ) = − 1 |G p | by Theorem 2.1. Proof. If H in normal in G we are done by Proposition 2.5.5. If H is not normal, then induct on the subnormal length.

Groups With Arbitrary p-Gradient Values
In this section we will prove the main result, that is, we construct a finitely generated group Γ with RG p (Γ) = α for each α > 0 ∈ R. To prove this, we need some technical results.
The following lemma is similar to Lemma 2.3 of Osin [7] concerning deficiency of a finitely presented group.  1. This is a standard computation.  m . A lower bound for the p-gradient when taking the quotient by the normal subgroup generated by an element raised to a p-power follows by the above lemma.

Therefore, a presentation for π(H)/([π(H), π(H)]π(H) p ) is obtained from a presentation for H/([H, H]H p ) by adding in
Theorem 3.2. Let G be a finitely generated group, p some fixed prime, and x ∈ G. Then RG p (G/ x p k ) ≥ RG p (G) − 1 p k . Proof. Case 1: There exists a normal subgroup H 0 of p-power index such that the order of x in G/H 0 is at least p k .
Since H 0 is a normal subgroup of p-power index, then without loss of generality we may assume that the order of x in G/H 0 is exactly p k . Let H be a normal subgroup of p-power index in G = G/ x p k . Let H ≤ G be the full preimage of H. Then H is a p-power index normal subgroup in G which contains x p k . Let L H = H ∩ H 0 . Then L H is a normal subgroup in G such that x p k ∈ L H , L H ⊆ H, and the order of x in G/L H is p k . Note that L H is normal and of p-power index in G since both H and H 0 are normal and of p-power index. Thus by Lemma 3.1, q(H) ≥ q(L H ) ≥ q(L H ) − 1 p k , which by definition is greater than or equal to RG p (G) − 1 p k . Therefore, It will be shown that RG p (G/ x p k ) = RG p (G) in this case. There exists an ℓ < k such that x p ℓ ∈ H for every normal subgroup H of ppower index in G. Then x p ℓ is in the kernel of natural map from G to its pro-p completion ϕ : G → G p . Therefore, Remark. The above theorem was independently stated and proved using slightly different language and a different method by Barnea and Schlage-Puchta (Theorem 3 in [3]). Corollary 3.3. Let G be a finitely generated group, p a fixed prime, and let x ∈ G. Then RG p (G/ x ) ≥ RG p (G) − 1.

p-Gradient and Direct Limits
Let (I, ≤) be a totally ordered set with smallest element 0 and let {G i | π ij } be a direct system of finitely generated groups with surjective homomorphisms π ij : G i → G j for every j ≥ i ∈ I.
Let G ∞ = lim − → G i be the direct limit of this direct system. Let π i : G i → G ∞ be the map obtained from the direct limit. Because all the maps in the direct system are surjective, then so are the π i . Let G = G 0 .
Another direct system {M i | µ ij } can be defined over the same indexing set I, where M i = G for each i and µ ij is the identity map. The direct limit of this set is clearly G = lim − → M i and the map obtained from the direct limit is finitely generated abelian and torsion and therefore is finite. Thus Q(H ′ ) can only have finitely many non-isomorphic subgroups. Since {M i } is an ascending set of subgroups, there must exist an n ∈ I such that M i = M n for every i ≥ n. Since ker ϕ i ⊆ ker ϕ j for each j ≥ i and ker ϕ i = ker ϕ, we know that M i ⊆ M j for every j ≥ i and The following lemma is similar to Pichot's related result for L 2 -Betti numbers where convergence is in the space of marked groups [8].
Proof. Fix a prime p. Let K G ∞ be a normal subgroup of p-power index. By Lemma 3.4 we obtain the subgroups H ′ and H ′ i for each i. Now, and by Lemma 3.4 Therefore, for each K G ∞ of p-power index, lim sup RG p (G i ) ≤ dp(K)−1

The Main Result
It is now possible to prove the main result that every nonnegative real number is realized as the p-gradient of some finitely generated group. Proof. Fix a prime p and a real number α > 0. Let F be the free group on ⌈α⌉ + 1 generators. Let Λ = {G | F surjects onto G, G is residually-p, and RG p (G) ≥ α}.
Since for any free group d(F ) = d p (F ) it is clear that RG p (F ) = rank(F )− 1 and therefore, Λ is not empty since F is in Λ. Λ can be partially ordered by G 1 G 2 if there is an epimorphism from G 1 to G 2 , denoted G 1 ։ G 2 . This order is antisymmetric since each group in this set is Hopfian.
Let C = {G i } be a chain in Λ. Each chain forms a direct system of groups over a totally ordered indexing set. Any chain can be extended so that it starts with the element Thus G ∞ ∈ Λ and G i G ∞ for each i. Thus, each chain C in Λ has a lower bound in Λ and therefore by Zorn's Lemma, Λ has a minimal element, call it Γ.
Since Γ and its p-residualization Γ have the same p-gradient and Γ surjects onto Γ, it implies that Γ ∈ Λ and Γ Γ. Thus Γ must be residually-p, otherwise Γ contradicts the minimality of Γ. Note: Γ does not have finite exponent.
If Γ had finite exponent then since Γ is finitely generated and residually finite it must be finite by the positive solution to the Restricted Burnside Problem [12]. This would imply RG p (Γ) < 0 by Corollary 2.8. This contradicts that Γ is in Λ.
Assume not. Then there exists a k ∈ N such that RG p (Γ) − 1 p k ≥ α. Since Γ is residually-p, the order of every element is a power of p and since Γ has infinite exponent, there exists an x ∈ Γ whose order is greater than p k .
If Γ ′ is not residually-p, replace it with its p-residualization, which will have the same p-gradient. Then Γ ′ ∈ Λ and Γ Γ ′ , which contradicts the minimality of Γ.
The result of Theorem 3.6 can be strengthened without much effort.
Theorem 3.7. Fix a prime p. For every real number α > 0 there exists a finitely generated residually-p torsion group Γ such that RG p (Γ) = α.
Proof. Barnea and Schlage-Puchta showed in Corollary 4 of [3], that for any α > 0 there exists a torsion group G with RG p (G) ≥ α. Applying the construction in Theorem 3.6, replacing the free group F with the presidualization of G, will result in a group Γ that is torsion, residually-p, and RG p (Γ) = α.
Y. Barnea and J.C. Schlage-Puchta [3] proved a result similar to Theorem 3.7 (inequality instead of equality) albeit in a slightly different way.

Applications
The construction given in Theorem 3.6 has a few immediate applications. First, it is noted that Theorem 3.7 gives a known counter example to the General Burnside Problem. The second application is more general and shows that there exist uncountably many pairwise non-commensurable groups that are finitely generated, infinite, torsion, non-amenable, and residuallyp.
This paper has been concerned with the (absolute) rank gradient and pgradient. There is, however, a related notion of rank gradient and p-gradient of a group with respect to a lattice of subgroups. A set of subgroups, {H i }, is called a lattice if the intersection of any two subgroups in the set is also in the set. In particular, a descending chain of subgroups is a lattice.

Definition.
1. The rank gradient relative to a lattice {H i } of finite index subgroups is defined as RG(G, 2. The p-gradient relative to a lattice {H i } of normal subgroups of p- The following theorem was proved by Abert, Jaikin-Zapirain, and Nikolov in [1]. Lackenby first proved the result for finitely presented groups in [5]. As a simple corollary, we provide a corresponding, albeit weaker, result concerning p-gradient. Proof. Let G be a finitely generated group with RG p (G) > 0. Let G be the p-residualization of G. Then 0 < RG p (G) = RG p ( G). Let {H i } be a descending chain of normal subgroups of p-power index in G which intersect in the identity. Then, Therefore, G is not amenable by Theorem 4.1. This implies that G is not amenable since a quotient of an amenable group is amenable.
The application of the construction used in Theorem 3.6 concerning commensurable groups is given below.
Definition. Two groups are called commensurable if they have isomorphic subgroups of finite index.
The following lemma is straightforward. Theorem 4.4. There exist uncountably many pairwise non-commensurable groups that are finitely generated, infinite, torsion, non-amenable, and residuallyp.
Proof. Let p be a fixed prime number. By Theorem 3.7 it is known that for every real number α > 0 there exists a finitely generated residually-p infinite torsion group, Γ, such that RG p (Γ) = α. By Corollary 4.2 these groups are all non-amenable. Since each of these groups is residually-p and torsion, they are all p-torsion. Thus, every subgroup of finite index in these groups is subnormal of p-power index.
By Theorem 2.9 if any two of these groups are commensurable, then the p-gradient of each group is a rational multiple of the other. Since there are uncountably many positive real numbers that are not rational multiples of each other, the result can be concluded.