We consider the natural projection
$p\mathrel{\mathop{\ordinarycolon}}P\rightarrow P/N$ and we will show that its
kernel $N$ is trivial. This projection induces a surjective map

$p_{n}\mathrel{\mathop{\ordinarycolon}}\,\frac{P}{\gamma_{n}P}\longrightarrow\frac{P/N}{\gamma_{n}(P/N)}\cong\frac{P}{(\gamma_{n}P)N},$

for every $n$. Since both of the previous groups are isomorphic to
$F/\gamma_{n}F$, for the same free group $F$, $p_{n}$ is a surjective
endomorphism of the Hopfian group $F/\gamma_{n}F$, so it is injective. This
implies that $N\subseteq\gamma_{n}P$ for every $n\geq 1$, so $N=1$. ∎

This is due to the fact that finitely generated nilpotent groups are Hopfian.
We have two immediate consequences from the last proposition.

######  Corollary 5.3.2.

Parafree groups of abelian rank 1 are isomorphic to $\mathbb{Z}$.

###### Proof.

Applying 5.3.1 to $N=[G,G]$ it follows that $G\cong G/N\cong\mathbb{Z}$. ∎

By similar reasons, we can also prove that the only abelian (equivalently,
nilpotent) parafree groups are the trivial group and $\mathbb{Z}$.

######  Proposition 5.3.3.

The free product of parafree groups is parafree.

######  Proposition 5.3.4.

Subgroups of parafree are not necessarily parafree.

###### Proof.

The group $G=\langle a,b,c|a^{2}b^{2}c^{3}\rangle$ is parafree. The subgroup
of $G$ generated by $a,b,cac^{-1},cbc^{-1}$ has presentation $\langle
x,y,z,t|x^{2}y^{2}z^{2}t^{2}\rangle\cong\pi_{1}(S_{3})$. So $G$ contains the
surface groups

$\pi_{1}(\Sigma_{g})\xhookrightarrow{}\pi_{1}(S_{3})\xhookrightarrow{}G$

for $g\geq 2$; which are not parafree. ∎

The following proposition is harder. Its proof can be found in [4].

######  Proposition 5.3.5.

The two-generator subgroups of parafree groups are free.

######  Proposition 5.3.6.

For every integer $n\geq 2$, there exists a parafree $G$ with
$r_{\operatorname{ab}}(G)=n$ and with a non-free three-generated subgroup.

###### Proof.

Let $n\geq 2$ and consider the 3-generated group $G_{3}=\langle
a,b,c|a^{2}b^{2}c^{3}\rangle$, which is a non-free parafree group. By taking
$G_{n}=G_{3}*\mathbb{Z}*\cdots\mathbb{Z}$, the free product of $G_{3}$ with
$n-2$ copies of $\mathbb{Z}$, we get a group $G_{n}$ with
$r_{\operatorname{ab}}(G_{n})=n$ that contains a subgroup isomorphic to
$G_{3}$. ∎

The previous group $G_{3}=\langle a,b,c|a^{2}b^{2}c^{3}\rangle$ is quite
interesting. It is not only a non-free parafree group, it is residually free
([2] and non-limit ([35]).

######  Corollary 5.3.7.

The centre of a parafree group is either trivial or the whole group. This last
case only occurs for $\mathbb{Z}$. More generally, any abelian subgroup of a
parafree group is free.

The following is [7, Theorem 7.1]. It uses the $L^{2}$-methods of chapter 8.

######  Theorem 5.3.8.

Let $G$ be a finitely generated parafree group and let $N$ be a finitely
generated normal subgroup of infinite index. Then $N=1$.

###### Proof.

The group $G$ has rank $n$. If $n=1$ then, by 5.3.2, $G\cong\mathbb{Z}$ and
the statement is trivial. Suppose that $n\geq 2$ and let $F$ be the free group
of rank $n$. Since $G$ is parafree, $G_{\widehat{p}}$ is pro-$p$ free by 5.2.2
and the canonical map $G\xhookrightarrow{}G_{\widehat{p}}\cong
F_{\widehat{p}}$ is injective and has dense image. By 8.1.5,
$b_{1}^{(2)}(G)\geq b_{1}^{(2)}(F)=n-1>0$. Let us suppose that $N$ is not
trivial. Then, since $G$ is torsion-free, $N$ is infinite. We have a short
exact sequence

$1\rightarrow N\rightarrow G\rightarrow G/N\rightarrow 1,$

where both $N$ and $G/N$ are infinite, $b_{1}^{(2)}(G)>0$ and
$b_{1}^{(2)}(N)<\infty$ (since $N$ is finitely generated). This contradicts
8.1.4. Thus $N=1$. ∎

##  Chapter 6 The theory of group rings

The main idea that underlies this chapter can be roughly outlined as follows.
Given a group homomorphism $G\longrightarrow H$, we want to “linearise” it by
studying instead the corresponding map $I_{G}\longrightarrow I_{H}$. In the
last setting, cohomological and ring-theoretical arguments are applicable.
From this, one wants to derive information about the initial group
homomorphism. The most important result of this chapter is 6.3.1.

###  6.1 The bar resolution

Let $G$ be a group. We review the construction of the bar resolution
$\overline{C}^{*}(G,k)\rightarrow k\rightarrow 0$, a resolution of $k$ over
$kG$. It is also named standard resolution and it allows us to explicitly
compute homology groups of low degree. Let $\overline{C}^{n}(G,k)$ be a
$kG$-module with free basis given by the symbols
$\\{|g_{1}|\cdots|g_{n}|\mathrel{\mathop{\ordinarycolon}}g_{i}\in G\\}$, a
collection that is naturally in bijection with $G^{n}$. Observe that
$\overline{C}^{0}(G,k)=kG$. We define
$d_{n}\mathrel{\mathop{\ordinarycolon}}\overline{C}^{n}(G,k)\rightarrow\overline{C}^{n-1}(G,k)$
by

$\displaystyle d_{n}(|g_{1}|\cdots|g_{n}|)=$
$\displaystyle\,g_{1}\,|g_{2}|\cdots|g_{n}|$
$\displaystyle-|g_{1}g_{2}|g_{3}|\cdots|g_{n}|$ $\displaystyle+\dots$
$\displaystyle+(-1)^{n-1}\,|g_{1}|\dots|g_{n-1}g_{n}|$
$\displaystyle+(-1)^{n}\,|g_{1}|\cdots|g_{n-1}|.$

We claim that the following is a resolution of projective (in fact, free)
$kG$-modules

$\xrightarrow{d_{n+2}}\overline{C}^{n+1}(G,k)\xrightarrow{d_{n+1}}\overline{C}^{n}(G,k)\xrightarrow{d_{n}}\cdots\xrightarrow{d_{3}}\overline{C}^{2}(G,k)\xrightarrow{d_{2}}\overline{C}^{1}(G,k)\xrightarrow{d_{1}}kG\xrightarrow{d_{0}}k\rightarrow
0.$

This particular resolution allows us to compute $H^{n}(G,k)$ as the homology
of the chain complex obtained after applying the functor
$\mathbin{\mathop{\otimes}\limits_{kG}}k$ to the previous resolution. This
sequence is natural from the point of view of topology.

Homology groups also appear as correction terms in sequences that are not
exact and homology groups of low degree are very meaningful. In fact,
$H^{0}(G,k)\cong k$ and $H_{1}$ verifies the following.

######  Lemma 6.1.1.

There is a natural isomorphism
$H_{1}(G,k)=k\mathbin{\mathop{\otimes}\limits_{\mathbb{Z}}}G_{\operatorname{ab}}$.

###### Proof.

We use the bar resolution of $k$ to compute $H_{1}(G,k)$. By applying to

$\xrightarrow{d_{3}}\overline{C}^{2}(G,k)\xrightarrow{d_{2}}\overline{C}^{1}(G,k)\xrightarrow{d_{1}}kG\xrightarrow{d_{0}}k\rightarrow
0$

the functor $\mathbin{\mathop{\otimes}\limits_{kG}}k$, we obtain the sequence
of $k$-modules

$\xrightarrow{{d}_{3}\otimes k}kG^{2}\xrightarrow{d_{2}\otimes
k}kG\xrightarrow{0}k\xrightarrow{\operatorname{id}}k\rightarrow 0,$

where $\widetilde{d}_{2}$ sends each $(g,h)$ to $g+h-gh$. So

$H_{1}(G,k)\cong\frac{k\,G^{2}}{\operatorname{im}\widetilde{d}_{2}}\cong
k\otimes_{\mathbb{Z}}G_{\operatorname{ab}}.\qed$

More importantly, this particular resolution, the bar resolution, allows us to
add explicitly an homology group to a non-exact resolutions of $k$ over $kG$,
to obtain an exact sequence. Of course, here underlies the fact that the
computation of the homology does not depend on the choice of projective
resolution of $k$ over $kG$, since any pair of such resolutions is
homotopically equivalent and the application of
$\mathbin{\mathop{\otimes}\limits_{kG}}k$ preserves chain homotopies, as it is
an additive functor. This fact will underlie the arguments of subsequent
sections.

###  6.2 The augmentation ideal

During section 4.3, we saw the importance of defining objects in terms of
universal properties. We worked with pro-$p$ completions in terms of neat
diagrams and, roughly, we could observe that it is functorial. The approach of
this section is similar. We introduce the notion of derivation to describe the
augmentation ideal $kI_{G}$ in terms of a universal property.

######  Definition 6.2.1 (Derivation).

Let $M$ be a $G$-module. We say that a function
$d\mathrel{\mathop{\ordinarycolon}}G\rightarrow M$ is a derivation if
$d(xy)=d(x)+xd(y)$ for every $x,y\in G$. We denote by
$\operatorname{Der}(G,M)$ the abelian group of derivations $G\rightarrow M$.

There is a universal derivation $D$ of $G$. We define
$D\mathrel{\mathop{\ordinarycolon}}G\rightarrow kI_{G}$ to be the derivation
such that $D(g)=g-1$. There is an isomorphism of abelian groups
$\operatorname{Der}(G,M)\cong\operatorname{Hom}_{G}(kI_{G},M)$ described as
follows. Given $f\in\operatorname{Hom}_{kG}(kI_{G},M)$, we define a derivation
$d=f\circ D\mathrel{\mathop{\ordinarycolon}}G\rightarrow M$.

${G}$${kI_{G}}$${M.}$$\scriptstyle{d}$$\scriptstyle{D}$$\scriptstyle{f}$ (6.1)

Reciprocally, given $d\in\operatorname{Der}(G,M)$, we can define the
associated $kG$-homomorphism
$f\mathrel{\mathop{\ordinarycolon}}kI_{G}\rightarrow M$ by $f(g-1)=d(g)$.
These correspondences are well-defined and are inverses of each other.

######  Lemma 6.2.2.

Let $F$ be a free group freely generated by $S\subseteq F$. Then
$\\{s-1\mathrel{\mathop{\ordinarycolon}}s\in S\\}$ is a free basis of the
$kF$-module $kI_{F}$.

###### Proof.

It is clear that $\operatorname{Der}(G,A)$ is in bijection with group
homomorphisms $\gamma\mathrel{\mathop{\ordinarycolon}}F\rightarrow M\rtimes F$
that are splittings of the exact sequence

$1\rightarrow M\rightarrow M\rtimes F\rightarrow F\rightarrow 1.$

In fact, those $\gamma$ are exactly those maps of the form
$\gamma(f)=(d(f),f)$ for some derivation $d$. In other words, for any
$\\{a_{s}\mathrel{\mathop{\ordinarycolon}}s\in S\\}\subseteq M$, there is
exactly one $d\in\operatorname{Der}(F,M)$ such that $d(s)=a_{s}$ for all $s\in
S$. Using the characterisation
$\operatorname{Der}(F,M)\cong\operatorname{Hom}_{kF}(kI_{F},M)$ of eq. 6.1, we
can also conclude that for any $kF$-module $M$ and any
$\\{a_{s}\mathrel{\mathop{\ordinarycolon}}s\in S\\}\subseteq M$, there is
exactly one $kF$-homomorphism
$f\mathrel{\mathop{\ordinarycolon}}I_{F}\rightarrow M$ such that
$f(s-1)=a_{s}$ for all $s\in S$. The conclusion follows. ∎

######  Definition 6.2.3 (Fox derivatives).

Let $F$ be a free group freely generated by $S\subseteq F$. We denote by
$\dfrac{\partial{}}{\partial{s}}\mathrel{\mathop{\ordinarycolon}}F\rightarrow
kF$ the maps defined by

$f-1=\sum_{s\in S}\dfrac{\partial{}f}{\partial{s}}\,(s-1),\,\,\,\mbox{for all
$f\in F$.}$

Our aim is to prove the following result.

######  Theorem 6.2.4.

Let $F$ be a free group freely generated by $S\subseteq F$. Let $T\subseteq F$
and consider the normal closure $N=\langle\langle T\rangle\rangle$. Consider
$G=F/N$. Given $f\in F$, we denote its image under the projection
$F\rightarrow G$ by $\overline{f}\in G$. There is an exact sequence of
$kG$-modules of the form

${0}$${k\mathbin{\mathop{\otimes}\limits_{\mathbb{Z}}}R_{\operatorname{ab}}}$${kG^{(S)}}$${kG}$${k}$${0,}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\varepsilon}$

where the maps $\delta_{1}$ and $\delta_{2}$ are defined as follows:

$\displaystyle\partial_{1}\left(g_{s}\,e_{s}\right)=$ $\displaystyle
g_{s}\,(s-1),\,\,\,\mbox{for all $s\in S$ and $g_{s}\in G$,}$
$\displaystyle\partial_{2}\left(c\otimes\overline{r}\,[G,G]\right)=$
$\displaystyle\sum_{s\in
S}c\,\overline{\dfrac{\partial{}r}{\partial{s}}}\,e_{s},\,\,\,\mbox{for all
$c\in k$, $r\in R$.}$

######  Definition 6.2.5.

In the context of the previous theorem, the group $R_{\operatorname{ab}}$, or
more broadly $k\otimes_{\mathbb{Z}}R_{\operatorname{ab}}$, is called the
relation module of $G$. Notice that it requires a choice of $F$ and a map
$F\rightarrow G$. It has a natural $kG$-structure described as follows. The
conjugation action of $F$ on $R$, leads to a right $kF$-module structure on
$k\otimes R_{\operatorname{ab}}$. However, the action of $R$ is trivial since
$r^{t}\equiv r\mod[R,R]$ for all $r,t\in R$. So this induces a natural
$kG$-modules structure on $k\otimes R_{\operatorname{ab}}$.

###### Proof.

The sequence

${0}$${kG^{(S)}}$${kG}$${k}$${0,}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\varepsilon}$

is not exact at $kG^{(S)}$. It needs a correction of $H_{1}(R,k)$ and we
compute this correction explicitly with the aid of the bar resolution.
Consider the following commutative diagram of $kR$-modules, whose rows are
resolutions of $k$ over $kR$,

${\cdots}$${\overline{C}_{3}(R,k)]}$${\overline{C}_{2}(R,k)}$${\overline{C}_{1}(R,k)}$${kR}$${k}$${0}$${\cdots}$${0}$${0}$${kF^{(S)}}$${kF}$${k}$${0.}$$\scriptstyle{d_{4}}$$\scriptstyle{d_{3}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{d_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{d_{1}}$$\scriptstyle{f_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$

Here $f_{0}$ is the natural inclusion, $f_{k}=0$ for $k\geq 2$ and

$f_{1}\left(r|t|\right)=\sum_{s\in
S}r\,\dfrac{\partial{}t}{\partial{s}}\,e_{s},\,\,\,\mbox{for all $r,t\in R$.}$

Since this the arrows are projective resolutions of $kR$-modules, the chain
map $f_{k}$ is an homotopy equivalence. We apply the functor
$k\mathbin{\mathop{\otimes}\limits_{kR}}$ and observe that the resulting
diagram is

${\cdots}$${kR^{3}}$${kR^{2}}$${kR}$${k}$${k}$${0,}$${\cdots}$${0}$${0}$${kG^{(S)}}$${kG}$${k}$${0,}$$\scriptstyle{k\otimes
d_{4}}$$\scriptstyle{k\otimes
d_{3}}$$\scriptstyle{0}$$\scriptstyle{0}$$\scriptstyle{k\otimes
d_{2}}$$\scriptstyle{k\otimes
d_{1}}$$\scriptstyle{0}$$\scriptstyle{\iota}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$

from where we can adapt the following exact sequence of abelian groups

${0}$${\frac{kR}{\operatorname{im}d_{2}\otimes
k}}$${kG^{(S)}}$${kG}$${k}$${0.}$$\scriptstyle{k\otimes
f_{1}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$

By substituting the first non-zero factor by
$k\otimes_{\mathbb{Z}}R_{\operatorname{ab}}$, which is isomorphic to
$H_{1}(R,k)$ according to the isomorphism that is implicit in the proof of
lemma 6.1.1, then the corresponding $k\otimes f_{1}$ turns into $\partial_{2}$
and the result is the following exact sequence of abelian groups

${0}$${k\otimes_{\mathbb{Z}}R_{\operatorname{ab}}}$${kG^{(S)}}$${kG}$${k}$${0,}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\partial_{0}}$

whose maps are also seen to be $kG$-homomorphisms. The only map for which this
is not obvious is the map $\partial_{2}$. Let $r\in R$ and let $f\in F$.
Notice that

$\displaystyle\partial_{2}\left(frf^{-1}\,[R,R]\right)=$
$\displaystyle\sum_{s\in
S}\overline{\dfrac{\partial{}frf^{-1}}{\partial{s}}}\,e_{s}$ $\displaystyle=$
$\displaystyle\sum_{s\in
S}\overline{\dfrac{\partial{}f}{\partial{s}}+f\dfrac{\partial{}r}{\partial{s}}-frf^{-1}\dfrac{\partial{}f}{\partial{s}}}\,e_{s}$
$\displaystyle=$ $\displaystyle\sum_{s\in
S}\overline{f\dfrac{\partial{}r}{\partial{s}}}\,e_{s}$ $\displaystyle=$
$\displaystyle\overline{f}\,\partial_{2}\left(r\,[R,R]\right).\qed$

One can view an algebraic proof of 6.2.4 that uses no homological machinery in
[30, Chapter 11]. A purely topological argument is given in [8, Chapter II,
Section 5], by means of the Hopf’s formula. Our argument is based on the
indications given in [8, Exercise 4, Chapter IV, Section 2].

We have given a description of the augmentation ideal of a group in terms of a
presentation. We are also interested in giving a description of the
augmentation ideal of an an amalgamated product and of an HNN extension.
However, we should first study free products.

######  Notation 6.2.6.

Given two groups $\Lambda\leq\Gamma$, we denote by $kI_{\Lambda}^{\Gamma}$ the
left ideal of $kG$ generated by $I_{\Lambda}$. Since $k\Gamma$ is a natural
free right $k\Lambda$-module, it follows that the canonical map of
$k\Gamma$-modules

$k\Gamma\mathbin{\mathop{\otimes}\limits_{k\Lambda}}kI_{\Lambda}\longrightarrow
kI_{\Lambda}^{\Gamma}$

is an isomorphism.

######  Lemma 6.2.7.

Let $\Gamma$ be the free product of two groups $A$ and $B$. Then the
$k\Gamma$-homomorphism

$kI_{A}^{\Gamma}\oplus kI_{B}^{\Gamma}\longrightarrow kI_{\Gamma},$

defined by $(x,y)\mapsto x+y$ for $x\in kI_{A}$ and $y\in kI_{B}$, is an
isomorphism.

###### Proof.

We name $\psi$ the given map. This statement can be verified by brute force
using Britton’s lemma. However, it is cleaner to refer to the correspondence
$\operatorname{Der}(\Gamma,M)\cong\operatorname{Hom}_{\mathbb{Z}\Gamma}(I_{\Gamma},M)$
of eq. 6.1. ∎

######  Theorem 6.2.8 (Swan).

Let $\theta_{1}\mathrel{\mathop{\ordinarycolon}}C\rightarrow A$ and
$\theta_{2}\mathrel{\mathop{\ordinarycolon}}C\rightarrow B$ be group
homomorphisms and consider the induced amalgamated product
$\Gamma=A\underset{C}{*}B$. There is an exact sequence of $k\Gamma$-modules

${0}$${kI_{C}^{\Gamma}}$${kI_{A}^{\Gamma}\oplus
kI_{B}^{\Gamma}}$${kI_{\Gamma}}$${0,}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

where $\alpha(z)=(\theta_{1}(z),-\theta_{2}(z))$ for all $z\in kI_{C}$ and
$\beta(x,y)=x+y$ for all $x\in I_{A}$, $y\in I_{B}$.

###### Proof.

It is clear that $\beta\circ\alpha=0$, that $\alpha$ is injective and that
$\beta$ is surjective. We simply have to check that
$\operatorname{im}\alpha=\ker\beta$.

Let $\Gamma_{0}=A*B$. We consider the normal subgroup $R$ of $\Gamma_{0}$
generated the elements of the set
$\\{\theta_{1}(c)^{-1}\,\theta_{2}(c)\mathrel{\mathop{\ordinarycolon}}c\in
C\\}$. Then $\Gamma\cong\Gamma_{0}/R$. The idea is to consider an exact
sequence using lemma 6.2.7 for the free product $\Gamma_{0}$, to take
$R$-coinvariants, and to study the correction term that arises in the new
sequence for $\Gamma$ with the aid of the bar resolution of $R$.

Consider a diagram of the following form

${\cdots}$${\overline{C}_{3}(R,k)}$${\overline{C}_{2}(R,k)}$${\overline{C}_{1}(R,k)}$${kR}$${k}$${0}$${\cdots}$${0}$${0}$${kI_{A}^{\Gamma_{0}}\oplus
kI_{B}^{\Gamma_{0}}}$${k\Gamma_{0}}$${k}$${0.}$$\scriptstyle{d_{4}}$$\scriptstyle{d_{3}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{d_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{d_{1}}$$\scriptstyle{f_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\psi}$$\scriptstyle{\partial_{0}}$

The map $\psi$ is the isomorphism $kI_{A}^{\Gamma_{0}}\oplus
kI_{B}^{\Gamma_{0}}\rightarrow I_{\Gamma_{0}}$ of lemma 6.2.7 while the maps
$f_{k}$ are zero for $k\geq 2$. Lastly, the map $f_{1}$ is the $kR$-linear
extension of $f_{1}(|r|)=\psi^{-1}(r-1)$. It is clear that $f_{0}\circ
d_{1}=\psi\circ f_{1}$. Moreover, since the map
$\psi^{-1}(r-1)\mathrel{\mathop{\ordinarycolon}}R\rightarrow
kI_{A}^{\Gamma_{0}}\oplus kI_{B}^{\Gamma_{0}}$ is a $R$-derivation, then
$f_{1}$ verifies that $f_{1}\circ d_{2}=0$. So the previous diagram is
commutative. The arrows are projective resolutions of $kR$-modules and the
maps $f_{i}$ define a chain homotopy. Applying the functor
$k\mathbin{\mathop{\otimes}\limits_{kR}}$ we obtain

${\cdots}$${kR^{3}}$${kR^{2}}$${kR}$${k}$${k}$${0}$${\cdots}$${0}$${0}$${kI_{A}^{\Gamma}\oplus
kI_{B}^{\Gamma}}$${k\Gamma}$${k}$${0.}$$\scriptstyle{k\otimes
d_{4}}$$\scriptstyle{k\otimes
d_{3}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{k\otimes
d_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{0}$$\scriptstyle{f_{0}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\beta}$$\scriptstyle{\partial}$

Arguing as in 6.2.4, we have an exact sequence of the form

${0}$${k\otimes R_{\operatorname{ab}}}$${kI_{A}^{\Gamma}\oplus
kI_{B}^{\Gamma}}$${k\Gamma}$${k}$${0,}$$\scriptstyle{f_{1}}$$\scriptstyle{\beta}$$\scriptstyle{\partial}$

where the image of $r+[R,R]\in R_{\operatorname{ab}}$ under $f_{1}$ is
$\phi^{-1}(r-1)$ for any $r\in R$. Since $R_{\operatorname{ab}}$ is generated
by the elements of the set
$\\{\theta_{1}^{-1}(c)\,\theta_{2}(c)\mathrel{\mathop{\ordinarycolon}}c\in
C\\}$, the image of $f_{1}$ is generated by their corresponding images. Let
$c\in C$.

Notice that

$\theta_{1}^{-1}(c)\,\theta_{2}(c)-1=\theta_{1}(c)^{-1}\,\left((\theta_{2}(c)-1)-(\theta_{1}(c)-1)\right),$

and so

$f_{1}(\theta_{1}^{-1}(c)\,\theta_{2}(c))=\theta_{1}(c)^{-1}\,(1-\theta_{1}(c),\theta_{2}(c)-1).$

The $k\Gamma$-span of these elements, for $c\in C$, generate the image of
$f_{1}$, which, by exactness, is the kernel of $\beta$. Crucially, this is the
image of $\alpha$. ∎

######  Theorem 6.2.9.

Let $A$ be a subgroup of $H$ and let
$\theta\mathrel{\mathop{\ordinarycolon}}A\rightarrow H$ be a group
monomorphism. Consider the corresponding HNN extension
$\Gamma=H\mathbin{\mathop{*}\limits_{A}}$. The $k\Gamma$-homomorphism

${kI_{H}^{\Gamma}\oplus k\Gamma}$${kI_{\Gamma},}$$\scriptstyle{\beta}$

where $\beta(x,y)=x+y(t-1)$ for all $x\in I_{H}$ and $y\in k\Gamma$, is
surjective and the kernel is $k\Gamma$-generated by the set
$\\{(\theta(a)-1-t(a-1),\theta(a)-1)\mathrel{\mathop{\ordinarycolon}}a\in
A\\}$.

Our argument is essentially the same as in 6.2.8.

###### Proof.

Let $\Gamma_{0}=H*(t)$ and let $R$ be the normal subgroup of $\Gamma_{0}$
generated by the elements
$\\{a^{-1}t^{-1}\theta(a)t\mathrel{\mathop{\ordinarycolon}}a\in A\\}.$ Then
$\Gamma\cong\Gamma_{0}/R$. Consider a diagram of the form

${\cdots}$${\overline{C}_{3}(R,k)}$${\overline{C}_{2}(R,k)}$${\overline{C}_{1}(R,k)}$${kR}$${k}$${0}$${\cdots}$${0}$${0}$${kI_{H}^{\Gamma_{0}}\oplus
k\Gamma_{0}}$${k\Gamma_{0}}$${k}$${0.}$$\scriptstyle{d_{4}}$$\scriptstyle{d_{3}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{d_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{d_{1}}$$\scriptstyle{f_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\psi}$$\scriptstyle{\partial_{0}}$

The map $\psi\mathrel{\mathop{\ordinarycolon}}kI_{H}^{\Gamma_{0}}\oplus
k\Gamma_{0}\rightarrow kI_{\Gamma_{0}}$ is the isomorphism of lemma 6.2.7,
defined by $(x,y)\mapsto x+y(t-1)$ for all $x\in kI_{H}$ and $y\in
k\Gamma_{0}$. Here we have implicitly identified $k\Gamma_{0}\cong I_{\langle
t\rangle}^{\Gamma_{0}}$ by means of the isomorphism

$z\mapsto z(t-1),\,\,\,\mbox{for all $z\in k\Gamma_{0}$.}$

On the other side, the map $f_{0}$ is the natural inclusion, the maps $f_{k}$
are zero for $k\geq 2$ and $f_{1}$ is the $k\Gamma_{0}$-extension of
$f_{1}(|r|)=\psi^{-1}(r-1),$ which defines, when restricted to $G$, a
derivation $G\rightarrow kI_{H}^{\Gamma_{0}}\oplus k\Gamma_{0}$. So
$f_{1}\circ d_{2}=0$. It is also direct to see that $f_{0}\circ
d_{1}=\psi\circ f_{1}$. So the previous diagram represents a chain homotopy
$f_{i}$ between projective resolutions of $k$ over $kR$. Applying
$k\mathbin{\mathop{\otimes}\limits_{kR}}$ we get the commutative diagram

${\cdots}$${kR^{3}}$${kR^{2}}$${kR}$${k}$${k}$${0}$${\cdots}$${0}$${0}$${kI_{H}^{\Gamma}\oplus
k\Gamma}$${k\Gamma}$${k}$${0,}$$\scriptstyle{k\otimes
d_{4}}$$\scriptstyle{k\otimes
d_{3}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{k\otimes
d_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{0}$$\scriptstyle{f_{0}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\beta}$$\scriptstyle{\partial}$

From this, we can obtain, arguing as in 6.2.4, an exact sequence of
$k\Gamma$-modules

${0}$${k\otimes R_{\operatorname{ab}}}$${kI_{A}^{\Gamma}\oplus
kI_{B}^{\Gamma}}$${k\Gamma}$${k}$${0,}$$\scriptstyle{f_{1}}$$\scriptstyle{\beta}$$\scriptstyle{\partial}$

where the image of $r+[R,R]\in R_{\operatorname{ab}}$ under $f_{1}$ is
$\phi^{-1}(r-1)$ for any $r\in R$. By exactness, the image of $f_{1}$ must be
the kernel of $\beta$, and we want to check that this can be
$k\Gamma$-generated by the elements of
$\\{(\theta(a)-1-t(a-1),\theta(a)-1)\mathrel{\mathop{\ordinarycolon}}a\in
A\\}$. The image of $f_{1}$ will be $k\Gamma$-generated by the images under
$f_{1}$ of $\\{a^{-1}t^{-1}\theta(a)t\mathrel{\mathop{\ordinarycolon}}a\in
A\\}$, since this set generates the abelian group $R_{\operatorname{ab}}$.
Lastly, observe that, for any $a\in A$,

$a^{-1}t^{-1}\theta(a)t-1=a^{-1}t^{-1}\,\left(\theta(a)-1-t(a-1)+(\theta(a)-1)(t-1)\right).$

Hence

$f_{1}(a^{-1}t^{-1}\theta(a)t+[R,R])=(\theta(a)-1-t(a-1),\theta(a)-1).$

and the conclusion follows. ∎

######  Corollary 6.2.10.

Let $A=\langle a\rangle$ be a cyclic subgroup of $H$ and let
$\theta\mathrel{\mathop{\ordinarycolon}}A\rightarrow H$ be a group
monomorphism. Consider the corresponding HNN extension
$\Gamma=H\mathbin{\mathop{*}\limits_{A}}$. Then

$\frac{kI_{H}^{\Gamma}\oplus
k\Gamma}{k\Gamma\,\left(\theta(a)-1-t(a-1),\theta(a)-1\right)}\cong
I_{\Gamma},$

where this isomorphism of $k\Gamma$-modules is given by

$[(x,y)]\mapsto x+y(t-1),\,\,\,\mbox{for all $x\in I_{H}$ and $y\in
k\Gamma$.}$

###  6.3 Back to group homomorphisms

We wanted to study the maps that group homomorphisms induce between their
augmentation ideals. Here will give two instances in which one can derive
information from these induced maps back to the initial group homomorphisms.

######  Lemma 6.3.1.

Let $\phi\mathrel{\mathop{\ordinarycolon}}\widetilde{G}\rightarrow G$ be a
surjective group homomorphism of kernel $K$. Suppose that the natural
homomorphism of $kG$-modules

$kI_{\widetilde{G}}\mathbin{\mathop{\otimes}\limits_{k\widetilde{G}}}kG\longrightarrow
kI_{G},$

defined by the $k$-linear extension of $a\otimes b\mapsto\phi(a)b$, is an
isomorphism. Then

$k\otimes_{\mathbb{Z}}K_{\operatorname{ab}}=0.$

###### Proof.

We can view any group as a quotient of a free group. We take a free group $F$
freely generated by some $S\subseteq F$ and a surjective map
$F\rightarrow\widetilde{G}$ of kernel $\widetilde{N}\unlhd F$. By composing
this projection with $\phi$, we have a surjective $F\rightarrow G$, of
(possibly) bigger kernel $N\unlhd F$. By using the functoriality of the exact
sequence of 6.2.4, we obtain a commutative diagram of $k$-modules, with exact
rows, of the form

${k\otimes\widetilde{N}_{\operatorname{ab}}}$${k\widetilde{G}^{(S)}}$${kI_{\widetilde{G}}}$${0}$${k\otimes
N_{\operatorname{ab}}}$${kG^{(S)}}$${kI_{G}}$${0,}$$\scriptstyle{\iota_{\operatorname{ab}}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\phi}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\phi}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$

where the unspecified arrows are the natural ones, induced either by
$\phi\mathrel{\mathop{\ordinarycolon}}\widetilde{G}\rightarrow G$ or the
natural inclusion
$\iota\mathrel{\mathop{\ordinarycolon}}\widetilde{N}\xhookrightarrow{}N$.

Applying the right-exact functor $\otimes_{k\widetilde{G}}kG$, where $kG$ has
the natural $k\widetilde{G}$ structure induced by $\phi$, we get a commutative
diagram of $kG$-modules with exact arrows

${\left(k\otimes\widetilde{N}_{\operatorname{ab}}\right)\otimes_{k\widetilde{G}}kG}$${kG^{(S)}}$${kI_{G}}$${0}$${k\otimes
N_{\operatorname{ab}}}$${kG^{(S)}}$${kI_{G}}$${0.}$$\scriptstyle{\iota_{\operatorname{ab}}\otimes
1}$$\scriptstyle{\partial_{2}\otimes
1}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{\partial_{1}}$

We look at the arrows that point downwards. Since the second arrow is
surjective and the third one is injective, then
$\iota_{\operatorname{ab}}\otimes 1$, the first arrow, is surjective. This is
one of the so called “four lemma” on diagram-chasing. We now rewrite the map
$\iota_{\operatorname{ab}}\otimes 1$. There is a commutative diagram of the
form

${\left(k\otimes\widetilde{N}_{\operatorname{ab}}\right)\otimes_{k\widetilde{G}}kG}$${k\otimes\frac{\widetilde{N}}{[\widetilde{N},N]}}$${k\otimes
N_{\operatorname{ab}}}$${k\otimes\frac{N}{[N,N]},}$$\scriptstyle{\iota_{\operatorname{ab}}\otimes
1}$$\scriptstyle{\psi}$$\scriptstyle{\alpha}$$\scriptstyle{\operatorname{id}}$
(6.2)

where $\psi$ is an isomorphism defined by

$\psi\left((1\otimes\widetilde{n}+[\widetilde{N},\widetilde{N}])\otimes
g\right)=1\otimes gng^{-1}+[\widetilde{N},N]$

and $\alpha$ is defined by

$\alpha(1\otimes\widetilde{n}+[\widetilde{N},N])=1\otimes\iota(\widetilde{n})+[N,N].$

The inverse of $\psi$ would be given by

$\psi^{-1}\left(\widetilde{n}+[\widetilde{N},N]\right)=(1\otimes\widetilde{n}+[\widetilde{N},\widetilde{N}])\otimes
1.$

For the previous map to be well-defined, we observe the following in the
module
$\left(k\otimes\widetilde{N}_{\operatorname{ab}}\right)\otimes_{k\widetilde{G}}kG$.
For any $n\in N$, $\widetilde{n}\in\widetilde{N}$,

$\displaystyle(1\otimes
n\widetilde{n}n^{-1}+[\widetilde{N},\widetilde{N}])\otimes 1=$
$\displaystyle(1\otimes(\widetilde{n}+[\widetilde{N},\widetilde{N}])\,n)\otimes
1$ $\displaystyle=$
$\displaystyle(1\otimes\widetilde{n}+[\widetilde{N},\widetilde{N}])\otimes\phi(n)\cdot
1$ $\displaystyle=$
$\displaystyle(1\otimes\widetilde{n}+[\widetilde{N},\widetilde{N}])\otimes 1.$

Hence, for any $n\in N$ and $\widetilde{n}\in\widetilde{N}$,

$(1\otimes[\widetilde{n},n]+[\widetilde{N},\widetilde{N}])\otimes 1=0.$

This verification says that $\psi^{-1}$ is also well-defined. It is easy to
see that $\psi$ and $\psi^{-1}$ are inverses of each other. So $\psi$ is an
isomorphism of $kG$-modules.

We proved that $\iota_{\operatorname{ab}}\otimes 1$ is surjective. So the map
$\alpha$ of the commutative square (6.2) must be surjective, too.

From the natural exact sequence of $\mathbb{Z}G$-modules

${\frac{\widetilde{N}}{[\widetilde{N},N]}}$${\frac{N}{[N,N]}}$${\frac{N}{\widetilde{N}[N,N]}}$${0.}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta^{\prime}}$

we obtain, by applying the right exact functor
$k\mathbin{\mathop{\otimes}\limits_{\mathbb{Z}}}$, an exact sequence of
$kG$-modules

${k\otimes\frac{\widetilde{N}}{[\widetilde{N},N]}}$${k\otimes\frac{N}{[N,N]}}$${k\otimes\frac{N}{\widetilde{N}[N,N]}}$${0.}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

We have already deduced that the map $\alpha$ is surjective, so the image of
$\beta$ is zero and then the third module must be zero. Observe that $K\cong
N/\widetilde{N}\leq F/\widetilde{N}\cong\widetilde{G}$, so

$k\otimes\frac{N}{\widetilde{N}[N,N]}\cong k\otimes K_{\operatorname{ab}}\cong
0.\qed$

A particular realisation of the previous lemma is a result due to Lewin.

######  Corollary 6.3.2 (Lewin).

Let $G$ be a group generated by two subgroups $A$ and $B$ with $C=A\bigcap B$.
Suppose that $kI_{C}^{G}=kI_{C}^{G}\bigcap kI_{B}^{G}$. Then the natural map
$A\underset{C}{*}B\longrightarrow G$ is an isomorphism.

###### Proof.

Denote $\widetilde{G}=A\underset{C}{*}B$ and name
$\phi\mathrel{\mathop{\ordinarycolon}}\widetilde{G}\rightarrow G$ the natural
map, which is surjective.

Consider a commutative diagram of $k$-modules of the form

${kI_{C}^{\widetilde{G}}}$${kI_{A}^{\widetilde{G}}\oplus
kI_{kB}^{\widetilde{G}}}$${kI_{\widetilde{G}}}$${0}$${kI_{C}^{G}}$${kI_{A}^{G}\oplus
kI_{B}^{G}}$${kI_{G}}$${0,}$$\scriptstyle{\alpha}$$\scriptstyle{\phi}$$\scriptstyle{\phi}$$\scriptstyle{\beta}$$\scriptstyle{\phi}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

where the arrows are the obvious ones. We define $\alpha(z)=(z,-z)$ for all
$z\in I_{C}$ and $\beta(x,y)=x+y$ for all $x\in I_{A}$ and $y\in I_{B}$. By
6.2.8, the first row is an exact sequence of $k\widetilde{G}$-modules.
Furthermore, by assumption, the second row is an exact sequence of
$kG$-modules.

We apply the right-exact functor
$kG\mathbin{\mathop{\otimes}\limits_{k\widetilde{G}}}$ to the first row and we
obtain a commutative diagram of $kG$-modules

${kI_{C}^{G}}$${kI_{A}^{G}\oplus
kI_{kB}^{G}}$${kG\otimes_{k\widetilde{G}}kI_{\widetilde{G}}}$${0}$${kI_{C}^{G}}$${kI_{A}^{G}\oplus
kI_{B}^{G}}$${kI_{G}}$${0,}$$\scriptstyle{\alpha}$$\scriptstyle{\phi}$$\scriptstyle{\phi}$$\scriptstyle{\beta}$$\scriptstyle{\phi}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

with exact rows.

Now we look at the arrows that point downwards. The first is an epimorphism,
and the second and fourth are monomorphism. A version of the four lemma
ensures that the third arrow
$1\otimes\phi\mathrel{\mathop{\ordinarycolon}}kG\otimes_{k\widetilde{G}}kI_{\widetilde{G}}\rightarrow
kI_{G}$ is a monomorphism. Moreover, this map is surjective, since $\phi$
itself is surjective. By 6.3.1, the map $\phi$ is an isomorphism. ∎

##  Chapter 7 Universal division rings of fractions

We want to emphasise our aim in talking about the concept of “universal
division rings of fractions” before going into details.

Let $\widetilde{G}$ be a candidate to being a parafree group. Suppose we
already know that its pro-$p$ completions $\widetilde{G}_{\widehat{p}}$ are
free. Suppose, in addition, we also know that, for a suitable prime $p$, the
kernel of the canonical map
$\phi\mathrel{\mathop{\ordinarycolon}}\widetilde{G}\longrightarrow\widetilde{G}_{\widehat{p}}$
is free. Let $G$ be the image of $\widetilde{G}$ under the previous map. We
consider the canonical $\mathbb{F}_{p}G$-module

$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}\widetilde{G}}}\mathbb{F}_{p}I_{\widetilde{G}}$

and the canonical map of $\mathbb{F}_{p}G$-modules

$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}\widetilde{G}}}\mathbb{F}_{p}I_{\widetilde{G}}\longrightarrow\mathbb{F}_{p}I_{G}.$
(7.1)

If we check that the previous map is an isomorphism, we would deduce, by
6.3.1, that

$\mathbb{F}_{p}\mathbin{\mathop{\otimes}\limits_{\mathbb{Z}}}\ker\phi=0,$

implying that $\ker\phi=1$, since it is free. In order to check that the map
of (7.1) is an isomorphism, we develop a “dimension function” on
$\mathbb{F}_{p}G$-modules. This map is clearly surjective and, assuming
reasonable properties about this “dimension function”, if we prove that the
dimension of both involved modules is the same, then the surjective map of
(7.1) would be an isomorphism, as we want.

This dimension function would be defined as follows. Let
$\mathbb{F}_{p}G\xhookrightarrow{}\mathcal{D}$ be an embedding of
$\mathbb{F}_{p}G$ into a division ring $\mathcal{D}$. A tentative “dimension
function” for the collection of $\mathbb{F}_{p}G$-modules $M$ goes as follows.
We can define

$\dim
M=\dim_{\mathcal{D}}\mathcal{D}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M.$

However, for this dimension function to enjoy desirable properties, the ring
$\mathcal{D}$ cannot be any division ring. It has to be “minimal” and
“universal” in some sense. We will now give a brief account of how these
notions are precised for some classes of rings, before introducing the
universal division ring of fractions of $\mathbb{F}_{p}G$.

###  7.1 Embeddability of domains into division rings

For a ring to be embeddable into a division ring, this ring must be a domain.
In the commutative setting, this condition is not only necessary but also
sufficient for the existence of such embedding. In fact, given a commutative
domain $A$, there is the standard construction of the field of fractions
$\operatorname{Frac}(A)$ of $A$ with a canonical embedding
$A\xhookrightarrow{}\operatorname{Frac}(A)$. It satisfies that, for any ring
homomorphism $f\mathrel{\mathop{\ordinarycolon}}A\rightarrow B$ such that
$f(A\setminus\\{0\\})\subseteq B^{\times}$, there exists a unique
$\widehat{f}\mathrel{\mathop{\ordinarycolon}}\operatorname{Frac}(A)\rightarrow
B$ such that $\widehat{f}\circ\iota=f$; that is, such that the diagram

${A}$${\operatorname{Frac}(A)}$${B}$$\scriptstyle{\iota}$$\scriptstyle{f}$$\scriptstyle{\widehat{f}}$
(7.2)

is commutative.

Underlying the construction of $\operatorname{Frac}(A)$, there is the more
general notion of localization, of fundamental importance in algebra and
geometry. There is also a non-commutative analogue of this notion, namely the
localization in the sense of Ore.

Given a ring $R$ and a (multiplicatively closed) subset $S\subseteq R$, we ask
for the existence of a ring $R_{S}$ equipped with an injective map
$\phi_{S}\mathrel{\mathop{\ordinarycolon}}R\longrightarrow R_{S}$ such that

  1. 1.

the map $\phi_{S}$ is injective;

  2. 2.

any element of $R_{S}$ can be written in the form
$\phi_{S}(r)\phi_{S}(s)^{-1}$ for some $r\in R$ and $s\in S$; and

  3. 3.

for any ring homomorphism $\psi\mathrel{\mathop{\ordinarycolon}}R\rightarrow
B$ with $\psi(S)\subseteq B^{\times}$, there exists a unique
$\widehat{\psi_{S}}\mathrel{\mathop{\ordinarycolon}}R_{S}\rightarrow B$ such
that $\psi=\widehat{\psi}\circ\phi_{S}$. In other words, the diagram

${A}$${S^{-1}A}$${B,}$$\scriptstyle{\phi_{S}}$$\scriptstyle{\psi}$$\scriptstyle{\widehat{\psi}}$
(7.3)

would be commutative.

The natural candidate for $R_{S}$, and, in fact, the only one, is constructed
from $R$ by attaching inverses $t_{s}$ to each element $s\in S$ as follows

$R_{S}=\frac{R\,\langle t_{s};s\in S\rangle}{\langle t_{s}s-1,st_{s}-1;s\in
S\rangle}.$ (7.4)

It is easy to check that it the universal property of the diagram (7.3). It
would remain to ask under which conditions can we ensure that the canonical
map $\phi_{S}\mathrel{\mathop{\ordinarycolon}}R\rightarrow R_{S}$ is injective
and that the elements of $R_{S}$ can be written as $a\,s^{-1}$. In this case,
we name $R_{S}$ the right Ore localisation of $R$ with respect to $S$. We give
sufficient conditions due to Asano and Gabriel, inspired by the work of Ore
[42].

######  Definition 7.1.1.

We say that $S\subseteq R$ is a right Ore set of regular elements if

  1. 1.

$S$ is closed under multiplication.

  2. 2.

For any $a\in R$ and $s\in S$, $as=0$ implies that $a=0$.

  3. 3.

For any $a\in R$ and $s\in S$, $aS\bigcap sR\neq\emptyset$.

The following statement is taken from the book of Cohn [12, Theorem 0.7.1].

######  Theorem 7.1.2 (Criterion for faithful localization).

Let $S\subseteq R$ be closed under multiplication. Then the canonical map
$R\rightarrow R_{S}$, defined in (7.4), is an embedding and every element of
$R_{S}$ can be written in the form $r\,s^{-1}$, for $r\in R$, $s\in S$, if and
only if $S$ is a right Ore set of regular elements.

######  Definition 7.1.3.

A ring $R$ is said to be a right Ore domain if $S=R\setminus\\{0\\}$ is a
right Ore set of regular elements. In this case, $R$ can be embedded into
$R_{S}$, which is a division ring generated by $R$. We name
$Q_{\text{ore}}(R)=R_{S}$ the (right) Ore division ring of fractions.

It is not very difficult to find a big family of non-commutative Ore domains.

######  Proposition 7.1.4 (“Little Goldie’s theorem”, 1957).

Let $R$ be a right Noetherian domain, then $S=R\setminus\\{0\\}$ is a right
Ore set of regular elements. In particular, $R$ can be embedded into its right
Ore ring of fractions, which is a division ring.

###### Proof.

To check that $S=R\setminus\\{0\\}$ is a right Ore set of regular elements, we
only have to verify that for any $a\in R$ and $s\in S$, $aS\bigcap
sR\neq\emptyset$. If $a=0$, the previous intersection contains zero. Suppose
that $a\neq 0$ and consider the chain of right ideals
$I_{m}=(s,as,a^{2}s,\dots,a^{m}s)$, which is increasing, so it has to
stabilise. In particular, there exists $n\geq 1$ so

$a^{n}s=\sum_{k=0}^{n-1}a^{k}\,s\,r_{k},\,\,\,\mbox{for some $r_{k}\in R$.}$

Consider the minimal $k_{0}$ so $r_{k_{0}}\neq 0$, then

$a\left(a^{n-k_{0}-1}s-\sum_{k=k_{0}}^{n-2}a^{k-k_{0}}\,s\,r_{k}\right)=sr_{k_{0}}\in
aS\bigcap sR.\qed$

This produces many examples of Ore domains, such as polynomial algebras. We
review the situation by saying that an Ore domain $R$ can be embedded into its
Ore division ring of fractions
$\iota_{\operatorname{ore}}\mathrel{\mathop{\ordinarycolon}}R\xhookrightarrow{}Q_{\text{ore}}(R)$
and that this embedding verifies the following universal property. For any
ring homomorphism $\psi\mathrel{\mathop{\ordinarycolon}}R\longrightarrow B$
such that $\psi(R\setminus\\{0\\})\subseteq B^{*}$, there exists a unique
$\widehat{\psi}$ such that the following diagram

${R}$${Q_{\text{ore}}(R)}$${B}$$\scriptstyle{\iota_{\operatorname{ore}}}$$\scriptstyle{\psi}$$\scriptstyle{\widehat{\psi}}$
(7.5)

is commutative. In addition, if $\psi$ is injective, the resulting
$\widehat{\psi}$ is injective, too. Moreover, $Q_{\text{ore}}(R)$ is a flat
$R$-module. This means that the functor
$\mathbin{\mathop{\otimes}\limits_{R}}Q_{\text{ore}}(R)$, from the category of
$R$-modules to the category of abelian groups; preserves exactness of short
exact sequences.

When discussing the question of embedability of domains into division rings,
it is natural, for historical reasons, to introduce the class of Ore domains.
Furthermore, these domains will be part of our arguments in section 7.6.
However, the class of Ore domains is still a very restrictive one. In fact, a
result of Bartholdi and Kielak [1] states that a group ring $KG$, with
coefficients in a field $K$, is an Ore domain if and only if $KG$ is a domain
and $G$ is amenable. For this reason, the previous methods will not be
directly applicable to our groups of interest, namely abstract subgroups $G$
of free pro-$p$ groups ${\bf F}$. The crucial aspect of these groups $G$ is
that they are orderable.

Malcev and Neumann discovered independently a way to embed the group ring of
an orderable group into a division rings of power series. For the group
$\mathbb{Z}$, notice that $K\mathbb{Z}\cong K[X,X^{-1}]$. The embedding
provided by Malcev and Neumann, where $\mathbb{Z}$ is viewed with the standard
ordering, would be $K[X,X^{-1}]\xhookrightarrow{}K((X))$, where $K((X))$
denotes the ring of one-variable Laurent series with coefficients in the field
$K$.

######  Definition 7.1.5 (Malcev-Neumann construction of $D((G))$).

Given a division ring $D$ and an orderable group $G$, we define the formal
power series of $G$ over $D$, denoted $D((G))$, as the set of formal sums

$\left\\{\sum_{g\in G}a_{g}\,g\mathrel{\mathop{\ordinarycolon}}\mbox{the
support $\\{g\mathrel{\mathop{\ordinarycolon}}a_{g}\neq 0\\}$ is well-
ordered}\right\\}.$

The operations on $D((G))$ are defined analogously as in $K((X)).$

$\left(\sum_{g}a_{g}\,g\right)+\left(\sum_{g}b_{g}\,g\right)=\sum_{g}(a_{g}+b_{g})\,g.$
(7.6)
$\left(\sum_{g}a_{g}\,g\right)\left(\sum_{h}b_{h}\,h\right)=\sum_{k}\left(\sum_{gh=k}a_{g}\,b_{h}\right)k.$
(7.7)

The difficult technical lemma about the order on $G$ which allows one to prove
that (7.7) is well defined and that $D((G))$ is a division ring is the
following.

######  Lemma 7.1.6.

Let $G$ be an orderable group and let $S\subseteq G$ be a well-ordered subset
such that $s>1$ for every $s\in S$. Consider

$S^{\omega}=\bigcup_{n\geq 1}S^{n}.$

Then $S^{\omega}$ is well-ordered and every $s\in S^{\omega}$ appears only in
finitely many sets $S^{n}$.

Given an element $0\neq\sum_{g\in G}a_{g}\,g\in D((G))$, we use the previous
lemma to explicitly write down its inverse. Since its support is well-ordered,
it has a minimum $g_{0}\in G$, and we can write

$0\neq\sum_{g\in G}a_{g}\,g=a_{g_{0}}\,g_{0}\,\left(1+\sum_{1<g\in
G}b_{g}\,g\right).$

This way, writing $c=\sum_{1<g\in G}b_{g}\,g$, its inverse will be

$\left(\sum_{k=0}^{\infty}(-1)^{k}c^{k}\right)a_{g_{0}}^{-1}\,g_{0}^{-1}.$
(7.8)

We refer to the paper of Neumann [40, Part I] for details.

This provides embeddings into division rings for group rings of orderable
groups. However, for these group rings, there may not exist a division ring
$\mathcal{D}$ and a monomorphism $R\xhookrightarrow{}\mathcal{D}$ verifying a
universal property in the sense of the commutative diagram (7.5).

There is still a suitable and weakened notion of universality for division
rings, due to Cohn, which we shall discuss next.

###  7.2 Definition and existence of the ring $\mathcal{D}_{R}$

######  Definition 7.2.1 (Division closure).

Given a ring extension $R_{0}\subseteq\mathcal{D}_{0}$, where
$\mathcal{D}_{0}$ is a division ring, we say that
$\mathcal{D}\subseteq\mathcal{D}_{0}$ is the division closure of $R_{0}$ in
$\mathcal{D}_{0}$ if $R_{0}$ generates rationally $\mathcal{D}_{0}$. In other
words, each element of $\mathcal{D}_{0}$ can be built up from the elements of
$R_{0}$ in stages, using addition, subtraction, multiplication, and division
by nonzero elements. We denote the division closure of $R_{0}$ in
$\mathcal{D}_{0}$ by $\mathcal{D}=\mathcal{D}(R_{0},\mathcal{D}_{0})$ or,
simply, by $\mathcal{D}(R_{0})$.

######  Definition 7.2.2.

A $R$-ring is a homomorphism
$\phi\mathrel{\mathop{\ordinarycolon}}R\rightarrow S$. Two $R$-rings
$\phi_{1}\mathrel{\mathop{\ordinarycolon}}R\rightarrow S_{1}$ and
$\phi_{2}\mathrel{\mathop{\ordinarycolon}}R\rightarrow S_{2}$ are said to be
isomorphic if there exists a ring isomorphism
$\alpha\mathrel{\mathop{\ordinarycolon}}S_{1}\rightarrow S_{2}$ such that
$\phi_{2}=\alpha\circ\phi_{1}$.

We now precise Cohn’s notion of universal division ring of fractions.

######  Definition 7.2.3.

Consider a $R$-ring
$\phi\mathrel{\mathop{\ordinarycolon}}R\longrightarrow\mathcal{D}$. We also
say that it is

  1. 1.

a division $R$-ring if $\phi$ is injective and $\mathcal{D}$ is a division
ring;

  2. 2.

a $R$-ring of fractions if $\mathcal{D}$ is generated by $\phi(R)$.

If the previous conditions hold, then we say that
$\phi\mathrel{\mathop{\ordinarycolon}}R\xhookrightarrow{}\mathcal{D}$ is a
division ring of fractions of $R$.

######  Definition 7.2.4.

Let $R$ be a ring and let $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ be two
division $R$-rings of fractions.

  * •

We say write $\dim_{\mathcal{D}_{1}}\leq\dim_{\mathcal{D}_{2}}$ if any
finitely presented module $M$, it follows that
$\dim_{\mathcal{D}}\mathcal{D}\mathbin{\mathop{\otimes}\limits_{R}}M\leq\dim_{\mathcal{D}^{\prime}}D^{\prime}\mathbin{\mathop{\otimes}\limits_{R}}M$.

  * •

We say that there is a specialization from $\mathcal{D}_{1}$ to
$\mathcal{D}_{2}$ if there exists a local subring $R\subseteq
B\subseteq\mathcal{D}_{1}$ with residual division ring isomorphic to
$\mathcal{D}_{2}$.

######  Definition 7.2.5.

Let $\mathcal{D}_{R}$ be a division $R$-ring of fractions. We say that it is a
universal division $R$-ring of fractions if, for any other division $R$-ring
of fractions $\mathcal{D}^{\prime}$,
$\dim_{\mathcal{D}}\leq\dim_{\mathcal{D}^{\prime}}.$

We can give a few families of examples.

######  Example 7.2.6.

The following rings $R$ have a universal division $R$-ring of fractions
$\mathcal{D}_{R}$.

  * •

Commutative domains $A$, for which
$\mathcal{D}_{A}\cong\operatorname{Frac}(A)$.

  * •

More generally, right (resp. left) Ore domains $R$, for which
$\mathcal{D}_{R}$ is their right (resp. left) Ore division ring of fractions
$\mathcal{D}_{R}\cong Q_{\text{ore}}(R)$.

  * •

Semifirs [10]: A ring $R$ is said to be a semifir if every finitely generated
left ideal of $R$ is free of fixed rank.

  * •

In particular, if $K$ is a field, then $R=K\langle\langle X\rangle\rangle$ has
universal $\mathcal{D}_{R}$ since $R$ is a semifir by [12, Proposition
2.9.19]).

Another significant example for us is the completed group algebra
$\mathbb{F}_{p}[[{\bf F}]]$. We know, from 4.4.2, that this ring is isomorphic
to $\mathbb{F}_{p}\langle\langle X\rangle\rangle$, where $|X|=d({\bf F})$.
From the previous examples, we know the following.

######  Proposition 7.2.7.

The ring $\mathbb{F}_{p}[[{\bf F}]]$ has a universal division ring of
fractions $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}.$

As we remarked before, there is no chance in having universal property for the
embedding $R\xhookrightarrow{}\mathcal{D}_{R}$ as in the setting of Ore
division rings of fractions (7.5). In fact, Herbera and Sánchez [22] prove
that, if $R=K\langle X\rangle$ with $|X|>1$, there are infinitely many non-
isomorphic $R$-division rings of fractions. However, for some favourable
situations one still expects to be able to compare different $R$-division
rings. For example, let $G$ be a subgroup of ${\bf F}$. Then $\mathbb{F}_{p}G$
embeds into $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$ and also into
$\mathbb{F}_{p}((G))$, after giving $G$ some ordering. It is not easy to
compare both embeddings nor to study whether one of them turns out to provide
a universal division $\mathbb{F}_{p}G$-ring. Both questions are answered
positively. Interestingly, Cohn and Hughes developed some methods that allow
us to compare different $R$-division rings and, in particular, to prove the
previous claim about $\mathbb{F}_{p}G-$division rings of fractions. First of
all, let us mention a more elementary criterion that can be used to compare
division $R$-rings for Ore domains $R$.

######  Lemma 7.2.8.

Suppose that $R$ is an (left or right) Ore domain and that
$R\xhookrightarrow{}\mathcal{D}$ is a division $R$-ring of fractions. Then
$\mathcal{D}$ and $Q_{\text{ore}}(R)$ are isomorphic $R$-rings.

###### Proof.

By the universal property (7.5) of the Ore division $R$-ring of fractions
$R\xhookrightarrow{}Q_{\text{ore}}(R)$, there is a commutative diagram

${R}$${Q_{\text{ore}}(R)}$${\mathcal{D},}$$\scriptstyle{\iota_{\operatorname{ore}}}$$\scriptstyle{\psi}$$\scriptstyle{\widehat{\psi}}$
(7.9)

with $\widehat{\psi}$ injective. Furthermore, since
$R\xhookrightarrow{}\mathcal{D}$ is a division $R$-ring of fractions, then
$\widehat{\psi}$ is surjective. So $\widehat{\psi}$ defines an isomorphism
between the $R$-rings $\mathcal{D}$ and $Q_{\text{ore}}(R)$. ∎

We will talk about Hughes criterion later, we start with two results of Cohn
[11, Theorem 4.4.1 and Subsection 4.1].

######  Proposition 7.2.9.

Let $R$ be a ring and let $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ be two
division $R$-rings of fractions. Then the following holds.

  * •

The rings $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are isomorphic, as
$R$-rings, if and only if $\dim_{\mathcal{D}_{1}}=\dim_{\mathcal{D}_{2}}$.

  * •

There is a specialization from $\mathcal{D}_{1}$ to $\mathcal{D}_{2}$ if and
only if $\dim_{\mathcal{D}_{1}}\leq\dim_{\mathcal{D}_{2}}$.

In particular, if the universal division ring of fractions $\mathcal{D}_{R}$
exists, then it is unique.

######  Definition 7.2.10.

Let $R\xhookrightarrow{}\mathcal{D}_{R}$ be a universal division $R$-ring of
fractions. Given a $R$-module $M$, we denote

$\dim_{R}M=\dim_{\mathcal{D}}\mathcal{D}\mathbin{\mathop{\otimes}\limits_{R}}M.$

By 7.2.9, this is well-defined.

In section 7.4, we will use the criterion of 7.2.9 to check that for any
closed subgroup ${\bf H}$ of ${\bf F}$, the division closure of
$\mathbb{F}_{p}[[{\bf H}]]$ in $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$ is
isomorphic to $\mathcal{D}_{\mathbb{F}_{p}[[{\bf H}]]}.$

Using this fact, we would later prove, in section 7.5, that
$\mathcal{D}_{\mathbb{F}_{p}G}$ exists and that the division closure of
$\mathbb{F}_{p}G$ in $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$ is isomorphic
to $\mathcal{D}_{\mathbb{F}_{p}G}$. However, in this case, we will not use
7.2.9, but a criterion of Hughes on group rings of locally indicable groups
(2.2.10).

###  7.3 Sylvester rank functions

Before establishing properties about the rings $\mathcal{D}_{\mathbb{F}_{p}G}$
and $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$, we discuss some properties
about their induced dimension functions in terms of Sylvester rank functions.

Let $M$ be a finitely presented $R$-module. This is, there is an exact
sequence of $R$-modules

$R^{m}\longrightarrow R^{n}\longrightarrow M\longrightarrow 0.$

In other words, there is a matrix $A\in M_{m\times n}(R)$ such that

$M\cong\frac{R^{n}}{R^{m}A}.$

The consideration of a reasonable “dimension function” $\dim$ on $R$-modules
$M$ is related to a reasonable “rank function” on matrices $A\in M_{m\times
n}(R)$.

For example, let $\mathcal{D}$ be a division $R$-ring. Then

$\dim_{\mathcal{D}}\mathcal{D}\mathbin{\mathop{\otimes}\limits_{R}}M=n-\text{rk}_{\mathcal{D}}(A),$
(7.10)

where $\text{rk}_{\mathcal{D}}(A)$ is the rank of $A$ as a matrix with
coefficients in the division ring $\mathcal{D}$.

For more general rings, that are not division rings, we do not expect a notion
of rank as well behaved as the one coming from linear algebra, though we are
interested in preserving some of them.

######  Definition 7.3.1.

Let $R$ be a ring. A Sylvester matrix rank function $\operatorname{rk}$ on $R$
is a function that assigns a non-negative real number to each matrix over $R$
and that satisfies the following conditions.

  1. 1.

$\operatorname{rk}(A)=0$ if $A$ is any zero matrix and
$\operatorname{rk}(1)=1$.

  2. 2.

For all matrices $A_{1},A_{2}$

$\operatorname{rk}\begin{pmatrix}A_{1}&0\\\
0&A_{2}\end{pmatrix}=\operatorname{rk}(A_{1})+\operatorname{rk}(A_{2}).$

  3. 3.

For all matrices $A_{1},A_{2},A_{3}$, of appropriate sizes,

$\operatorname{rk}\begin{pmatrix}A_{1}&A_{3}\\\
0&A_{2}\end{pmatrix}\geq\operatorname{rk}(A_{1})+\operatorname{rk}(A_{2}).$

Given a Sylvester matrix rank function $\operatorname{rk}$ on $R$, we have a
“dimension” function on finitely-presented $R$-modules by setting
$\dim(R^{n}/R^{m}A)=n-\operatorname{rk}(A)$, as occurs in (7.10). This
$\operatorname{rk}$ enjoys the following properties.

######  Definition 7.3.2.

A Sylvester module rank function on $R$ is a function that assings a non-
negative real number to each finitely presented $R$-module and that satisfies
the following conditions.

  1. 1.

$\dim(\\{0\\})=0$ and $\dim R=1$.

  2. 2.

$\dim(M_{1}\oplus M_{2})=\dim M_{1}+\dim M_{2}$, and

  3. 3.

if $M_{1}\longrightarrow M_{2}\longrightarrow M_{3}\longrightarrow 0$ is
exact, then

$\dim M_{1}+\dim M_{3}\geq\dim M_{2}\geq\dim M_{3}.$

By previous comments, we know there is a natural correspondence between
Sylvester matrix rank functions and Sylvester matrix module functions. Our
next aim is to discuss a few aspects about the convergence of these Sylvester
rank functions. However, these notions uniquely provide a “dimension function”
for finitely presented modules. Recall that we are also interested in finitely
generated modules. Given a Sylvester matrix rank function, Li [32] provides a
way to extend the induced dimension function to finitely generated modules so
it verifies some favourable properties, as those listed in 7.3.2.

######  Definition 7.3.3.

Given a Sylvester module rank function $\dim$, we define its extension to
finitely generated modules $M$ as follows.

$\dim M=\inf\\{\dim\widetilde{M}\mathrel{\mathop{\ordinarycolon}}\mbox{
$\widetilde{M}$ is finitely presented and $M$ is a quotient of
$\widetilde{M}$}\\}.$

The following lemma reflects that this is a natural extension for the type of
module rank functions we are interested in.

######  Lemma 7.3.4.

Let $R\xhookrightarrow{}\mathcal{D}$ be a division $R$-ring. We consider the
Sylvester module rank function $\dim$ defined by

$\dim M=\dim_{\mathcal{D}}\mathcal{D}\mathbin{\mathop{\otimes}\limits_{R}}M.$

Then the extension $\widetilde{\dim}$ to finitely generated modules $M$
verifies that

$\widetilde{\dim}M=\dim_{\mathcal{D}}\mathcal{D}\mathbin{\mathop{\otimes}\limits_{R}}M.$

This exhibits that in order to study these dimension functions, it suffices to
restrict ourselves to the study of finitely presented modules. However, for
other Sylvester rank functions, their extensions are not as easy to depict.
Before describing more precise examples about this phenomena, we introduce the
useful notion of convergence in Sylvester rank functions.

######  Definition 7.3.5.

Let $\dim_{1}$ and $\dim_{2}$ be two Sylvester module rank functions on $R$.
We write $\dim_{1}\geq\dim_{2}$ if, for every finitely presented $R$-module
$M$, $\dim_{1}M\geq\dim_{2}M$.

By 7.2.9, we can say that the universal division $R$-ring of fractions
$\mathcal{D}_{R}$ is the the division $R$-ring of fractions which induces the
minimal Sylvester module rank function $\dim_{R}$.

######  Definition 7.3.6 (Convergence).

Let $\dim,\dim_{i}$ be two Sylvester module rank functions on $R$. We write
$\operatorname{rk}=\lim_{i\rightarrow\infty}\operatorname{rk}_{i}$ if, for
every finitely presented $R$-module $M$, $\dim
M=\lim_{i\rightarrow\infty}\dim_{i}M$.

Equivalently, one could define the previous notion of convergence by setting a
right topology in the collection of Sylvester matrix rank functions.
Precisely, we denote by $\mathbb{P}(R)$ the collection of Sylvester matrix
rank functions on $R$ and we view $\mathbb{P}(R)$ as a subspace of the space
of real-valued functions on matrices, $\mathbb{R}^{\mathrm{Mat}}(R)$. This
bigger space is endowed with the topology of pointwise convergence. Then
$\mathbb{P}(R)$ is a compact convex subset of $\mathbb{R}^{\mathrm{Mat}(R)}$.

There are many interesting problems, for example, in the surroundings of Lück
approximation conjectures, which can be rephrased, and generalised, in terms
of convergence principles of certain Sylvester matrix rank functions.

######  Lemma 7.3.7.

Let $M$ be finitely generated $R$-module and suppose that
$\dim=\lim_{i\rightarrow\infty}\dim_{i}$. Then

$\dim M\geq\limsup_{i\rightarrow\infty}\dim M_{i}.$

If, in addition, $\dim_{i}\geq\dim$ for all $i$, then

$\dim M=\lim_{i\rightarrow\infty}\dim M_{i},\,\,\,\,\mbox{for all finitely
generated $R$-modules $M$.}$

###### Proof.

Let $W$ be the set of all finitely presented $R$-modules $\widetilde{M}$ such
that $M$ is a quotient of $\widetilde{M}$. Then

$\displaystyle\dim M=$ $\displaystyle\inf_{\widetilde{M}\in
W}\dim\widetilde{M}$ $\displaystyle=$ $\displaystyle\inf_{M\in
W}\limsup_{i\rightarrow\infty}\dim_{i}\widetilde{M}$ $\displaystyle\geq$
$\displaystyle\limsup_{i\rightarrow\infty}\inf_{M\in W}\dim_{i}\widetilde{M}$
$\displaystyle=$ $\displaystyle\limsup_{i\rightarrow\infty}\dim_{i}M,$

and the first part of the statement follows. If, in addition,
$\dim_{i}\geq\dim$, then it is clear that

$\liminf_{i\rightarrow\infty}\dim_{i}M\geq\dim M,$

so it follows that $\lim_{i\rightarrow\infty}\dim_{i}M$ exists and equals
$\dim M$. ∎

###  7.4 The ring $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$

Our next aim is to study concrete convergence of Sylvester module rank
functions of $\mathbb{F}_{p}[[{\bf F}]]$, following the treatment of [26,
Section 3.1].

######  Proposition 7.4.1.

Let ${\bf F}={\bf N}_{1}>{\bf N}_{2}>\cdots$ be a chain of open normal
subgroups with trivial intersection. Let $M$ be a finitely generated
$\mathbb{F}_{p}[{\bf F}]]$-module. Then

$\dim_{\mathbb{F}_{p}[[{\bf
F}]]}M=\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf
F}/{\bf N}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}M\right)}{|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf N}_{i}|}.$

###### Proof.

Let ${\bf N}$ be a normal open subgroup of ${\bf F}$, we define the Sylvester
module rank function $\dim_{\mathbb{F}_{p}[{\bf F}/{\bf N}]}$ on
$\mathbb{F}_{p}[[{\bf F}]]$ by

$\dim_{\mathbb{F}_{p}[{\bf F}/{\bf
N}]}M=\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf F}/{\bf
N}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}M\right)}{|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf N}|}.$

We claim that

$\dim_{\mathbb{F}_{p}[{\bf F}/{\bf N}]}\geq\dim_{\mathbb{F}_{p}[[{\bf F}]]}.$
(7.11)

In fact, let $M$ be a finitely generated $\mathbb{F}_{p}[[{\bf F}]]$-module
with

$\dim_{\mathbb{F}_{p}[[{\bf F}]]}=\dim_{\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}}\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf F}]]}}M=k.$

Then there exist $k$ elements $\\{m_{1},\dots,m_{k}\\}\subseteq M$ such that
$\\{1\otimes m_{1},\dots,1\otimes m_{k}\\}$ are
$\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$-linearly independent in
$\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf F}]]}}M$. In
particular, the $\mathbb{F}_{p}[[{\bf F}]]$-submodule $N$ spanned by
$\\{m_{1},\dots,m_{k}\\}$ is free of rank $k$. So $\mathbb{F}_{p}[{\bf F}/{\bf
N}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}N\cong\mathbb{F}_{p}[{\bf F}/{\bf N}]^{k}$ is a
$\mathbb{F}_{p}$-subspace of $\mathbb{F}_{p}[{\bf F}/{\bf
N}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf F}]]}}M$ of
dimension $k|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf N}|$, implying that

$\dim_{\mathbb{F}_{p}[{\bf F}/{\bf
N}]}(M)\geq\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf F}/{\bf
N}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}N\right)}{|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf
N}|}=k=\dim_{\mathbb{F}_{p}[[{\bf F}]]}M.$

This proves (7.11).

Furthermore, if $M$ is a finitely presented $\mathbb{F}_{p}[[{\bf
F}]]$-module, then [24, Theorem 1.4] implies that

$\dim_{\mathbb{F}_{p}[[{\bf
F}]]}M=\lim_{i\rightarrow\infty}\dim_{\mathbb{F}_{p}[{\bf F}/{\bf N}_{i}]}M.$
(7.12)

With (7.11) and (7.12), we can apply lemma 7.3.7 to conclude that for all
finitely generated $\mathbb{F}_{p}[[{\bf F}]]$-module $M$,

$\dim_{\mathbb{F}_{p}[[{\bf
F}]]}M=\lim_{i\rightarrow\infty}\dim_{\mathbb{F}_{p}[{\bf F}/{\bf
N}_{i}]}M.\qed$

The next proposition will allow us to view $\mathcal{D}_{\mathbb{F}_{p}[[{\bf
H}]]}$ inside $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$ whenever ${\bf H}$ is
an open subgroup of ${\bf F}$.

######  Proposition 7.4.2.

Let ${\bf H}$ be a closed finitely generated subgroup of ${\bf F}$. The
following holds.

  * (a)

Let $\mathcal{D}_{{\bf H}}$ be the division closure of $\mathbb{F}_{p}[[{\bf
H}]]$ in $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$. Then $\mathcal{D}_{H}$ and
$\mathcal{D}_{\mathbb{F}_{p}[[{\bf H}]]}$ are isomorphic $\mathbb{F}_{p}[[{\bf
H}]]$-rings.

  * (b)

If $M$ is a finitely generated $\mathbb{F}_{p}[[{\bf H}]]$-module, then

$\dim_{\mathbb{F}_{p}[[{\bf H}]]}M=\dim_{\mathbb{F}_{p}[[{\bf
F}]]}\left(\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf H}]]}}M\right).$

  * (c)

If ${\bf H}$ is open, then

$\dim_{\mathcal{D}_{{\bf H}}}\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}=|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf H}|.$

  * (d)

If ${\bf H}$ is open, then

$\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}\cong\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}\mathcal{D}_{\mathbb{F}_{p}[[{\bf H}]]}$

canonically as $(\mathbb{F}_{p}[[{\bf F}]],\mathbb{F}_{p}[[{\bf
H}]])$-bimodules.

###### Proof.

By 4.5.8, the pro-$p$ group ${\bf H}$ is free. Let ${\bf F}={\bf N}_{1}>{\bf
N}_{2}>\cdots$ be a chain of open normal subgroups with trivial intersection.
Define ${\bf H}_{i}={\bf N}_{i}\bigcap{\bf H}$. Then ${\bf H}={\bf H}_{1}>{\bf
H}_{2}>\cdots$ is a chain of open normal subgroups with trivial intersection.
Let $M$ be a finitely generated $\mathbb{F}_{p}[[{\bf H}]]$-module.

(a) By 7.4.1,

$\dim_{\mathbb{F}_{p}[[{\bf
H}]]}M=\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf
H}/{\bf H}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}M\right)}{|{\bf H}\mathrel{\mathop{\ordinarycolon}}{\bf H}_{i}|}.$
(7.13)

There is an injection of finite $p$-groups ${\bf H}/{\bf
H}_{i}\xhookrightarrow{}{\bf F}/{\bf N}_{i}$ and the image can be identified
with ${\bf H}{\bf N}_{i}/{\bf N}_{i}$. This yields to an isomorphism

$({\bf F}/{\bf N}_{i})/({\bf H}{\bf N}_{i}/{\bf N}_{i})\cong{\bf F}/{\bf
H}{\bf N}_{i}.$

Therefore, considering $\mathbb{F}_{p}[[{\bf F}/{\bf N}_{i}]]$ as a right
$\mathbb{F}_{p}[[{\bf H}]]$-module, we obtain that

$\mathbb{F}_{p}[{\bf F}/{\bf N}_{i}]\cong\mathbb{F}_{p}[{\bf H}/{\bf
H}_{i}]^{|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf H}{\bf N}_{i}|}$

as right $\mathbb{F}_{p}[[{\bf H}]]$-modules. So

$\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf H}/{\bf
H}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}M\right)=\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf F}/{\bf
N}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}M\right)}{|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf H}{\bf
N}_{i}|}.$

Combining the latter equation; the application of 7.4.1 to the finitely
generated $\mathbb{F}_{p}[[{\bf F}]]$-module $\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf H}]]}}M$; and
(7.13), we conclude that

$\displaystyle\dim_{\mathbb{F}_{p}[[{\bf F}]]}\left(\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf H}]]}}M\right)=$
$\displaystyle\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf
F}/{\bf N}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}(\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}M)\right)}{|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf N}_{i}|}$
$\displaystyle=$
$\displaystyle\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf
F}/{\bf N}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}M\right)}{|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf N}_{i}|}$
$\displaystyle=$
$\displaystyle\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf
H}/{\bf H}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}M\right)}{|{\bf H}\mathrel{\mathop{\ordinarycolon}}{\bf H}_{i}|}$
$\displaystyle=$ $\displaystyle\dim_{\mathbb{F}_{p}[[{\bf H}]]}M.$

On the other hand, since $\mathcal{D}_{{\bf H}}$ is the division closure of
$\mathbb{F}_{p}[[{\bf H}]]$ in $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$; then

$\displaystyle\dim_{\mathcal{D}_{\mathbb{F}_{p}[[{\bf
H}]]}}\left(\mathcal{D}_{\mathbb{F}_{p}[[{\bf
H}]]}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf H}]]}}M\right)=$
$\displaystyle\dim_{\mathbb{F}_{p}[[{\bf H}]]}M$ $\displaystyle=$
$\displaystyle\dim_{\mathbb{F}_{p}[[{\bf F}]]}\left(\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf H}]]}}M\right)$
$\displaystyle=$ $\displaystyle\dim_{\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}}\left(\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}\left(\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}M\right)\right)$ $\displaystyle=$
$\displaystyle\dim_{\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}}\left(\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf H}]]}}M\right)$
$\displaystyle=$ $\displaystyle\dim_{\mathcal{D}_{{\bf
H}}}\left(\mathcal{D}_{{\bf
H}}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf H}]]}}M\right).$

By Cohn’s criterion (7.2.9), the division $\mathbb{F}_{p}[[{\bf H}]]$-rings of
fractions $\mathcal{D}_{{\bf H}}$ and $\mathcal{D}_{\mathbb{F}_{p}[[{\bf
H}]]}$ are isomorphic.

(b) This was verfied during the proof of (a).

(c) Given a $\mathbb{F}_{p}[[{\bf F}]]$-module $M$, there are canonical
isomorphisms of $\mathbb{F}_{p}$-vector spaces $\mathbb{F}_{p}[{\bf H}/{\bf
H}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}M\cong\mathbb{F}_{p}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}_{i}]]}}M$ and $\mathbb{F}_{p}[{\bf F}/{\bf
N}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}M\cong\mathbb{F}_{p}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
N}_{i}]]}}M$. Moreover, since ${\bf H}$ is open then by compactness there
exists $N$ such that, for all $i\geq N$, ${\bf N}_{i}\subseteq{\bf H}$ and
hence ${\bf N}_{i}={\bf H}_{i}$. Now using the latter observations and (a) we
get

$\displaystyle\dim_{\mathcal{D}_{{\bf H}}}\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}$ $\displaystyle=\dim_{\mathbb{F}_{p}[[{\bf
H}]]}\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$
$\displaystyle=\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf
H}/{\bf N}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}\right)}{|{\bf
H}\mathrel{\mathop{\ordinarycolon}}{\bf N}_{i}|}$
$\displaystyle=\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
N}_{i}]]}}\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}\right)}{|{\bf
H}\mathrel{\mathop{\ordinarycolon}}{\bf N}_{i}|}$ $\displaystyle=|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf
H}|\,\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
N}_{i}]]}}\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}\right)}{|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf N}_{i}|}$ $\displaystyle=|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf
H}|\,\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}[{\bf F}/{\bf
N}_{i}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}\right)}{|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf N}_{i}|}$ $\displaystyle=|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf H}|\,\dim_{\mathbb{F}_{p}[[{\bf
F}]]}\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}=|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf H}|.$

(d) We first prove this when we additionally have that ${\bf H}$ is normal. In
this case, the automorphism-actions of ${\bf F}$ on $\mathcal{D}_{{\bf F}}$,
induced by conjugation, leave the subring $\mathbb{F}_{p}[[{\bf H}]]$
invariant; and, then, ${\bf F}$ also normalizes its division closure
$\mathcal{D}_{{\bf H}}$ in $\mathcal{D}_{N}$. As a consequence, the
$\mathbb{F}_{p}$ subspace ${\bf F}\mathcal{D}_{{\bf H}}$ spanned by all the
products $\\{f\cdot d\mathrel{\mathop{\ordinarycolon}}f\in{\bf
F},a\in\mathcal{D}_{{\bf H}}\\}$ is a subring of
$\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$. Moreover, it is clear that
$\dim_{\mathcal{D}_{H}}({\bf F}\mathcal{D}_{{\bf H}})\leq|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf H}|<\infty$. This implies that ${\bf
F}\mathcal{D}_{{\bf H}}$ is a domain and a finite dimensional
$\mathcal{D}_{{\bf H}}$-algebra; hence it is a division ring. Therefore, since
${\bf F}\mathcal{D}_{{\bf H}}$ contains $\mathbb{F}_{p}[[{\bf F}]]$ , ${\bf
F}\mathcal{D}_{{\bf H}}=\mathcal{D}_{\mathbb{F}_{p}{\bf F}}$. We deduce that
the canonical map of $(\mathbb{F}_{p}[[{\bf F}]],\mathcal{D}_{{\bf
H}})$-bimodules

$\alpha\mathrel{\mathop{\ordinarycolon}}\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}\mathcal{D}_{{\bf H}}\longrightarrow\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}$

can be read as a surjective homomorphism of right $\mathcal{D}_{{\bf
H}}$-modules of the same finite dimension by (b) and (c). So $\alpha$ is an
isomorphism.

Having verified the property of $(d)$ for open normal ${\bf H}\unlhd_{o}{\bf
F}$, we want to extend this property to all open ${\bf H}\leq_{o}{\bf F}$ by
induction on the index $|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf H}|$. If
$|{\bf F}\mathrel{\mathop{\ordinarycolon}}{\bf H}|\leq p$, then ${\bf
H}\unlhd_{o}{\bf F}$ satisfies the property. If $|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf H}|>p$, then there exists an open
normal ${\bf H}\leq_{o}{\bf H}_{1}\unlhd_{o}{\bf F}$ with $|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf H}_{1}|=p$. By the inductive
hypothesis,

$\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}\cong\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}\mathcal{D}_{{\bf H}}\cong\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}_{1}]]}}\left(\mathbb{F}_{p}[[{\bf
H}_{1}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}\mathcal{D}_{{\bf H}}\right)\cong\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}\mathcal{D}_{\mathbb{F}_{p}[[{\bf H}]]},$

with canonical isomorphisms of $(\mathbb{F}_{p}[[{\bf F}]],\mathcal{D}_{{\bf
H}})$-modules in each step. The induction is complete. ∎

From now on, we can identify $\mathcal{D}_{{\bf H}}$ and
$\mathcal{D}_{\mathbb{F}_{p}[[{\bf H}]]}$ as $\mathbb{F}_{p}[[{\bf H}]]$-rings
without explicit mention of it.

###  7.5 The ring $\mathcal{D}_{\mathbb{F}_{p}G}$

There are some convenient features about group rings $KG$ of locally indicable
groups. A result of Higman says that these rings are domains and that
$(KG)^{\times}$ is $K^{*}\times G$. We will talk about a specific type of
division rings of fractions of these group rings, introduced by Hughes.

######  Definition 7.5.1.

Let $\Gamma$ be a group and let $K\Gamma\xhookrightarrow{}\mathcal{D}$ be a
division $K\Gamma$-ring of fractions with the following property. For every
finitely generated $S\unlhd T\leq\Gamma$, such that $T=S\rtimes\mathbb{Z}$,
where $\mathbb{Z}\leq T$ is generated by $t\in T$, we have that

$\sum_{i\in\mathbb{Z}}KS\,t^{i}$ (7.14)

is a direct sum of $KS$-submodules of $KT\subseteq\mathcal{D}$. Then we say
that $\mathcal{D}$ is Hughes-free.

The following is proven in [23].

######  Theorem 7.5.2 (Hughes).

Let $\Gamma$ be a locally indicable group. There exists at most one Hughes-
free division $K\Gamma$-ring of fractions up to $K\Gamma$-isomorphism.

The following result is a consequence of [27, Theorem 1.1.].

######  Theorem 7.5.3 (Jaikin-Zapirain).

Let $G$ be a residually-(torsion-free nilpotent) group and let $E$ be a
division ring. Then there exists the universal division $EG$-ring of fractions
$\mathcal{D}_{EG}$ and it is Hughes-free.

As a consequence of the last two results, we can prove that a priori different
$\mathbb{F}_{p}G$-division rings of fractions are $\mathbb{F}_{p}G$-isomorphic
and universal, as the following corollary reflects. Let $G$ be an abstract
subgroup of ${\bf F}$, then $G$ is residually-(torsion-free nilpotent) by
4.3.7 and 7.5.3 applies.

######  Corollary 7.5.4.

Let ${\bf F}$ be a free pro-$p$ group and let $G\leq{\bf F}$ be a subgroup.

  1. 1.

Let $\mathcal{D}_{1}$ be the division closure of $\mathbb{F}_{p}G$ in
$\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$.

  2. 2.

Take any group order $\leq$ on $G$ and let $\mathcal{D}_{2}$ be the division
closure of $\mathbb{F}_{p}G$ in the corresponding $\mathbb{F}_{p}((G))$.

Then $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are both isomorphic, as
$\mathbb{F}_{p}G$-rings, to $\mathcal{D}_{\mathbb{F}_{p}G}$.

We divide the verification of this corollary in two lemmas. The first lemma
will correspond to the verification of the first part and relies on an
improvement of the Hughes-freeness criterion (7.5.2) that is due to Sánchez
[46, Theorem 6.3]. An alternative approach of the first part of 7.5.4 based on
Cohn’s criterion (7.2.9) is taken in [26, Proposition 3.4].

######  Lemma 7.5.5.

Let ${\bf F}$ be a free pro-$p$ group of finite rank and let $G\leq{\bf F}$ be
a subgroup. Let $\mathcal{D}_{G}$ be the division closure of $\mathbb{F}_{p}G$
in $\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$. Then $\mathcal{D}_{G}$ is
Hughes-free.

###### Proof.

Since $\mathbb{F}_{p}G$ has a Hughes-free division $\mathbb{F}_{p}G$-ring of
fractions (7.5.3), then in order to conclude that $\mathcal{D}_{G}$ is
isomorphic to $\mathcal{D}_{\mathbb{F}_{p}G}$ (and hence that it is Hughes-
free), it suffices to ensure by [46, Remark 6.4(a)] the following property:
for all finitely generated $T\leq G$, there exists a subgroup $N_{0}\unlhd T$
with $T=\langle N_{0},t\cong N_{0}\rtimes\mathbb{Z}$ such that (7.14) is
satisfied; or, in other words, such that the canonical map of
$(\mathbb{F}_{p}T,\mathcal{D}_{N_{0}})$-bimodules

$\delta\mathrel{\mathop{\ordinarycolon}}\mathbb{F}_{p}[T]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N_{0}}}\mathcal{D}_{N_{0}}\xhookrightarrow{}\mathcal{D}_{T}$
(7.15)

is injective.

Consider the pro-$p$ group $\overline{T}$. It is a closed subgroup of ${\bf
F}$. By 4.5.8, it is free of positive rank. As a consequence, there is a
continuous epimorphism
$f\mathrel{\mathop{\ordinarycolon}}\overline{T}\longrightarrow\mathbb{Z}_{p}$.
Its kernel ${\bf K}$ is a closed subgroup of $\overline{T}$. We consider
$K={\bf K}\bigcap T$. Notice that $T/K\cong\mathbb{Z}^{m}$ for some $m\geq 1$.

We want to prove that the canonical map of
$(\mathbb{F}_{p}T,\mathcal{D}_{K})$-bimodules

$\alpha\mathrel{\mathop{\ordinarycolon}}\mathbb{F}_{p}[T]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}K}}\mathcal{D}_{K}\longrightarrow\mathcal{D}_{T}$
(7.16)

is injective.

Recall that $\mathbb{F}_{p}T$ is a free right $\mathbb{F}_{p}K$-module with a
free basis given by a choice $\\{c_{i}\\}\subseteq T$ of representatives of
the left-cosets of $K$ in $T$. As a consequence,
$\mathbb{F}_{p}T\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}K}}\mathcal{D}_{K}$
is a free right $\mathcal{D}_{K}$-module of basis $\\{c_{i}\otimes 1\\}$.
Similarly, let ${\bf H}$ be an open subgroup of $\overline{T}$. Then
$\mathbb{F}_{p}[[\overline{T}]]$ is a free (profinite) right
$\mathbb{F}_{p}[[{\bf H}]]$-module with basis given by a choice of left coset
representatives $\\{c_{i}^{\prime}\\}$ of ${\bf H}$ in $\overline{T}$. In
particular, viewing
$\mathbb{F}_{p}[[\overline{T}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}\mathcal{D}_{{\bf H}}$ as a right $\mathcal{D}_{{\bf H}}$-module; the
set $\\{c_{i}^{\prime}\otimes 1\\}$ is $\mathcal{D}_{{\bf H}}$-linearly
independent.

Let us take some $0\neq x=\sum_{i=1}^{n}c_{i}\otimes
d_{i}\in\mathbb{F}_{p}T\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}K}}\mathcal{D}_{K}$,
where $n\geq 1$ and $0\neq d_{i}\in\mathcal{D}_{K}$. First of all, since
$K={\bf K}\bigcap T$, the elements $\\{c_{i}\\}$ represent different left-
cosets of ${\bf K}$ in $\overline{T}$. For these fixed $c_{1},\dots,c_{n}$;
there must exist an open subgroup ${\bf H}\leq\overline{T}$ such that the
elements $\\{c_{i}\\}_{1}^{n}$ represent different left-cosets of ${\bf H}$ in
$\overline{T}$.

There is a commutative diagram of canonical homomorphisms of
$(\mathbb{F}_{p}T,\mathcal{D}_{K})$-bimodules

${\mathbb{F}_{p}T\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}K}}D_{K}}$${\mathcal{D}_{T}}$${\mathbb{F}_{p}[[\overline{T}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
H}]]}}\mathcal{D}_{{\bf
H}}}$${\mathcal{D}_{\overline{T}},}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

where the map $\gamma$ is an isomorphism by 7.4.2.

By the choice of ${\bf H}$, $\beta(x)=\sum_{i=1}^{n}c_{i}\otimes d_{i}\neq 0$.
So, due to the commutativity of the above diagram, $\alpha(x)\neq 0$; hence
$\alpha$ is injective.

We select any $K\leq N\unlhd T$ such that $T/N\cong\mathbb{Z}$.

There is a commutative diagram of canonical homomorphisms of
$(\mathbb{F}_{p}T,\mathcal{D}_{K})$-bimodules

${\mathbb{F}_{p}N\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}K}}D_{K}}$${\mathcal{D}_{N}}$${\mathbb{F}_{p}T\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}K}}\mathcal{D}_{K}}$${\mathcal{D}_{T},}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\alpha}$

where $\alpha$ has been proven to be injective; and $\beta_{1}$ is injective
because $\mathbb{F}_{p}N$ is a direct summand of $\mathbb{F}_{p}T$ as right
$\mathbb{F}_{p}K$-modules. It follows that $\alpha_{1}$ is injective.

We denote $\mathcal{D}_{K}N$ to be the $\mathbb{F}_{p}$-linear span of
elements of the form $ab$ with $a\in\mathcal{D}_{K}$ and $b\in N$. This is a
subring of $\mathcal{D}_{T}$ because the automorphism-actions of $N$ on
$\mathcal{D}_{N}$, induced by conjugation, leave the subring $\mathbb{F}_{p}K$
invariant; and, then, $N$ also normalizes its division closure
$\mathcal{D}_{K}$ in $\mathcal{D}_{N}$.

We claim that $\mathcal{D}_{K}N$ is an Ore domain.

Notice that there is a natural ring homomorphism from the crossed product
$\mathbb{F}_{p}K*N/K\longrightarrow\mathcal{D}_{K}N$. By the injectivity of
$\alpha_{1}$, the previous is, in fact, an isomorphism. Since
$\mathcal{D}_{K}N$ is a domain and $\mathbb{F}_{p}K*N/K$ is Noetherian, due to
$N/K$ being abelian; then $\mathcal{D}_{K}N$ is an Ore domain by Goldie’s
theorem (7.1.4). Since $\mathcal{D}_{K}N\xhookrightarrow{}\mathcal{D}_{N}$ is
a division $\mathbb{F}_{p}N$-ring of fractions of the Ore domain
$\mathcal{D}_{K}N$; then, by (7.2.8), $\mathcal{D}_{N}$ is isomorphic to
$Q_{\text{ore}}(\mathcal{D}_{K}N)$.

Observe that $\mathcal{D}_{K}N$ can also be seen as the isomorphic image of
the $(\mathbb{F}_{p}N,\mathcal{D}_{K})$-bimodule
$\mathbb{F}_{p}N\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}K}}\mathcal{D}_{K}$
under $\alpha_{1}$. There is a commutative diagram of
$(\mathbb{F}_{p}N,\mathcal{D}_{K})$-bimodules

${\mathbb{F}_{p}T\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}K}}\mathcal{D}_{K}}$${\mathbb{F}_{p}T\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathcal{D}_{K}N}$${\mathcal{D}_{T},}$$\scriptstyle{\alpha}$$\scriptstyle{\cong}$$\scriptstyle{\alpha_{2}}$

from which we deduce that $\alpha_{2}$ is injective. Recall that
$\mathbin{\mathop{\otimes}\limits_{\mathcal{D}_{K}N}}Q_{\text{ore}}(\mathcal{D}_{K}N)$
is an exact functor, so the induced map

$\alpha_{3}\mathrel{\mathop{\ordinarycolon}}\left(\mathbb{F}_{p}T\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathcal{D}_{K}N\right)\mathbin{\mathop{\otimes}\limits_{\mathcal{D}_{K}N}}Q_{\text{ore}}(\mathcal{D}_{K}N)\longrightarrow\mathcal{D}_{T}\mathbin{\mathop{\otimes}\limits_{\mathcal{D}_{K}N}}Q_{\text{ore}}(\mathcal{D}_{K}N)\cong\mathcal{D}_{T}$

is injective, too. We recall that
$Q_{\text{ore}}(\mathcal{D}_{K}N)\cong\mathcal{D}_{N}$ canonically; so we have
an injective canonical homomorphism of
$(\mathbb{F}_{p}T,\mathcal{D}_{N})$-bimodules

$\alpha_{4}\mathrel{\mathop{\ordinarycolon}}\mathbb{F}_{p}[T]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathcal{D}_{N}\xhookrightarrow{}\mathcal{D}_{T}.$
(7.17)

This proves that we can take $N_{0}=N$ on (7.15) and the proof is complete. ∎

######  Remark 7.5.6.

In the previous lemma, the use of the criterion of J. Sánchez to establish the
Hughes-free property of $\mathcal{D}_{G}$ was crucial. It might be tempting to
directly prove this property for any pair $(S,T)$, with $S\unlhd T\unlhd G$
and $T/S\cong\mathbb{Z}$; by reducing it to the “pro-$p$ Hughes-free property”
that was established in 7.4.2. However, one may not be able to separate $S$
and $T$ with pro-$p$ closed subgroups. In other words, it may happen that
$T\subseteq\overline{S}$. For instance, one can take ${\bf F}=\mathbb{Z}_{p}$;
$a=1+\sum_{k=0}^{\infty}p^{2k+2}$; $b=1+\sum_{k=0}^{\infty}p^{2k+1}$;
$S=\langle a\rangle$; and $T=\langle a,b\rangle$. It is clear that $a,b$ are
$\mathbb{Z}$-linearly independent in $\mathbb{Z}_{p}$, so
$T/S\cong\mathbb{Z}$. Furthermore, $\overline{S}=\overline{T}={\bf F}$.

######  Lemma 7.5.7.

Let $G$ be an orderable group, with a choice of $\leq$, and let
$\mathcal{D}_{G}$ be the division closure of $\mathbb{F}_{p}G$ in the division
$\mathbb{F}_{p}G$-ring $\mathbb{F}_{p}((G))$. Then for any $H\leq G$, the
canonical map of $(\mathbb{F}_{p}G,\mathbb{F}_{p}H)$-bimodules

$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{H}\longrightarrow\mathcal{D}_{G}$

is injective.

###### Proof.

The canonical map
$\alpha\mathrel{\mathop{\ordinarycolon}}\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{H}\longrightarrow\mathcal{D}_{G}$
is defined by $a\otimes b\mapsto ab$, for all $a\in\mathbb{F}_{p}G$,
$b\in\mathcal{D}_{H}$. We take a collection of representatives
$\\{t_{i}\mathrel{\mathop{\ordinarycolon}}i\in I\\}$ of the left cosets of $H$
in $G$. This way, $G$ is equal to the disjoint union $\sqcup_{i\in I}t_{i}H$
and $\mathbb{F}_{p}G$ admits, as right $\mathbb{F}_{p}H$-module, the
decomposition $\mathbb{F}_{p}G=\oplus_{i\in I}t_{i}\mathbb{F}_{p}H.$ So
$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{H}$
admits, as right $\mathcal{D}_{H}$-vector space, the basis $\\{t_{i}\otimes
1\mathrel{\mathop{\ordinarycolon}}i\in I\\}$. We gave a description of the
inverses of elements of $\mathbb{F}_{p}((G))$ in (7.8). In particular, it
implies that any element of $\mathcal{D}_{H}$ is supported in $H$. Let us take
$x=\sum_{i\in I}t_{i}\otimes
x_{i}\in\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{H}$,
whose image equals $\alpha(x)=\sum_{i\in I}t_{i}x_{i}$. If $x\neq 0$, then
$x_{j}\neq 0$ for some $j$. Write $x_{j}=\sum_{h\in H}\lambda_{h}\,h$, with
$\lambda_{h}\in\mathbb{F}_{p}$ and some $\lambda_{h_{0}}\neq 0$. Since
$t_{i}x_{i}$ is supported in the coset $t_{i}H$, then the coefficient of the
power series $\sum_{i\in I}t_{i}x_{i}$ corresponding to $t_{j}h_{0}$ is equal
to $\lambda_{h_{0}}\neq 0$. So $\alpha(x)\neq 0$ and $\alpha$ is injective. ∎

With these two lemmas, 7.5.4 follows directly from previous results.

###### Proof of 7.5.4.

By 4.3.7, $G$ is residually-(torsion-free nilpotent) and hence, by 7.5.3,
there exists a universal division $\mathbb{F}_{p}G$-ring of fractions
$\mathcal{D}_{\mathbb{F}_{p}G}$ and this is Hughes-free. Both
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are Hughes-free division
$\mathbb{F}_{p}G$-rings by lemmas 7.5.5 and 7.5.7. We deduce by 7.5.2 that
these division $\mathbb{F}_{p}G$-rings are isomorphic, since all of them are
Hughes-free. ∎

######  Corollary 7.5.8 (Strong Hughes-free property).

Let $G$ be a residually-(torsion-free nilpotent) group. Then
$\mathcal{D}_{\mathbb{F}_{p}G}$ is strongly Hughes-free, this is, for any
$H\leq G$, the canonical map of $(\mathbb{F}_{p}G,\mathbb{F}_{p}H)$-bimodules

$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{\mathbb{F}_{p}H}\longrightarrow\mathcal{D}_{\mathbb{F}_{p}G}$

is injective.

###### Proof.

By 2.2.9, $G$ is orderable. After taking an ordering $\leq$ on $G$, we know
from lemma 7.5.7 that the division closure $D_{G}$ of $\mathbb{F}_{p}G$ in
$\mathbb{F}_{p}((G))$ is Hughes-free. By 7.5.2 and 7.5.3, $\mathcal{D}_{G}$
and $\mathcal{D}_{\mathbb{F}_{p}G}$ are isomorphic $\mathbb{F}_{p}G$-rings.
There is a commutative diagram of canonical homomorphisms of
$(\mathbb{F}_{p}G,\mathbb{F}_{p}H)$-bimodules

${\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{H}}$${\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{\mathbb{F}_{p}H}}$${\mathcal{D}_{G}}$${\mathcal{D}_{\mathbb{F}_{p}G},}$$\scriptstyle{\alpha}$$\scriptstyle{\cong}$$\scriptstyle{\beta}$$\scriptstyle{\cong}$

where the horizontal arrows are isomorphisms. We already verified in lemma
7.5.7 that $\alpha$ is injective. So $\beta$ must be injective, too. ∎

We finish this section with a formula that allow us to estimate dimensions of
$\mathbb{F}_{p}G$-modules by approximating it with the aid of finite-index
subgroups of $G$.

######  Proposition 7.5.9 (Lück-type approximation).

Let $G$ be an abstract finitely generated subgroup of a free pro-$p$ group of
finite rank ${\bf F}$. Let ${\bf F}={\bf N}_{1}>{\bf N}_{2}>\dots$ be a chain
of open normal subgroups of ${\bf F}$ with trivial intersection. Define
$G_{j}=G\bigcap{\bf N}_{j}$. Then for every finitely generated
$\mathbb{F}_{p}G$-module $M$,

$\dim_{\mathbb{F}_{p}G}M=\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[G_{i}]}}M\right)}{|G\mathrel{\mathop{\ordinarycolon}}G_{i}|}.$

###### Proof.

Let $\mathcal{D}_{G}$ be the division closure of $\mathbb{F}_{p}G$ in
$\mathcal{D}_{\mathbb{F}_{p}[[{\bf F}]]}$. By 7.5.4, $\mathcal{D}_{G}$ and
$\mathcal{D}_{\mathbb{F}_{p}G}$ are isomorphic as $\mathbb{F}_{p}G$-rings. So

$\dim_{\mathbb{F}_{p}G}M=\dim_{\mathcal{D}_{G}}(\mathcal{D}_{G}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M).$

In addition,

$\dim_{\mathcal{D}_{G}}(\mathcal{D}_{G}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M)=\dim_{\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}}(\mathcal{D}_{\mathbb{F}_{p}[[{\bf
F}]]}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M)=\dim_{\mathbb{F}_{p}[[{\bf
F}]]}(\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M).$

Since there is a canonical isomorphism $G/G_{i}\longrightarrow{\bf F}/{\bf
N}_{i}$, we have that

$\mathbb{F}_{p}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G_{i}}}M\cong\mathbb{F}_{p}[G/G_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M\cong\mathbb{F}_{p}[{\bf
F}/{\bf
N}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M\cong\mathbb{F}_{p}[{\bf
F}/{\bf N}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}\left(\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M\right).$

Thus, by 7.4.1,

$\dim_{\mathbb{F}_{p}G}M=\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\mathbb{F}_{p}[{\bf
F}/{\bf N}_{i}]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[[{\bf
F}]]}}\left(\mathbb{F}_{p}[[{\bf
F}]]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M\right)}{|{\bf
F}\mathrel{\mathop{\ordinarycolon}}{\bf
N}_{i}|}=\lim_{i\rightarrow\infty}\frac{\dim_{\mathbb{F}_{p}}\left(\mathbb{F}_{p}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}[G_{i}]}}M\right)}{|G\mathrel{\mathop{\ordinarycolon}}G_{i}|}$

and the conclusion follows. ∎

###  7.6 Examples of $\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free modules

The importance of the existence of a universal division rings of fractions of
a ring $R$, denoted by $\mathcal{D}_{R}$, is that it yields to a dimension
theory of certain $R$-modules by extending them to scalars in
$\mathcal{D}_{R}$. However, we need our $\mathcal{D}_{R}$-modules to be
torsion-free for an statement of the type “surjective homomorphisms between
$R$-modules of the same dimension are isomorphisms”.

We start by defining the concept of $\mathcal{D}_{R}$-torsion-free modules. We
then give some general methods to construct them; and, finally, we give some
explicit families of $\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free modules.

######  Definition 7.6.1.

Let $R\xhookrightarrow{}\mathcal{D}$ be an embedding of the ring $R$ into a
division ring $\mathcal{D}$. Let $M$ be a $R$-module. We say that $M$ is
$\mathcal{D}$-torsion-free if the canonical map
$M\longrightarrow\mathcal{D}\mathbin{\mathop{\otimes}\limits_{R}}M$ is
injective.

The following provides a more flexible criterion for verifying whether a
module is torsion-free.

######  Lemma 7.6.2 (Characterisations of torsion-freeness).

Let $M$ be a $\mathbb{F}_{p}G$-module. Then $M$ is
$\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free if and only if there exists a
$\mathcal{D}_{\mathbb{F}_{p}G}$-module $N$ and an injective homomorphism of
$\mathbb{F}_{p}G$-modules $M\xhookrightarrow{}N$.

We leave stated other two lemmas which are not hard to prove. We refer to [26,
Section 4.1] for details.

######  Lemma 7.6.3.

Let $\mathcal{D}$ be a division $R$-ring and let $M$ be a
$\mathcal{D}$-torsion-free $R$-module of finite $\mathcal{D}$-dimension. Let
$L$ be a non-trivial $R$-submodule of $M$. Then
$\dim_{\mathcal{D}}(M/L)<\dim_{\mathcal{D}}(M)$. Moreover, if
$\dim_{\mathcal{D}}L=1$, then $\dim_{\mathcal{D}}(M/L)=\dim_{\mathcal{D}}M-1$.

######  Lemma 7.6.4.

Let $\mathcal{D}$ be a division $R$-ring and let $1\rightarrow
M_{1}\rightarrow M_{2}\rightarrow M_{3}\rightarrow 0$ be an exact sequence of
$R$-modules. Assume that

  1. 1.

$M_{1}$ and $M_{3}$ are $\mathcal{D}$-torsion-free;

  2. 2.

$\dim_{\mathcal{D}}M_{1}$ and $\dim_{\mathcal{D}}M_{3}$ are finite; and

  3. 3.

$\dim_{\mathcal{D}}M_{1}+\dim_{\mathcal{D}}M_{3}=\dim_{\mathcal{D}}M_{2}$.

Then $M_{2}$ is also $\mathcal{D}$-torsion-free.

We denote by $H\leq G$ subgroups of a free pro-p group ${\bf F}$. we
alternatively classify what is it to be torsion free.

######  Lemma 7.6.5.

Let $M$ be a $\mathbb{F}_{p}H$-module. Then $M$ is
$\mathcal{D}_{\mathbb{F}_{p}H}$-torsion-free if and only if
$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M$ is
$\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free.

###### Proof.

Suppose that $M$ is $\mathcal{D}_{\mathbb{F}_{p}H}$-torsion-free. Then the
canonical map

$M\longrightarrow\mathcal{D}_{\mathbb{F}_{p}H}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M$

is injective. Notice that $\mathbb{F}_{p}G$ is a free right
$\mathbb{F}_{p}H$-module. Thus the canonical map

$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M\overset{\alpha}{\longrightarrow}\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\left(\mathcal{D}_{\mathbb{F}_{p}H}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M\right)$

is also injective. The right-hand side $\mathbb{F}_{p}G$-module is canonically
isomorphic to

$\left(\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{\mathbb{F}_{p}H}\right)\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M.$

We want to embed this $\mathbb{F}_{p}G$-module into a
$\mathcal{D}_{\mathbb{F}_{p}G}$-module. To do so, we recall the strong Hughes-
free property of 7.5.8, which says that the canonical map of
$(\mathbb{F}_{p}G,\mathbb{F}_{p}H)$-bimodules

$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{\mathbb{F}_{p}H}\longrightarrow\mathcal{D}_{\mathbb{F}_{p}G}$

is injective. Since $\mathcal{D}_{\mathbb{F}_{p}G}$ is a free right
$D_{\mathbb{F}_{p}H}$-module and $D_{\mathbb{F}_{p}H}$ is a free right
$\mathbb{F}_{p}H$-module, we deduce that both
$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{\mathbb{F}_{p}H}$
and $\mathcal{D}_{\mathbb{F}_{p}G}$ are free right $\mathbb{F}_{p}H$-modules.
It follows that the canonical map of $\mathbb{F}_{p}G$-modules

$\left(\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}\mathcal{D}_{\mathbb{F}_{p}H}\right)\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M\overset{\beta}{\longrightarrow}\mathcal{D}_{\mathbb{F}_{p}G}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M.$

Composing $\alpha$ and $\beta$, we construct an injective
$\mathbb{F}_{p}G$-homomorphism between
$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M$ and the
$\mathcal{D}_{\mathbb{F}_{p}G}$-module
$\mathcal{D}_{\mathbb{F}_{p}G}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M$.

By lemma 7.6.2, we conclude that
$\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}H}}M$ is a
$\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free module. ∎

######  Proposition 7.6.6.

Let $H\leq G\leq{\bf F}$ and let $A$ be a maximal abelian subgroup of $H$.
Then the $\mathbb{F}_{p}G$-module

$\mathbb{F}_{p}I_{H}^{G}\,\big{/}\,\mathbb{F}_{p}I_{A}^{G}$

is $\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free.

###### Proof.

Omitted. It is proven in [26, Proposition 4.8]. ∎

This directly leads to the following consequence.

######  Corollary 7.6.7.

Let $u\in G$. Suppose that the cyclic group $\langle u\rangle$ that $u$
generates is maximal abelian in $H$. Then the $\mathbb{F}_{p}G$-module

$\frac{\mathbb{F}_{p}I_{H}^{G}}{\mathbb{F}_{p}G(u-1)}$

is $\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free.

The next proposition produces torsion-free modules that have the form of
augmentation ideals of amalgamated products with cyclic amalgam.

######  Proposition 7.6.8.

Let $H_{1}$ and $H_{2}$ be two finitely generated subgroups of a finitely
generated free pro-$p$ group ${\bf F}$. Consider $A=H_{1}\bigcap H_{2}$ and
suppose that $A$ is a maximal abelian subgroup of $H_{1}$. Let $G=\langle
H_{1},H_{2}\rangle$ and let

$J=\\{(x,-x)\mathrel{\mathop{\ordinarycolon}}x\in\mathbb{F}_{p}I_{A}^{G}\\}\leq\mathbb{F}_{p}I_{H_{1}}^{G}\oplus\mathbb{F}_{p}I_{H_{2}}^{G}.$

Then the $\mathbb{F}_{p}G$-module

$M=\frac{\mathbb{F}_{p}I_{H_{1}}^{G}\oplus\mathbb{F}_{p}I_{H_{2}}^{G}}{J}$

is $\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free and

$\dim_{\mathbb{F}_{p}G}M=\dim_{\mathbb{F}_{p}H_{1}}\mathbb{F}_{p}I_{H_{1}}+\dim_{\mathbb{F}_{p}H_{2}}\mathbb{F}_{p}I_{H_{2}}-1.$

Before giving the proof, we shall make an observation. If $A$ is finitely
torsion-free abelian, as occurs in the previous proposition, then
$A\cong\mathbb{Z}^{n}$ and the universal division $\mathbb{F}_{p}A$-ring of
fractions is the field of fractions of the commutative domain
$\mathbb{F}_{p}A$. Given a nontrivial finitely generated ideal
$I=(a_{1},\dots,a_{n})$ of $\mathbb{F}_{p}A$, we see that the
$\operatorname{Frac}(\mathbb{F}_{p}A)$-vector space
$\operatorname{Frac}(\mathbb{F}_{p}A)\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}A}}I$
can be generated by any $1\otimes a_{i}$ with $a_{i}\neq 0$. In fact, if
$a_{1}\neq 0$, then $1\otimes a_{i}=a_{1}^{-1}a_{1}\otimes
a_{i}=a_{1}^{-1}\otimes a_{1}a_{i}=a_{1}^{-1}a_{i}\otimes
a_{1}=a_{1}^{-1}a_{i}\,(1\otimes a_{1})$. This implies that

$\dim_{\mathbb{F}_{p}A}I=\dim_{\operatorname{Frac}(\mathbb{F}_{p}A)}\operatorname{Frac}(\mathbb{F}_{p}A)\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}A}}I=1.$
(7.18)

###### Proof of 7.6.8.

We consider the $\mathbb{F}_{p}G$-submodule of $M$ defined by
$L=(\mathbb{F}_{p}I_{A}^{G}\oplus\mathbb{F}_{p}I_{H_{2}}^{G})/J$.

We want to apply lemma 7.6.4 to the short exact sequence $0\rightarrow
L\rightarrow M\rightarrow M/L\rightarrow 0$ of $\mathbb{F}_{p}G$-modules.

Since $L\cong\mathbb{F}_{p}I_{H_{2}}^{G}\leq\mathbb{F}_{p}G$, then it is
$\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free and

$\dim_{\mathbb{F}_{p}G}L=\dim_{\mathbb{F}_{p}H_{2}}\mathbb{F}_{p}I_{H_{2}}.$

By the observation (7.18),
$1=\dim_{\mathbb{F}_{p}A}\mathbb{F}_{p}I_{A}=\dim_{\mathbb{F}_{p}G}\mathbb{F}_{p}I_{A}^{G}=\dim_{\mathbb{F}_{p}G}J$.
So it is clear, by lemma 7.6.3, that

$\dim_{\mathbb{F}_{p}G}M=\dim_{\mathbb{F}_{p}G}\mathbb{F}_{p}I_{H_{1}}^{G}+\dim_{\mathbb{F}_{p}G}\mathbb{F}_{p}I_{H_{2}}^{G}-1=\dim_{\mathbb{F}_{p}H_{1}}\mathbb{F}_{p}I_{H_{1}}+\dim_{\mathbb{F}_{p}H_{2}}\mathbb{F}_{p}I_{H_{2}}-1.$

On the other side, the quotient $M/L$ is isomorphic to
$\mathbb{F}_{p}I_{H_{1}}^{G}/\mathbb{F}_{p}I_{A}^{G}$, which is
$\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free by 7.6.6. Again, by lemma 7.6.3,

$\dim_{\mathbb{F}_{p}G}M/L=\dim_{\mathbb{F}_{p}G}\mathbb{F}_{p}I_{H_{1}}^{G}-\dim_{\mathbb{F}_{p}G}\mathbb{F}_{p}I_{A}^{G}=\dim_{\mathbb{F}_{p}H_{1}}\mathbb{F}_{p}I_{H_{1}}-1.$

Therefore, the short exact sequence $0\rightarrow L\rightarrow M\rightarrow
M/L\rightarrow 0$ of $\mathbb{F}_{p}G$-modules satisfies the requirements of
lemma 7.6.4; and the conclusion follows. ∎

We now turn to the study of torsion-free modules that have the form of an
augmentation ideal of a cyclic HNN extension.

Our point is to prove that certain augmentation ideals, which may have the
form

$R^{m}/R(u_{1},\dots,u_{m})$

for some group ring $R$; are $\mathcal{D}_{R}$-torsion-free. When extending
this $R$-module with coefficients in a bigger ring, some $u_{i}$ may become
invertible. The following elementary lemma simply studies this scenario, which
we shall encounter many times.

######  Lemma 7.6.9.

Let $R$ be a unital ring and $M$ be a $R$-module. Let $m_{0}\in M$ and let $u$
be a unit of $R$. Then there is an isomorphism of $R$-modules

$\gamma\mathrel{\mathop{\ordinarycolon}}\frac{M\oplus
R}{(m_{0},u)}\longrightarrow M$

given by

$\gamma([m,r])=m-ru^{-1}m_{0},$

with inverse

$\gamma^{-1}(m)=[m,0].$

We can now state the last proposition of this section. The proof is relatively
long and, for convenience, it is divided in several intermediate lemmas.

######  Proposition 7.6.10.

Let $H\leq G$ be subgroups of ${\bf F}$. Suppose that we can write
$G=N\rtimes(t)=\langle N,t\rangle$ for some $H\leq N\leq G$ and some $t\in G$.
Let $u\in H$ be an element which generates a maximal abelian group $\langle
u\rangle$ in $H$ and suppose that $v=tut^{-1}\in H$. Then the
$\mathbb{F}_{p}G$-module $M$ defined by

$M=\frac{\mathbb{F}_{p}I_{H}^{G}\oplus\mathbb{F}_{p}G}{\mathbb{F}_{p}G\,(v-1-t(u-1),v-1)}$

is $\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free.

###### Proof.

Since $G\cong N\rtimes(t)$, we can write
$\mathbb{F}_{p}G\cong\mathbb{F}_{p}N[t^{\pm 1},\sigma]$ as a skew-polynomial
ring in one indeterminate $t$ with coefficients in $\mathbb{F}_{p}N$. Here
$\sigma$ is an automorphism of the ring $\mathcal{D}_{\mathbb{F}_{p}N}$ such
that $\sigma(\mathbb{F}_{p}N)=\mathbb{F}_{p}N$, since it is induced from the
automorphism of conjugation-by-$t$ in the subring $\mathbb{F}_{p}N$ of
$\mathbb{F}_{p}G$.

######  Lemma 7.6.11.

Let $R=\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm},\sigma]$. Then $R$ is a (right)
Ore domain and its (right) Ore division ring of fractions
$Q_{\operatorname{ore}}(R)$ is isomorphic to $\mathcal{D}_{\mathbb{F}_{p}G}$.

To check this lemma, we start noting that $R$ is a domain. In addition, by the
Hilbert’s basis theorem, the ring $R$ is also Noetherian; and hence, by Little
Goldie’s theorem 7.1.4, $R$ is an Ore domain. We want to compare the resulting
Ore division $R$-ring of fractions $R\xhookrightarrow{}Q_{\text{ore}}(R)$ with
$\mathcal{D}_{\mathbb{F}_{p}G}$. By 7.5.4, we can identify
$\mathcal{D}_{\mathbb{F}_{p}G}$ and $\mathcal{D}_{\mathbb{F}_{p}H}$ with the
division closures of $\mathbb{F}_{p}G$ and $\mathbb{F}_{p}H$, respectively, in
$\mathbb{F}_{p}((G))$. In this setting, there is a natural monomorphism
$\mathcal{D}_{\mathbb{F}_{p}N}\xhookrightarrow{}\mathcal{D}_{\mathbb{F}_{p}G}$.
We want to extend this to a monomorphism of rings
$R\xhookrightarrow{}\mathcal{D}_{\mathbb{F}_{p}G}$. Observe that, as
$(\mathbb{F}_{p}G,\mathbb{F}_{p}H)$-bimodules, there is a canonical injection

$R=\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm},\sigma]\cong\mathbb{F}_{p}G\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathcal{D}_{\mathbb{F}_{p}N}\xhookrightarrow{}\mathcal{D}_{\mathbb{F}_{p}G}$

according to lemma 7.5.7. In addition, we can observe that this map is a ring
homomorphism. So we have an ring monomorphism
$\iota\mathrel{\mathop{\ordinarycolon}}R\xhookrightarrow{}\mathcal{D}_{\mathbb{F}_{p}G}$.
Since $\mathcal{D}_{\mathbb{F}_{p}G}$ is a division ring, we have, by the
universal property (7.5) of the embedding
$\iota_{\operatorname{ore}}\mathrel{\mathop{\ordinarycolon}}R\xhookrightarrow{}Q_{\text{ore}}(R)$,
a commutative diagram of ring homomorphisms

${R}$${Q_{\text{ore}}(R)}$${D_{\mathbb{F}_{p}G},}$$\scriptstyle{\iota_{\operatorname{ore}}}$$\scriptstyle{\iota}$$\scriptstyle{\widehat{\iota}}$
(7.19)

where $\iota$ and $\iota_{\operatorname{ore}}$ are injective.

Moreover, $R$ contains $\mathbb{F}_{p}N[t^{\pm 1},\sigma]=\mathbb{F}_{p}G$, so
$\iota\mathrel{\mathop{\ordinarycolon}}R\xhookrightarrow{}\mathcal{D}_{\mathbb{F}_{p}G}$
is a division $R$-ring of fractions. Arguing as in lemma 7.2.8,
$\widehat{\iota}\mathrel{\mathop{\ordinarycolon}}Q_{\text{ore}}(R)\longrightarrow\mathcal{D}_{\mathbb{F}_{p}G}$
is an isomorphism of $R$-rings. So lemma 7.6.11 is proven.

There is a division $R$-ring
$R\xhookrightarrow{}\mathcal{D}_{\mathbb{F}_{p}N}((t^{\pm},\sigma))$, where
the latter is the ring of Skew Laurent series. The universal property of
$Q_{\text{ore}}(R)$ and the isomorphism
$Q_{\text{ore}}(R)\cong\mathcal{D}_{\mathbb{F}_{p}G}$ of lemma 7.6.11 induces
a commutative diagram of rings

${R}$${\mathcal{D}_{\mathbb{F}_{p}G}}$${D_{\mathbb{F}_{p}N}((t^{\pm},\sigma)).}$$\scriptstyle{\iota_{\operatorname{ore}}}$
(7.20)

In particular, we can upgrade this diagram to one of the following form

${\mathbb{F}_{p}G}$${\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm
1},\sigma]}$${\mathcal{D}_{\mathbb{F}_{p}G}}$${\mathcal{D}_{\mathbb{F}_{p}N}((t^{\pm
1},\sigma)),}$ (7.21)

where all the arrows are injective. This induces the following commutative
diagram of $\mathbb{F}_{p}G$-modules

${\mathbb{F}_{p}I_{H}^{G}}$${\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm
1},\sigma]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G}}$${\mathcal{D}_{\mathbb{F}_{p}G}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G}}$${\mathcal{D}_{\mathbb{F}_{p}N}((t^{\pm
1},\sigma))\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G}}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\delta}$
(7.22)

######  Lemma 7.6.12.

The map $\beta$ of the diagram (7.22) is injective.

We claim that both $\gamma$ and $\delta$ are injective. On the one hand, the
$\mathbb{F}_{p}G$-module $\mathbb{F}_{p}I_{H}^{G}$ is a submodule of
$\mathbb{F}_{p}G$, so it is $\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free and
$\gamma$ is injective. Furthermore, $\mathcal{D}_{\mathbb{F}_{p}N}((t^{\pm
1},\sigma))$ is a free right $\mathcal{D}_{\mathbb{F}_{p}G}$-module, since
$\mathcal{D}_{\mathbb{F}_{p}G}$ is a division ring; so
$\mathcal{D}_{\mathbb{F}_{p}G}$ is a direct summand of
$\mathcal{D}_{\mathbb{F}_{p}N}((t^{\pm 1},\sigma))$ as a right
$\mathbb{F}_{p}G$-module. Hence $\delta$ is injective. Since the diagram (eq.
7.22) is commutative, $\beta$ must be injective and 7.6.12 is proven.

We will now view this injective map $\beta$ in another diagram.

The commutative diagram of rings (7.21) induces the following commutative
diagram of $\mathbb{F}_{p}G$-modules

${M}$${\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm
1},\sigma]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M}$${\mathcal{D}_{\mathbb{F}_{p}G}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M}$${\mathcal{D}_{\mathbb{F}_{p}N}((t^{\pm
1},\sigma))\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M}$$\scriptstyle{\gamma^{\prime}}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta^{\prime}}$$\scriptstyle{\delta^{\prime}}$
(7.23)

Now let $R$ be either $\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm 1},\sigma]$ or
$\mathcal{D}_{\mathbb{F}_{p}N}((t^{\pm 1},\sigma)).$ Observe that

${R\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M}$${\frac{\left(R\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G}\right)\oplus
R}{R\,(v-1-t(u-1),v-1)}}$${R\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G},}$$\scriptstyle{\cong}$$\scriptstyle{\gamma\,\cong}$

where both arrows are isomorphisms of $\mathbb{F}_{p}$-modules and $\gamma$ is
given by lemma 7.6.9, since $v-1$ is invertible in
$\mathcal{D}_{\mathbb{F}_{p}N}$ and so also in $R$. We now integrate these
isomorphisms $\gamma$ into the commutative diagram of (7.23) as follows.

${M}$${\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm
1},\sigma]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M}$${\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm
1},\sigma]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G}}$${\mathcal{D}_{\mathbb{F}_{p}G}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M}$${\mathcal{D}_{\mathbb{F}_{p}N}((t^{\pm
1},\sigma))\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}M}$${\mathcal{D}_{\mathbb{F}_{p}N}((t^{\pm
1},\sigma))\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G}}$$\scriptstyle{\gamma^{\prime}}$$\scriptstyle{\alpha^{\prime}}$$\scriptstyle{\beta^{\prime}}$$\scriptstyle{\gamma_{1}\,\cong}$$\scriptstyle{\beta}$$\scriptstyle{\delta^{\prime}}$$\scriptstyle{\gamma_{2}\,\cong}$
(7.24)

We already know, from 7.6.12, that $\beta$ is injective. So $\beta^{\prime}$
must be injective since $\gamma_{1}$ and $\gamma_{2}$ are both isomorphisms.

We claim that $\alpha^{\prime}$ is also injective. If we proved this, then, by
inspecting again the commutative diagram (7.24), $\gamma^{\prime}$ would also
be injective and hence we could conclude that $M$ is
$\mathcal{D}_{\mathbb{F}_{p}G}$-torsion-free, as we want.

So it rests to prove the following lemma.

######  Lemma 7.6.13.

The map $\alpha^{\prime}$ of the diagram (7.24) is injective.

We will prepare a bit the ground before checking this lemma.

By assumption, $\langle u\rangle$ is a maximal abelian subgroup of $H$. By
7.6.7, the $\mathbb{F}_{p}N$-module

$M_{0}=\frac{\mathbb{F}_{p}I_{H}^{N}}{\mathbb{F}_{p}N(u-1)}$

is $\mathcal{D}_{\mathbb{F}_{p}N}$-torsion-free. This implies that the
canonical map

$M_{0}\longrightarrow\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}M_{0}$

is injective. This translates into the following equality of subsets of
$\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{N}$,

$1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{N}\,\,\bigcap\,\,\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}(u-1)=1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}N(u-1).$
(7.25)

From this, we want to prove the following lemma. We denote
$H^{t^{n}}=t^{n}Ht^{-n}$, which are also subgroups of $N$ having $\langle
u^{t^{n}}\rangle$ as maximal abelian subgroup.

######  Lemma 7.6.14.

For all $n\in\mathbb{Z}$, we have the following equality of subsets of
$\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{G}$,

$1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}t^{n}I_{H}^{N}\,\,\bigcap\,\,\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}t^{n}(u-1)=1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}t^{n}N(u-1).$
(7.26)

The same way we had the equality (7.25), we can derive, for the same reasons,

$1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H^{t^{-n}}}^{N}\,\,\bigcap\,\,\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}(u^{t^{-n}}-1)=1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}N(u^{t^{-n}}-1).$
(7.27)

Notice that $\mathbb{F}_{p}I_{G}$ has a right $\langle t\rangle$-module
structure by multiplication. This induces a right $\langle t\rangle$-module
structure on
$\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{G}$.
Since $N\unlhd G$, then $t$ normalises $N$. We have the equations of subsets
of
$\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{G}$:

$\left(1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}N(u-1)\right)\,t^{m}=1\otimes\mathbb{F}_{p}\,t^{m}N(u^{t^{-m}}-1),$

and

$\left(1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{N}\right)\,t^{m}=1\otimes\mathbb{F}_{p}\,t^{m}I_{H^{t^{-m}}}(u^{t^{-m}}-1).$

As a consequence, applying the multiplication-by-$t^{n}$ automorphism of
$\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{G}$
to the equation (7.27); we get (7.26).

Notice the following decomposition of $\mathbb{F}_{p}N$-modules

$\mathbb{F}_{p}I_{H}^{G}=\oplus_{n\in\mathbb{Z}}\,\,\mathbb{F}_{p}\,t^{n}I_{H}^{N},$

which yields to the following decomposition of $\mathbb{F}_{p}$-vector spaces

$\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{G}\cong\oplus_{n\in\mathbb{Z}}\,\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\,\mathbb{F}_{p}\,t^{n}I_{H}^{N}.$

######  Lemma 7.6.15.

We have the following equation of subsets of
$\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{G}$,

$1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{G}\,\,\bigcap\,\,\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}G(v-1-t(u-1))=1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}G(v-1-t(u-1)).$
(7.28)

Let us take an element $w$ that belongs to the left-hand side. This element
will have the form

$w=\sum_{k=n_{1}}^{n_{2}}c_{k}\otimes t^{k}(v-1-t(u-1)),\,\,\mbox{for some
$c_{k}\in\mathcal{D}_{\mathbb{F}_{p}N}$,}$

and will also belong to
$1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{G}$.
We rewrite

$w=c_{n_{1}}\otimes t^{n_{1}}(v-1)+\sum_{k=n_{1}+1}^{n_{2}}\left(c_{k}\otimes
t^{k}(v-1)-c_{k-1}\otimes t^{k}(u-1)\right)+c_{n_{2}}\otimes
t^{n_{2}+1}(u-1).$

Since $w\in
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{G}$,
we can look at the highest power $t^{n_{2}+1}$ to deduce that

$c_{n_{2}}\otimes t^{n_{2}+1}(u-1)\in
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}t^{n_{2}+1}I_{H}^{N}\,\,\bigcap\,\,\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}t^{n_{2}+1}(u-1).$

By (7.26), this implies that

$c_{n_{2}}\otimes t^{n_{2}+1}(u-1)\in
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}t^{n_{2}+1}N(u-1),$

so $c_{n_{2}}\in\mathbb{F}_{p}N$. Let $n_{1}+1\leq k\leq n_{2}$. Inspecting
again the expression of $w$ at the component with power $t^{k}$, we have that

$c_{k}\otimes t^{k}(v-1)-c_{k-1}\otimes t^{k}(u-1)\in
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}t^{k}I_{H}^{N}.$

If we knew that $c_{k}\in\mathbb{F}_{p}N$, then it would follow that

$c_{k-1}\otimes t^{k}(u-1)\in
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}t^{k}I_{H}^{N}\,\,\bigcap\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}t^{k}(u-1).$

By (7.26), this means that

$c_{k-1}\otimes t^{k}(u-1)\in
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}t^{n}N(u-1),$

and this implies that $c_{k-1}\in\mathbb{F}_{p}N$.

We have proven that if $c_{k}\in\mathbb{F}_{p}N$, for $n_{1}<k\leq n_{2}$,
then $c_{k-1}\in\mathbb{F}_{p}N$. Since we also know that
$c_{n_{2}}\in\mathbb{F}_{p}N$, an inductive argument gives that
$c_{k}\in\mathbb{F}_{p}N$ for every $k$, meaning that

$w\in
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}N[t^{\pm},\sigma]\,(v-1-t(u-1))=1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}G(v-1-t(u-1)).$

This proves that

$1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{G}\,\,\bigcap\,\,\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}G(v-1-t(u-1))\subseteq
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}G(v-1-t(u-1)),$

one inclusion of (7.28). The reverse inclusion is trivial, so equation (7.28)
is proven.

Observe that there is a canonical isomorphism of $\mathbb{F}_{p}N$-modules

$\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm},\sigma]\cong\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}N[t^{\pm},\sigma]=\mathcal{D}_{pN}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}G.$

This extends to a canonical isomorphism of $\mathbb{F}_{p}N$-modules

$\psi\mathrel{\mathop{\ordinarycolon}}\frac{\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm},\sigma]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G}}{\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm},\sigma]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}G(v-1-t(u-1))}\longrightarrow\frac{\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{G}}{\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}G(v-1-t(u-1))}.$

Lastly, there is a commutative triangle of canonical
$\mathbb{F}_{p}N$-homomorphisms

${\frac{\mathbb{F}_{p}I_{H}^{G}}{\mathbb{F}_{p}G(v-1-t(u-1))}}$${\frac{\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm},\sigma]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G}}{\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm},\sigma]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}G(v-1-t(u-1))}}$${\frac{\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}I_{H}^{G}}{\mathcal{D}_{\mathbb{F}_{p}N}\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}N}}\mathbb{F}_{p}G(v-1-t(u-1))},}$$\scriptstyle{\alpha^{\prime\prime}}$$\scriptstyle{\eta}$$\scriptstyle{\psi}$

The canonical map $\eta$ is injective due to (7.28). Since $\psi$ is an
isomorphism, this implies that $\alpha^{\prime\prime}$ is injective. From the
injectivity of $\alpha^{\prime\prime}$, the injectivity of $\alpha^{\prime}$
follows directly. In fact, let
$(x,y)\in\mathbb{F}_{p}I_{H}^{G}\oplus\mathbb{F}_{p}G$ belong to the kernel of
$\alpha$. Then $1\otimes(x,y)=c\,(v-1-t(u-1),v-1)$ for some
$c\in\mathcal{D}_{\mathbb{F}_{p}G}$. Notice that

$1\otimes x=c\otimes(v-1-t(u-1))\in
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}I_{H}^{G}\bigcap\mathcal{D}_{\mathbb{F}_{p}N}[t^{\pm},\sigma]\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}(v-1-t(u-1)).$

From the injectivity of $\alpha^{\prime\prime}$, this implies that

$c\otimes(v-1-t(u-1))\in
1\mathbin{\mathop{\otimes}\limits_{\mathbb{F}_{p}G}}\mathbb{F}_{p}G(v-1-t(u-1)),$

so $c\in\mathbb{F}_{p}G$ and then $(x,y)\in\mathbb{F}_{p}G(v-1-t(u-1),v-1).$
Thus $\alpha^{\prime}$ is injective and lemma 7.6.13 is demonstrated. The
proof of 7.6.10 is complete. ∎

##  Chapter 8 Homology and $L^{2}$-Betti numbers

$L^{2}$-Betti numbers have played an important role in the solution of many
problems in group theory. They were originally introduced by M. Atiyah in the
context of Riemannian manifolds in 1974. J. Dodziuk extended the notion of
$L^{2}$-Betti numbers to free cocompact actions of discrete groups $G$ on CW-
complexes $X$ in 1977. Later on, in 1998, W. Lück extended this notion to
arbitrary $G$-CW-complexes.

We shall not discuss here how these analytical and topological notions are
defined. We are more interested in the algebraic approach that P. Linnell [33]
introduced in 1993. If $G$ is residually-(torsion-free nilpotent), we can
combine Linnell’s definition and a description of
$\mathcal{D}_{\mathbb{F}_{p}G}$ that is based on the work of A. Jaikin-
Zapirain and D. López-Álvarez [28] to redefine the $L^{2}$-Betti numbers as
follows.

Let $k\geq 0$. We define the $k$-th $L^{2}$-Betti number of $G$, denoted
$b_{k}^{(2)}(G)$, by

$b_{k}^{(2)}(G)=\dim_{\mathcal{D}_{\mathbb{Q}G}}H_{k}(G;\mathcal{D}_{\mathbb{Q}G}).$

By analogy, we can define in characteristic $p$ the $k$-th mod $p$
$L^{2}$-Betti number of $G$ as

$\beta_{k}^{\text{mod}p}(G)=\dim_{\mathcal{D}_{\mathbb{F}_{p}G}}H_{k}(G;\mathcal{D}_{\mathbb{F}_{p}G}).$

We are most interested in $\beta_{1}^{\mathtt{mod\,p}}(G)$ and we will later
see how these numbers are related to the computations of dimensions of the
previous chapter.

First, we discuss how the most elementary version of Betti numbers can be used
to study discrete groups.

######  Definition 8.0.1.

Given a finitely generated group $G$, its first Betti number is

$b_{1}(G)=\dim_{\mathbb{Q}}\left(G/[G,G]\right)\mathbin{\mathop{\otimes}\limits_{\mathbb{Z}}}\mathbb{Q}.$

If $G$ is finitely generated, we understand very well the dimension of the
previous $\mathbb{Q}$-vector space. Since $G/[G,G]$ is a finitely generated
abelian group, it must have the form $\mathbb{Z}^{r}\oplus T$, where $T$ is
finite (the torsion subgroup). Tensoring by $\mathbb{Q}$ forgets about the
torsion part of the $\mathbb{Z}$-module $G/[G,G]$, so

$\left(G/[G,G]\right)\mathbin{\mathop{\otimes}\limits_{\mathbb{Z}}}\mathbb{Q}\cong\mathbb{Q}^{r}.$

In particular, $r=b_{1}(G)$. Interestingly, one can find $b_{1}(G)$ just by
looking at certain finite quotients of $G$.

######  Lemma 8.0.2.

Let $G$ be finitely generated and let $p$ be a prime. Then $b_{1}(G)$ is the
biggest $m$ such that $G$ surjects $(C_{p^{k}})^{m}$ for every $k\geq 1$.

We will see how one can exploit this observation to estimate Betti numbers of
dense subgroups of free pro-$p$ groups. Later, this will have consequences for
the first $L^{2}$-Betti number of a parafree group.

######  Lemma 8.0.3.

Let $G$ and $G^{\prime}$ be finitely generated and let $p$ be a prime. If
there is a group homomorphism
$h\mathrel{\mathop{\ordinarycolon}}G\longrightarrow G_{\widehat{p}}^{\prime}$
of dense image, then $b_{1}(G)\geq b_{1}(G^{\prime})$.

###### Proof.

Let $m=b_{1}(G^{\prime})$. Recall that, given a finite $p$-group, there is a
correspondence (4.4) between the epimorphisms $G^{\prime}\longrightarrow P$
and the epimorphisms $G_{\widehat{p}}^{\prime}\longrightarrow P$; given by
$f\mapsto f_{\widehat{p}}$. Similarly, given any epimorphism
$g\mathrel{\mathop{\ordinarycolon}}G^{\prime}_{\widehat{p}}\longrightarrow P$,
there is an epimorphism $g\circ
f\mathrel{\mathop{\ordinarycolon}}G\longrightarrow P$. This is because $P$ is
finite, $g$ is continuous and $f(G)$ is dense in $G^{\prime}_{\widehat{p}}.$

Therefore, every finite $p$-quotient of $G^{\prime}$ is also a finite
$p$-quotient of $G^{\prime}$; hence $b_{1}(G)\geq b_{1}(G^{\prime})$ by 8.0.2.
∎

In particular, the first Betti number is a pro-$p$ invariant.

######  Corollary 8.0.4.

If $G$ and $G^{\prime}$ are finitely generated and $G_{\widehat{p}}\cong
G_{\widehat{p}}^{\prime}$, then $b_{1}(G)=b_{1}(G^{\prime})$.

###  8.1 Estimations and computations of $b_{1}^{(2)}(G)$

In order to compute the $L^{2}$-Betti numbers of a finitely presented
residually finite group, it suffices to have information about the Betti
numbers of its finite-index subgroups. This is precised by the following
important result of Lück [37].

######  Theorem 8.1.1 (Lück’s Approximation Theorem).

Let $G$ be finitely presented and let $G=N_{1}>N_{2}>\dots>N_{m}>\dots$ be a
sequence of finite-index normal subgroups with $\bigcap_{m}N_{m}=1$. Then

$\lim_{m\rightarrow\infty}\frac{b_{1}(N_{m})}{|G\mathrel{\mathop{\ordinarycolon}}N_{m}|}=b_{1}^{(2)}(G).$

A favourable family of groups for which we can estimate or compute their
$L^{2}$-Betti numbers are residually finite groups for which we have some
structural information of their finite-index subgroups in relation to their
index. This is the case of fundamental groups of some topological spaces, such
as free groups and surface groups.

######  Example 8.1.2 (First $L^{2}$-Betti numbers of surface groups).

Finitely generated free groups $F_{n}$ have $b_{1}^{(2)}(F_{n})=n-1$. If $S$
is a closed surface of genus $g\geq 1$, then

$b_{1}^{(2)}(\pi_{1}(S))=-\chi(S)=2g-2.$

We will use Lück’s approximation theorem to verify this. The case of $F_{n}$
follows from Schreier’s index-rank formula (2.0.1) and the fact that a
$b_{1}(F_{k})=k$ for every $k\geq 1$. Let
$F_{n}=N_{1}>N_{2}>\dots>N_{m}>\dots$ be a sequence of finite-index normal
subgroups with $\bigcap_{m}N_{m}=1$. The existence of this chain is ensured by
because $F_{n}$ is residually finite. By 8.1.1,

$b_{1}^{(2)}(F_{n})=\lim_{m\rightarrow\infty}\frac{b_{1}(N_{m})}{|F_{n}\mathrel{\mathop{\ordinarycolon}}N_{m}|}=\lim_{m\rightarrow\infty}\frac{1+|F_{n}\mathrel{\mathop{\ordinarycolon}}N_{m}|(n-1)}{|F_{n}\mathrel{\mathop{\ordinarycolon}}N_{m}|}=n-1.$

Let $S$ be a closed orientable surface. As we verified in remark 2.1.2, if $S$
is orientable, then $b_{1}(\pi_{1}(S))=2-\chi(S)$, and, if $S$ is non-
orientable, then $b_{1}(\pi_{1}(S))=1-\chi(S)/2$. The computation of their
$b_{1}^{(2)}$ is analogous and we only precise the case when $S$ is
orientable. Let $\pi_{1}(S)=N_{1}>N_{2}>\dots>N_{m}>\dots$ be a sequence of
finite-index normal subgroups with $\bigcap_{m}N_{m}=1$. The existence of this
chain is ensured by 2.1.4. By 2.1.3,
$b_{1}(N_{m})=2-|\pi_{1}(S)\mathrel{\mathop{\ordinarycolon}}N_{m}|\chi(S)$, so

$b_{1}^{(2)}(\pi_{1}(S))=\lim_{m\rightarrow\infty}\frac{b_{1}(N_{m})}{|\pi_{1}(S)\mathrel{\mathop{\ordinarycolon}}N_{m}|}=\lim_{m\rightarrow\infty}\frac{2-|\pi_{1}(S)\mathrel{\mathop{\ordinarycolon}}N_{m}|\chi(S)}{|\pi_{1}(S)\mathrel{\mathop{\ordinarycolon}}N_{m}|}=-\chi(S)=2g-2.$

We have also used that
$|\pi_{1}(S)\mathrel{\mathop{\ordinarycolon}}N_{m}|\rightarrow\infty$ because
$\pi_{1}(S)$ is infinite.

In the previous example we saw the relation between the Euler characteristic
of a surface and the first $L^{2}$-Betti number of its fundamental group. This
relation is extended to one-relator groups and surface-plus-one-relation
groups by Dicks and Linnell [14]. The Euler characteristic also appears in the
computation of the first $L^{2}$-Betti numbers of compact 3-manifold groups
[34]. As opposed to the case of surface groups, their first $L^{2}$-Betti
numbers are not typically nonzero.

In the case of finitely generated groups that a priori are not finitely
presented, one can still estimate its first $L^{2}$-Betti number in a similar
way as Lück’s approximation theorem. The following is a result of Lück and
Osin [38].

######  Theorem 8.1.3.

Let $G$ be finitely generated and let $G=N_{1}>N_{2}>\dots>N_{m}>\dots$ be a
sequence of finite-index normal subgroups with $\bigcap_{m}N_{m}=1$. Then

$\limsup_{m\rightarrow\infty}\frac{b_{1}(N_{m})}{|G\mathrel{\mathop{\ordinarycolon}}N_{m}|}\leq
b_{1}^{(2)}(G).$

In order to understand finitely generated normal subgroups of parafree groups,
we will also use the following theorem from of Gaborian [18, Theorem 6.8].

######  Theorem 8.1.4.

Suppose that

$1\rightarrow N\rightarrow G\rightarrow G^{\prime}\rightarrow 1$

is an exact sequence of groups where $N$ and $G^{\prime}$ are infinite. If
$b_{1}^{(2)}(N)<\infty$, then $b_{1}^{(2)}(G)=0$.

The following result is due to Bridson and Reid [7, Proposition 7.5].

######  Proposition 8.1.5.

Let $G$ be a finitely generated group and let $F$ be a finitely presented
group that is residually-$p$ for some prime $p$. Suppose that there is an
embedding $G\xhookrightarrow{}F_{\widehat{p}}$ of dense image. Then
$b_{1}^{(2)}(G)\geq b_{1}^{(2)}(F)$.

This is proven by estimating $b_{1}^{(2)}(F)$ and $b_{1}^{(2)}(G)$ with Lück
approximations’ of theorems 8.1.1 and 8.1.3, respectively.

In particular, the first $L^{2}$-Betti number is a pro-$p$ invariant among
finitely presented residually-$p$ groups.

######  Corollary 8.1.6.

Let $G$ and $G^{\prime}$ be finitely presented residually-$p$ groups such that
$G_{\widehat{p}}\cong G^{\prime}_{\widehat{p}}$. Then
$b_{1}^{(2)}(G)=b_{1}^{(2)}(G^{\prime})$.

Before proving 8.1.5, we make a few observations about the pro-$p$ topology.

In general, given abstract groups $\Lambda\leq\Gamma$; it may not happen that
the canonical map
$\Lambda_{\widehat{p}}\longrightarrow\overline{\Lambda}\subseteq\Gamma_{\widehat{p}}$
is an isomorphism. We will implicitly search for this favourable scenario
during chapter 9. For the moment, we can give a simpler sufficient criterion
to ensure that it holds.

######  Lemma 8.1.7.

Let $\Lambda\xhookrightarrow{}\Gamma$ be an inclusion of abstract groups,
where $\Lambda$ has finite $p$-power index in $\Gamma$. Then the canonical map
${\Lambda}_{\widehat{p}}\longrightarrow\Gamma_{\widehat{p}}$ is injective. In
other words, by denoting
$\iota_{\widehat{p}}\mathrel{\mathop{\ordinarycolon}}\Gamma\longrightarrow\Gamma_{\widehat{p}}$,
the canonical map
${\Lambda}_{\widehat{p}}\longrightarrow\overline{\iota_{\widehat{p}}(\Lambda)}\leq\Gamma_{\widehat{p}}$
is an isomorphism of pro-$p$ groups.

###### Proof.

By [43, Proposition 3.2.6], the conclusion of this lemma holds exactly when
the pro-$p$ topology of $\Gamma$ induces the pro-$p$ topology on its subspace
$\Lambda$. In addition, by [43, Lemma 3.1.4(a)], the previous condition is
ensured if $\Lambda$ has $p$-power index in $\Gamma$. ∎

This observation can be used to compare first $L^{2}$-Betti numbers of
abstract groups with the rank of their pro-$p$ completions.

###### Proof of 8.1.5.

Since $F$ is finitely generated, for every integer $D\geq 1$ there are only a
finite number of subgroups of $F$ of index at most $D$. Let $F(d)$ denote the
intersection of all normal subgroups of $F$ with $p$-power index at most
$p^{d}$. Alternatively, $F(d)$ is the finite intersection of all kernels of
surjective maps $F\longrightarrow P$ for any finite $p$-group of order at most
$p^{d}$. By the correspondence discussed in eq. 4.4, it is clear that the
closure $\overline{F(d)}$ of $F(d)$ in $F_{\widehat{p}}$ is equal to the
finite intersection of all the open normal subgroups of $F_{\widehat{p}}$
(here we have implicitly used the fact that $\overline{X\bigcap
Y}=\overline{X}\bigcap\overline{Y}$ for $p$-power index subgroups of $F$). In
particular, each $\overline{F(d)}$ has finite index in $F$ and
$\bigcap_{d}\overline{F(d)}=\\{1\\}$.

We introduce $G(d)=G\bigcap\overline{F(d)}$, which does also verify that
$\bigcap_{d}G(d)=\\{1\\}$. The canonical map $G\longrightarrow
F_{\widehat{p}}/\overline{F(d)}$ has dense image so it is surjective. By the
choice of $G(d)$, it factors through an isomorphism $G/G(d)\longrightarrow
F_{\widehat{p}}/\overline{F(d)}$. On the other side, the canonical maps
$F/F(d)\longrightarrow(F/F(d))_{\widehat{p}}\longrightarrow
F_{\widehat{p}}/\overline{F(d)}$ are isomorphisms since $F/F(d)$ is a finite
$p$-groups. This results in the equalities

$|G\mathrel{\mathop{\ordinarycolon}}G(d)|=|F_{\widehat{p}}\mathrel{\mathop{\ordinarycolon}}\overline{F(d)}|=|F\mathrel{\mathop{\ordinarycolon}}F(d)|.$

By lemma 8.1.7, the canonical map
$F(d)_{\widehat{p}}\longrightarrow\overline{F(d)}\subseteq F_{\widehat{p}}$ is
an isomorphism of pro-$p$ groups. Since $G$ is dense in $F_{\widehat{p}}$,
then $G(d)$ is dense in $\overline{F(d)}\cong F(d)_{\widehat{p}}$, so
$b_{1}(G(d))\geq b_{1}(F(d))$ for all $d\geq 1$ by 8.0.3.

Lastly, we can compare $b_{1}^{(2)}(F)$ and $b_{1}^{(2)}(G)$ by using the
sequences $\\{F(d)\\}_{d}$ and $\\{G(d)\\}_{d}$ by means of 8.1.1 and 8.1.5.
Notice that

$b_{1}^{(2)}(G)\geq\limsup_{d\rightarrow\infty}\frac{b_{1}(G(d))}{|G\mathrel{\mathop{\ordinarycolon}}G(d)|}\geq\limsup_{d\rightarrow\infty}\frac{b_{1}(F(d))}{|F\mathrel{\mathop{\ordinarycolon}}F(d)|}=b_{1}^{(2)}(F).\qed$

We mentioned in the introduction that
$b_{1}^{(2)}(G)=\dim_{\mathcal{D}_{\mathbb{Q}G}}H_{1}(G;\mathcal{D}_{\mathbb{Q}G})$.
The group ring $\mathbb{Q}G$ of zero characteristic admits some analytical