1 Introduction

# Structure of normally and finitely non-co-Hopfian groups

## Abstract.

A group is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to . We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two applications: First, we show any characteristic proper finite-index subgroup isomorphic to arises by pulling back a finite-index subgroup of the abelianization, and secondly, we prove special cases (for normal subgroups) of conjectures of Benjamini and Nekrashevych-Pete regarding the classification of scale-invariant groups.

## 1. Introduction

### 1.1. Main result

A group is co-Hopfian if it does not contain any proper subgroup isomorphic to itself. A classification of groups that are not co-Hopfian seems extremely difficult because a free group is not co-Hopfian, and similarly, any nontrivial free product is not co-Hopfian. Indeed, contains the subgroup which is isomorphic to but is a proper subgroup whenever . See e.g. [Bel03, DP03, KW01] and references therein for further information on and examples of co-Hopfian groups.

Following [BGHM10], we say a group is finitely co-Hopfian if does not contain any proper finite-index subgroup isomorphic to . The finite co-Hopf property seems far more tractable than the co-Hopf property in general, at least when is finitely generated. Indeed it seems failure of the finite co-Hopf property is closely related to the presence of nilpotent subgroups. De Cornulier has studied which finitely generated nilpotent groups are finitely non-co-Hopfian [Cor16].

In part this is suggested by an analogous problem of topological nature: Namely, if is a closed manifold, we say is (topologically) self-covering if there exists a finite cover with degree greater than 1 and such that is diffeomorphic (homeomorphic) to . If is self-covering, then is not finitely co-Hopfian.

Historically, a situation of particular interest has been the classification of expanding maps. A map is expanding if there is a Riemannian metric on such that for any unit tangent vector . Franks observed that if admits an expanding map, then has polynomial growth [Fra70]. Gromov proved that a group with polynomial growth is virtually nilpotent, and used this to prove that any manifold admitting an expanding map is infranil [Gro81]. See [Van] for more information on self-covering manifolds.

Let us now return to the group-theoretical setting. Finitely non-co-Hopfian groups also have an obvious connection to profinite groups: Indeed if is a proper finite-index isomorphic to , then one has a chain

 Γ=Γ0⊋Γ1⊋Γ2⊋… (1.1)

of finite-index subgroups of , each of which is isomorphic to . If is in addition finitely generated, then for every there is a finite-index normal subgroup of contained in . It follows that the chain (1.1) is detected in the profinite completion of . In this case, work of Reid easily implies that the profinite completion contains a nontrivial pronilpotent normal subgroup [Rei14] (see Theorem 3.10 and Proposition 3.12).

We give a classification of finitely generated finitely non-co-Hopfian groups in the case that we can take , i.e. if the groups that are isomorphic to , are already normal. The obvious examples are given by free abelian groups. Our main result is that any example arises in this way:

###### Theorem 1.1.

Let be a finitely generated group that admits a decreasing chain of subgroups

 Γ=Γ0⊋Γ1⊋Γ2⊋…

where are finite-index normal subgroups with . Set . Then is nilpotent and, modulo torsion, it is free abelian.

###### Remark 1.2.

Equivalently, is nilpotent and virtually abelian (see Proposition 3.16).

In light of Theorem 1.1, it is natural to ask:

###### Question 1.3.

Can one classify finitely non-co-Hopfian groups in terms of nilpotent groups in a manner similar to Theorem 1.1?

In any example we know of, a finitely non-co-Hopfian group admits a quotient that contains a nontrivial normal nilpotent subgroup. It seems plausible that this is the case for any finitely non-co-Hopfian group because, as already remarked above, its profinite completion contains a closed pronilpotent normal subgroup. In general the normal nilpotent subgroup can only be found in a quotient and might be a proper subgroup and not finitely generated (e.g. in the case of a solvable Baumslag-Solitar group, see Example 2.2).

Let us now return to the setting of Theorem 1.1. It is immediate from Theorem 1.1 that any chain of subgroups as in Theorem 1.1 comes from pullback of a chain of subgroups of a virtually abelian group:

###### Corollary 1.4.

Let be finitely generated and suppose admits a decreasing chain of of finite-index normal subgroups with . Then there exist

• a finitely generated nilpotent group such that is abelian,

• a surjection , and

• finite-index normal subgroups ,

such that for every .

Further, it is not necessarily true that is torsion-free (see Example 2.1). However, in this example the torsion in already appears in but does not increase thereafter. This is a general phenomenon: Since the torsion subgroup of any finitely generated nilpotent group is finite, and the chain of subgroups of in Corollary 1.4 is decreasing with trivial intersection, we see that for , the group is free abelian. Hence the torsion can be avoided by passing to a finite-index subgroup of that is compatible with the chain of subgroups:

###### Corollary 1.5.

Let be a finitely generated group that admits a chain of finite-index normal subgroups with . Set . Then for , the group is free abelian.

In particular admits a surjection onto a free abelian group such that there are finite-index subgroups with for any .

###### Remark 1.6.

In general, one may not be able to realize the free abelian group as the factor of any finite-index subgroup of (see Example 2.2). In that example, is given by a semi-direct product , and the self-embedding respects this decomposition. We do not know whether it is always the case that a finite-index subgroup of decomposes as a semi-direct product.

###### Remark 1.7.

In [Van], we studied a topological analogue of a special case of Theorem 1.1, namely self-covers of a closed manifold such that every iterate is a regular cover. We proved that any such self-cover is induced by a linear endomorphism of a compact torus in the following sense: There is a map such that is homotopic to . This is proved using a mixture of the theory of topological transformation groups and structural results for locally finite groups. Unfortunately, neither of these is applicable in the setting of Theorem 1.1.

### 1.2. Characteristic self-embeddings

Recall that a subgroup of is characteristic if it is invariant under all automorphisms of . In particular, a characteristic subgroup is invariant under all inner automorphisms, and hence it is normal.

Obvious examples of groups with characteristic finite-index subgroups isomorphic to are given by free abelian groups. The following result shows that all examples come from free abelian groups.

###### Corollary 1.8.

Let be a finitely generated group. If admits a finite-index characteristic copy of itself, then there is a free abelian group with characteristic finite-index subgroup , and a surjection such that . In particular, we have .

###### Remark 1.9.

Note that Corollary 1.8 applies as soon as one has a single finite-index characteristic subgroup isomorphic to , whereas in the setting of normal subgroups, we need a chain for Theorem 1.1 to hold.

### 1.3. Scale-invariant groups

Benjamini proposed the following algebraic analogue to the existence of an expanding map on a closed manifold [Ben06].

###### Definition 1.10 (Benjamini).

A finitely generated group is scale-invariant if there exists a nested decreasing sequence

 Γ=Γ0⊇Γ1⊇Γ2⊇…

such that for all , and is finite.

###### Remark 1.11.

The motivation for the introduction of scale-invariant groups comes from percolation theory on graphs: In the traditional setting of percolation, the underlying graph is the grid , and one has access to the very powerful renormalization method. Unfortunately this does not generalize to the Cayley graph of a general group , but it seems plausible that renormalization techniques generalize to Cayley graphs of scale-invariant groups. See [NP11] for more information on the relationship between percolation theory and scale-invariant groups.

Recall that Gromov proved that any closed manifold admitting an expanding map is infranil, and in particular is virtually nilpotent [Gro81]. Similarly, Benjamini conjectured scale-invariant groups are virtually nilpotent [Ben06]. Many counterexamples to this conjecture were produced by Nekrashevych-Pete [NP11]:

###### Theorem 1.12 (Nekrashevych-Pete [Np11]).

There exist scale-invariant groups that are not virtually nilpotent. Indeed, if is a scale-invariant group with a nested decreasing sequence as in Definition 1.10, and is a group of automorphisms such that each is -invariant, then is scale-invariant.

For example, for any subgroup , the group is scale-invariant.

In these examples, the sequence of subgroups is never generated by a single source, such as an expanding map. To recapture this aspect, Nekrashevych-Pete proposed the following definition:

###### Definition 1.13 (Nekrashevych-Pete [Np11]).

A finitely generated group is strongly scale-invariant if there exists an embedding with image of finite index and such that is finite.

Any strongly scale-invariant group is clearly scale-invariant. However, the counterexamples to Benjamini’s conjecture produced in Theorem 1.12 are never strongly scale-invariant. Indeed, these groups have a self-similar action on a rooted tree to produce, and the subgroups are constructed by fixing a suitable geodesic ray and taking to be the vertex stabilizer at the th level. The self-similarity of the action implies that and since these trees have finite degree, it follows that .

In their examples, it is essential that the geodesic ray is aperiodic. This implies that there is no embedding such that . Based on this observation, Nekrashevych-Pete proposed the following variant of Benjamini’s conjecture:

###### Conjecture 1.14 (Nekrashevych-Pete [Np11]).

Any strongly scale-invariant group is virtually nilpotent.

We observe that Theorem 1.1 implies both Benjamini’s conjecture and Nekrashevych-Pete’s conjecture 1.14 under the additional assumption that are all normal:

###### Corollary 1.15.

Let be a finitely generated group that admits a descending chain of finite-index normal subgroups such that and is finite. Then is virtually abelian.

### 1.4. Outline of the proof of Theorem 1.1

Let be a finitely generated group with a descending chain of normal finite-index subgroups, each of which is isomorphic to . We start by observing that the profinite completion of admits proper open self-embeddings (Proposition 3.3). Work of Reid implies that contains a normal pronilpotent subgroup, and that each of the finite groups is nilpotent. Most of the proof is devoted to obtaining a uniform bound on the nilpotency class of .

To this end, we first prove Theorem 1.1 assuming is nilpotent. This is done in Section 4, and relies on Lie theoretic considerations.

In the general case, we write the nilpotent group as a product of -groups where runs over the set of primes. Hence the problem of uniformly bounding the nilpotency class separates out into a problem over each prime . In Section 5, we give a uniform bound at almost every prime. Roughly, the idea is to apply the nilpotent case of Theorem 1.1 to suitable nilpotent quotients of and conclude that modulo torsion, these are abelian. We will show the torsion is concentrated at a finite set of primes, away from which this argument yields the desired uniform bound on nilpotency class.

Finally in Section 6 we obtain a bound on nilpotency class of the localization of at a fixed prime . Combined with the above bound at almost every prime, this yields a uniform bound on the nilpotency class of . We finish the proof by once again appealing to the nilpotent case of the main theorem (Section 7). Finally we establish the applications to characteristic finite-index subgroups and scale-invariant groups at the end of Section 7.

### Acknowledgments:

I would like to thank Ralf Spatzier for countless helpful discussions. I am thankful to David Fisher for making me aware of the notion of scale-invariant groups and the results of [NP11].

## 2. Two examples

In this section we discuss two examples. The first shows that in Theorem 1.1, the quotient is not necessarily abelian or torsion-free.

###### Example 2.1.

Let be the three-dimensional integer Heisenberg group, i.e. the group with presentation

 Δ:=⟨x,y,z∣[x,y]=z,[x,z]=[y,z]=e⟩.

Write for the dilation defined by and and . It is easy to see that is not normal in . However, set . Then is a normal subgroup and is nonabelian.

Now define and set for any . Then form a chain of finite-index normal subgroups of , each of which is isomorphic to , and we have

 Γ/Γ∞≅(Λ/φ(Λ))×Z.

In the above example has a free abelian factor. However, in general, even though is virtually free abelian, we may not be able to realize it as a factor of any finite-index subgroup of , as the next example shows.

###### Example 2.2.

Let and consider the corresponding Baumslag-Solitar group

 Λ:=⟨x,t∣txt−1=xd⟩.

Take two copies of with generators . Define an automorphism of the free product by

 φx(gi):=xigix−1i

whenever for . Likewise define the automorphism . Now consider the group

 Γ:=(Λ1∗Λ2)⋊φxZ.

Write for the generator of that acts by . Finally, define by and . Using that it is easy to verify that is a homomorphism. For any , set . We have

 Γk=(Λ1∗Λ2)⋊dkZ⊆Γ.

Hence the group admits a decreasing chain of finite-index normal subgroups, each of which is isomorphic to . However, it is easy to see has no free abelian factor.

## 3. Preliminaries

### 3.1. Profinite groups

We briefly review basic definitions and facts related to profinite groups, and establish notation. For a more thorough discussion, see for example [RZ10, Wil98]. A group is profinite if it is the inverse limit of an inverse system of finite groups. By equipping each of the finite groups with the discrete topology, we obtain the profinite topology on .

The profinite topology makes into a compact Hausdorff space. A local base for the profinite topology of at the identity is given by open subgroups of finite index. This gives rise to a useful finiteness property for profinite groups.

###### Definition 3.1.

A profinite group is said to be of type (F) if for every , there are only finitely many open subgroups of of index .

If is any group, we denote by the profinite completion of . There is a natural map with dense image. The map is universal for maps of to profinite groups: If is any profinite group and is a homomorphism, then there is a unique homomorphism such that .

We will be particularly interested in finitely generated groups . Even though an infinite profinite group is never finitely generated (because it is uncountable), there is a useful concept of finite generation:

###### Definition 3.2.

We say a profinite group is finitely generated if is topologically finitely generated, i.e. there exists a finite set such that is a dense subgroup of .

Any finitely generated profinite group is of type (F), see e.g. [RZ10, 2.5.1]. Further, because the map has dense image, we see that if is finitely generated, then is finitely generated as a profinite group.

The universal property of the profinite completion has the following consequence in the context of finitely non-co-Hopfian groups:

###### Proposition 3.3.

Let be a finitely generated group and suppose is a finite-index subgroup of with . Fix such an isomorphism and consider the composition

 φ:Γ→Λ↪Γ.

Then there exists a unique open embedding with .

###### Proof.

By the universal property of profinite completions, the composition

 φ:Γ≅⟶φ(Γ)↪Γ

(where the first map is ) gives rise to a map

 ˆφ:ˆΓ≅⟶ˆφ(Γ)→ˆΓ.

Here the first map is an isomorphism of profinite groups and in particular a homeomorphism. Therefore we only need to show the second map is an open embedding. But if is any finitely generated group and is a finite-index subgroup of , then the map (induced by inclusion) is an open embedding. This is immediate from the fact that there is a characteristic finite-index subgroup such that . ∎

### 3.2. Contraction groups

We are therefore led to study open self-embeddings of profinite groups. To give a good description of these, we will first recall the notion of a contraction groups, originally introduced by Müller-Römer [MR76]. Here we will follow [BW04], who give a slightly different definition that is less general.

###### Definition 3.4.

Let be a locally compact topological group and be an automorphism. Then the pair is a contraction group if for any , we have .

We have the following three essential examples:

###### Example 3.5.

1. Let . Then multiplication by some with is a contraction.

2. Let . Then multiplication by is a contraction.

3. Let be a prime and consider the ring of formal Laurent series over . We view as an additive group. Then multiplication by is a contraction.

The classification of contraction groups began with the work of Siebert [Sie89], who separated the classification into a problem for connected groups and totally disconnected groups, and completely classified connected contraction groups. In the connected case, is a simply-connected real unipotent Lie group (compare Example 3.5.(1)).

The totally disconnected case was further developed by the work of Baumgartner-Willis [BW04] and solved by Glöckner-Willis [GW10].

###### Theorem 3.6 (Glöckner-Willis [Gw10]).

Let be a totally disconnected contraction group. Then the set of torsion elements Tor and the set of divisible elements are -invariant closed subgroups of , and

 G≅Tor(G)×Div(G).

Further the divisible part is described as follows: there exists a finite set of primes and unipotent -adic Lie groups such that

 Div(G)≅∏iUpi.

### 3.3. Endomorphisms of profinite groups

The above work on contraction groups was used in the context of open self-embeddings of profinite groups by Reid [Rei14]. However, note that if is a contraction group, then is never compact. Therefore we make the following analogous definition in the setting of compact groups:

###### Definition 3.7.

Let be a compact topological group and be a morphism. We say is contracting if for any , we have .

For the totally disconnected contraction groups of Examples 3.5.(2) and (3), we can take compact subgroups that are preserved by and restricts to a contracting endomorphism of , e.g. in Example 3.5.(2) and in Example 3.5.(3). In the case of open contracting embeddings, it is straightforward that one can also recover the contraction group from the embedding of the compact subgroup:

###### Proposition 3.8.

Let be a compact topological group and be an open contracting embedding. Set

 H:=lim−→(Gα↪Gα↪…).

Then is a locally compact group and naturally induces an automorphism of such that is a contraction group.

###### Remark 3.9.

Instead of constructing as an inductive limit, we can also consider the ascending HNN-extension

 G∗α:=⟨G,t∣∀g∈G:tgt−1=α(g)⟩.

The group is then obtained as

 H=⋃n≥0t−nGtn.

Using the theory of contraction groups, we have the following result by Reid that describes open embeddings of finitely generated profinite groups.

###### Theorem 3.10 (Reid [Rei14]).

Let be a profinite group of type (F) and an open embedding. Then there exist closed subgroups and of such that

• ,

• is -invariant and restricts to an open contracting embedding on ,

• We can write where is a compact open subgroup of a product of finitely many -adic unipotent Lie groups (for finitely many primes ) and is a bounded exponent solvable group that is residually nilpotent, and

• is -invariant and restricts to an automorphism on .

###### Remark 3.11.

and are explicitly given in terms of , namely we have

 C:={g∈G∣ˆφn(g)n→∞⟶e}

and

In fact Reid proves a more general version of the decomposition of Theorem 3.10, where one can considers a collection of endomorphisms . We will need a stronger conclusion than Theorem 3.10 provides in the restricted case of a single endomorphism, which however is immediate from the proof of Theorem 3.10 in [Rei14].

To state this stronger version, recall that a group is pronilpotent if it is the inverse limit of a system of finite nilpotent groups. In particular, any pronilpotent group is residually nilpotent. The stronger version of Theorem 3.10 that we need is as follows.

###### Proposition 3.12.

Let be a profinite group of type (F) and an open embedding. Let be as constructed in Theorem 3.10. Then is pronilpotent.

###### Proof.

This slightly stronger result is immediate from Reid’s proof of [Rei14, Theorem 4.3.(i)]. Indeed, after the reduction to the case , Reid proceeds to construct a sequence of finite-index open normal subgroups with and such that is nilpotent for each . Hence is pronilpotent.∎

### 3.4. Nilpotent groups

The previous result leads us from open embeddings of profinite groups to pronilpotent groups. To exploit this connection later, we will now recall some classical facts about nilpotent groups. Recall that the lower central series of a group is inductively defined by and for any . We say is nilpotent of class if is the first term of the lower central series such that . We say is nilpotent if is nilpotent of class for some .

We start by recalling the following elementary result that gives a splitting of a finite nilpotent group over primes.

###### Proposition 3.13 (See e.g. [Rob96, 5.2.4]).

Let be a finite nilpotent group. Then

1. For any prime dividing , there exists a unique -Sylow subgroup of , and

2. We have , where the product is over all primes.

The corresponding result holds for profinite groups:

###### Proposition 3.14 (see e.g. [Rz10, 2.3.8]).

Let be a pronilpotent group. Then for any prime , there exist closed pro subgroups of such that

 G≅∏pG(p),

where the product is over all primes.

We continue by studying finitely generated nilpotent groups. The following result allows us to reduce to the torsion-free case.

###### Proposition 3.15 (See e.g. [Rob96, 5.2.7]).

Let be a finitely generated nilpotent group. Then the set of torsion elements is a finite normal subgroup of .

Therefore the study of a finitely generated nilpotent group breaks up into studying its torsion subgroup and the torsion-free part . Occasionally it will be useful to us to work with a finite-index torsion-free subgroup rather than the torsion-free quotient . We can do so by the following result, which follows immediately from Proposition 3.15 and the fact that nilpotent groups are residually finite, a result of Gruenberg (see e.g. [Rob96, 5.2.21]):

###### Proposition 3.16.

Let be a finitely generated nilpotent group. Then is virtually torsion-free. In particular, is abelian if and only if is virtually abelian.

The theory of finitely generated torsion-free nilpotent groups is intimately connected to Lie theory because of the following result of Mal’cev.

###### Theorem 3.17 (Mal’cev (see e.g. [Rag72, Cor. 2 of 2.11 and 2.18])).

Let be a finitely generated torsion-free nilpotent group. Then there exists a unique simply-connected nilpotent Lie group such that there is an embedding as a cocompact lattice.

Given a finitely generated torsion-free nilpotent group , we say the simply-connected nilpotent Lie group given by Theorem 3.17 is the Lie hull of . The Lie hull of captures an incredible amount of the algebraic structure of . One of the strongest instances is the following superrigidity result of Mal’cev, showing that homomorphisms of to other simply-connected nilpotent Lie groups are controlled by .

###### Theorem 3.18 (Mal’cev (see e.g. [Rag72, 2.17])).

Let be a finitely generated torsion-free nilpotent group with Lie hull . Then for any simply-connected nilpotent Lie group and any homomorphism , the map uniquely extends to a homomorphism .

## 4. Nilpotent case of the main theorem

The goal of this section is to prove Theorem 1.1 under the additional assumption that is nilpotent, i.e. the following result.

###### Theorem 4.1.

Suppose is a finitely generated and nilpotent group and admits a descending chain of finite-index normal subgroups with . Set . Then modulo its torsion, is abelian.

This result will be used several times in the subsequent sections to establish the general version of Theorem 1.1. Before starting the proof, we provide a brief outline:

First we reduce the statement to the case of torsion-free nilpotent groups (Step 1 below). Any such group is a lattice in its Lie hull . Using that the Lie hull is the same for and its subgroup , we obtain automorphisms of with (Step 2).

Then are cocompact lattices in contained in . This allows us to consider an (archimedean) limit of , namely . Contrary to the profinite limit , we can control because it is a compact nilpotent Lie group. We use this to show that have normal abelian subgroups of uniformly bounded index (Step 3). Finally we conclude from this that is abelian modulo torsion (Step 4).

### Step 1 (reduction to torsion-free case)

We claim that it suffices to establish the theorem for torsion-free nilpotent groups. Indeed, let be any finitely generated group with a descending chain of finite-index normal subgroups with . Choose such isomorphisms and view these as embeddings (with image ). Recall that the set of form a finite normal subgroup, and likewise for (see Proposition 3.15).

For any , we clearly have

 φk(Tor(Γ))⊆Tor(Γk)⊆Tor(Γ). (4.1)

On the other hand, since , we have . Since is finite, both inclusions in Equation 4.1 are equalities. Hence descends to an embedding . Further note that is torsion-free and nilpotent, and if it is abelian modulo its torsion, then so is .

For the remainder of the proof, we assume satisfies the hypotheses of Theorem 4.1 and is torsion-free. We choose isomorphisms as above, and view them as self-embeddings with image .

### Step 2 (constructing automorphisms of the Lie hull)

Since is finitely generated, torsion-free and nilpotent, it admits a Lie hull , i.e. the unique simply-connected nilpotent Lie group such that embeds into as a cocompact lattice (see Theorem 3.17). Henceforth we will identify with its image in .

By Mal’cev’s Superrigidity Theorem 3.18, the embeddings uniquely extend to continuous homomorphisms , which we will also denote by . We claim each is an automorphism of .

Indeed, since is of finite index in , we know that is also a lattice in . Therefore we can also apply Mal’cev’s Superrigidity Theorem 3.18 to the inverse map

 φ−1k|φk(Γ):φk(Γ)→Γ.

The uniqueness in Mal’cev’s superrigidity theorem implies that the extension of the inverse is inverse to . Hence extends to an automorphism of .

### Step 3 (control on Γ/Γk)

We claim that there exists such that for any , the group has a normal abelian subgroup of index at most .

To see this, set (as a subgroup of ), where . Set

 H:=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⋃k≥0Γ−k.

Since is cocompact and is contained in the closed subgroup , we must have that is also cocompact in . Further since is a normal subgroup of for any , we have that is a normal subgroup of for any . Since is closed, it follows that also normalizes .

Hence is a compact nilpotent Lie group containing for any . Note that has finitely many connected components and its identity component is a compact connected nilpotent group and hence is a torus. Let be the number of connected components of .

Let and consider the composition

 f:Γ/Γkφ−1k⟶Γ−k/Γ⊆H/Γ=K↠K/K0.

Note that has index at most . Further embeds into the compact torus (by omitting the last map in the above composition) and hence is abelian. This proves the claim.

### Step 4 (end of the proof)

We claim that is virtually abelian, which will finish the proof (using Proposition 3.16). Let be as in Step 3. Since is finitely generated, it has only finitely many subgroups of index at most . Define to be the intersection of all of these subgroups. We will show that the image of in is abelian.

To see this, note that is a finite-index normal subgroup of with the following property: Whenever is a finite group of order at most and is a homomorphism, .

Hence for any , the image of the composition

 Λ→¯¯¯¯Γ→Γ/Γkφ−1k⟶Γ−k/Γ↪K

is contained in (because maps trivially to ). We conclude that the image of in is abelian, so that for all . Hence

 [Λ,Λ]⊆∩kΓk=Γ∞.

Therefore the image of in is abelian, as desired.∎

## 5. Almost global bound on nilpotency class

We will now start the proof of the general case of Theorem 1.1. Let be a finitely generated group with a descending chain of finite-index normal subgroups such that . We start by using the structure of the profinite completion of to obtain information about the finite groups .

###### Proposition 5.1.

is nilpotent for every .

###### Proof.

Fix isomorphisms and view these as self-embeddings

 φk:Γ↪Γ

with image . These maps induce

 ˆφk:ˆΓ→ˆΓ,

on the profinite completion of . By Proposition 3.3, the map is an open embedding for each . By Reid’s work on open self-embeddings of profinite groups (see Theorem 3.10), there are closed subgroups and such that

 ˆΓ=Ck⋊Qk

and restricts to a contracting endomorphism of and an automorphism of . In particular, we have

 Γ/Γk≅ˆΓ/ˆφk(ˆΓ)=Ck/ˆφk(Ck).

By Proposition 3.12, the group is pronilpotent, and since is open, it follows that the finite quotient is nilpotent.∎

In order to use the nilpotency of to obtain information about , we will show the nilpotency class of is uniformly bounded. Most of the next two sections is devoted to the proof of this. Once this uniform bound has been established, the proof of Theorem 1.1 will be finished at the end of Section 6.

To bound the nilpotency class of , write it as a product over primes (see Proposition 3.13):

 Γ/Γk=∏p(Γ/Γk)(p)

where is a finite -group. In this way, the problem of uniformly bounding the nilpotency class of separates out into a problem over each prime. The desired uniform bound on nilpotency class will be obtained in two steps: First we give a uniform bound on the nilpotency class of for almost all primes (see Theorem 5.2 below). Finally, in Theorem 6.1, we will obtain a local bound on nilpotency class at any fixed prime . The rest of this section will be devoted to obtain the bound at almost all primes.

###### Theorem 5.2.

There is a finite set of primes and such that for any and , the group is nilpotent of class at most .

Let us first introduce some notation. Recall that is the th term in the lower central series of . Set . Note that is a finitely generated nilpotent group of class at most . Set , so is a finitely generated torsion-free nilpotent group.

Let us now give an outline of the proof of Theorem 5.2. We will first show that descend to self-embeddings of (Claims 5.3 and 5.4). We can therefore apply the nilpotent case of the Main Theorem 1.1, i.e. Theorem 4.1. This easily shows that any nonabelian subgroup of comes from the torsion of a certain nilpotent group of class , where is the nilpotency class of . We will show that if a prime contributes nonabelian torsion at some class , then it already did so at (Lemma 5.11), and hence nonabelian subgroups of are only located at the finite set of primes visible at . We will use this to complete the proof of Theorem 5.2 at the end of the section.

###### Claim 5.3.

descend to embeddings .

###### Proof.

It is clear the descend to maps

 ¯¯¯¯φk:Nc→Nc

such that the image is normal and finite-index in . Since the image of any torsion element is torsion, we see that in turn descend to maps

 ¯¯¯¯φk:N(∞)c→N(∞)c

with finite-index and normal image. It remains to show is injective.

Let be the Lie hull of , i.e. the unique simply-connected nilpotent Lie group containing as a cocompact lattice (see Theorem 3.17). By Mal’cev’s Superrigidity Theorem 3.18, the maps extend to continuous homomorphisms

 ¯¯¯¯φk:G→G.

We claim these are isomorphisms. Indeed, is a finite-index subgroup of and hence a cocompact lattice in . Therefore is a connected closed cocompact subgroup of . But since is a simply-connected nilpotent Lie group, the only connected closed cocompact subgroup of is itself (see e.g. [Rag72, 2.1]). It follows that is surjective.

By a dimension count, we see that , i.e. is discrete, so that is a covering map. Since is simply-connected, it follows that is an automorphism of , as desired. ∎

Consider the set of primes such that for some , we have . If is finite, then the conclusion of Theorem 5.2 is immediately satisfied. Henceforth we assume is infinite.

###### Claim 5.4.

For any , we have

###### Remark 5.5.

At this point we do not even know that , or even , is infinite, but this will follow from the above claim.

###### Proof.

Since the map has finite kernel , we have for any and that

 [N(∞)c:¯¯¯¯φk(N(∞)c)]≥1|Tor(Nc)|[Nc:¯¯¯¯φk(Nc)].

Therefore it suffices to show that for any , we have

 [Nc:¯¯¯¯φk(Nc)]k→∞⟶∞. (5.1)

Further, it suffices to establish (5.1) for because the abelianization map satisfies so that

 [Nc:¯¯¯¯φk(Nc)]≥[N1:¯¯¯¯φk(N1)].

It remains to show that . It is straightforward to see that

 N1/¯¯¯¯φk(N1)=(Γ/Γk)ab

so Let be the set of prime divisors of . Since splits as a product of nontrivial -groups for , and any nontrivial -group has nontrivial abelianization, we find

 |(Γ/Γk)ab|≥∏p∈Pkp.

By assumption, is infinite, so we have , as desired.∎

Since , we can assume (after passing to a subsequence if necessary) that

 N(∞)c⊇¯¯¯¯φ1(N(∞)c)⊇¯¯¯¯φ2(N(∞)c)⊇…

is a descending chain of subgroups. By Theorem 4.1 (the nilpotent case of Theorem 1.1), we see that is abelian modulo torsion. In fact, we can conclude the following slightly stronger result.

###### Lemma 5.6.

is abelian its modulo torsion.

###### Proof.

Taking the quotient by Tor gives a surjective map

 π(∞)c:Nc→N(∞)c

that maps to . Hence descends to a surjective map

 ¯¯¯¯¯¯Nc→¯¯¯¯¯¯¯¯¯¯¯¯N(∞)c

with finite kernel Tor. Since is abelian modulo its torsion, the same holds for .∎

Next we show the torsion-free part of does not depend on .

###### Lemma 5.7.

Let . Then the natural maps descend to isomorphisms

 ¯¯¯π(∞)c:¯¯¯¯¯N(∞)c≅⟶¯¯¯¯¯N(∞)d.

In particular, is torsion.

###### Proof.

Surjectivity of the quotient map immediately yields that is surjective. To show it is injective, it suffices to consider the case . Consider the composition

 Γ→¯¯¯¯¯Nc→¯¯¯¯¯N(∞)c. (5.2)

Since is contained in the kernel of (5.2) and is torsion-free abelian, we see (5.2) factors through the quotient and hence descends to a map , which is easily checked to be inverse to . ∎

Our next step is most conveniently phrased in terms of subvarieties of groups and verbal subgroups associated to the lower central series. Let us therefore first recall these notions and fix some notation.

###### Definition 5.8.

Let be a collection of words on some collection of symbols. Let denote the corresponding verbal subgroup of , and write for the largest quotient of that satisfies all the group laws given by .

Note that terms of the lower central series of a group give examples of verbal subgroups. For the th term, we can consider the verbal subgroup , and we will write . For example, in the special case we have been writing . We need the following general fact, the proof of which is straightforward.

###### Lemma 5.9.

Let be any group and suppose is a normal subgroup. Write for the image of in . Then

 V(G)/VG(H)≅V(G/H).

Our goal will now be to show that any nonabelian subgroup of is located at divisors of . More precisely, if is nilpotent of class , we have . The latter surjects onto , which is abelian by Lemma 5.6. Therefore any nonabelian subgroup of is located at divisors of . Our goal is to show such a subgroup in fact needs to be located at divisors of .

Unfortunately in general the torsion of can grow as . We remedy this in the following way. Since is finitely generated and nilpotent, is virtually torsion-free (See Proposition 3.16). Let be a finite-index normal torsion-free subgroup of . Write

 ¯¯¯¯¯N(fin)c:=¯¯¯¯¯Nc/¯¯¯¯¯N(tf)c

for the quotient. Set . Since is torsion-free, projects isomorphically onto its image in .

Consider the image of under . Since is torsion (by Lemma 5.7), we see that is a torsion-free finite-index normal subgroup of . Write

 ¯¯¯¯¯N(fin)2:=¯¯¯¯¯N2/¯¯¯¯¯N(tf)2

for the quotient. Unlike the situation for torsion groups of at different values of , there is an easy description of the relationship between and as follows.

We have

1. , and

###### Proof.

In our situation, we apply Lemma 5.9 to the group with normal subgroup and verbal subgroup . Since is the image of in , we obtain that

 ¯¯¯¯¯Nc=Nc/∩k¯¯¯¯φk(Nc)=Nilc(Γ/Γ∞).

This proves (i).

It follows that . Applying Lemma 5.9 again to the group with normal subgroup and words (so for any group ), we find

 ¯¯¯¯¯N2/¯¯¯¯¯N(tf)2≅Nil2(¯¯¯¯¯Nc/¯¯¯¯¯N(tf)c).

Write

 ¯¯¯¯¯N(fin)c=∏p¯¯¯¯¯N(fin,p)c

as a product of -groups. We show that any nonabelian contributions are located at a finite set of primes that does not depend on :

###### Lemma 5.11.

Let be a prime that does not divide . Then is abelian.

###### Proof.

By Lemma 5.10.(ii), we have