# On the profinite rigidity of free and surface groups

Ismael Morales

###### Abstract.

Let $S$ be a free or surface group. We establish a Tits alternative for groups
in the finite, soluble and $p$-genus of $S$. This generalises (and gives a new
proof of) the analogous result of Baumslag for parafree groups. Then we study
applications to profinite rigidity. A well-known question of Remeslennikov
asks whether a finitely generated residually finite $G$ with profinite
completion $\widehat{G}\cong\widehat{S}$ is necessarily $G\cong S$. We give a
positive answer when $G$ belongs to a class of groups ${\bf\mathcal{H}_{\bf
ab}}$ that has a finite abelian hierarchy starting with finitely generated
residually free groups. This strengthens previous results of Bridson, Corner,
Reid and Wilton. Then we show that one-relator groups in the genus of $S$ are
hyperbolic and virtually special. Lastly, we prove that
$S\times\mathbb{Z}^{n}$ is profinitely rigid within finitely generated
residually free groups.

##  1\. Introduction

A fundamental idea in the study of infinite groups consists of understanding
how much algebraic structure of a group $G$ can be detected in its collection
of finite quotients, or if, in fact, the isomorphism type of $G$ can be
determined out of these quotients. This sort of question broadly belongs to
the area of profinite rigidity.

To expand on the above, we briefly review a few definitions. A non-empty class
of finite groups $\mathcal{C}$ is a formation if it is closed under taking
homomorphic images and subdirect products. Given a finitely generated group
$\Lambda$, we say that it is residually-$\mathcal{C}$ if, for all
$1\neq\lambda\in\Lambda$, there exists $N\unlhd\Lambda$ such that
$\lambda\notin N$ and $\Lambda/N\in\mathcal{C}$. In this case, following the
terminology of [Gru11], we define the $\mathcal{C}$-genus of $\Lambda$ as the
collection of finitely generated residually-$\mathcal{C}$ groups $\Gamma$ with
isomorphic pro-$\mathcal{C}$ completion
$\Gamma_{\widehat{\mathcal{C}}}\cong\Lambda_{\widehat{\mathcal{C}}}$.
Equivalently, the latter happens if $\Gamma$ and $\Lambda$ have the same
collection of isomorphism types of finite quotients belonging to $\mathcal{C}$
(see Section 2 for more details). We will mostly think of $\mathcal{C}$ as
being the formations of all finite groups, all finite soluble groups, all
finite nilpotent groups or all finite $p$-groups (for some prime $p$). In
these cases, we refer to the $\mathcal{C}$-genus as the finite, soluble,
nilpotent and $p$-genus, respectively. Lastly, we say that a finitely
generated residually finite $\Lambda$ is profinitely rigid if the only
$\Gamma$ in its finite genus is $\Gamma\cong\Lambda$.

It is easy to see that finitely generated abelian groups are profinitely
rigid. However, for groups satisfying more complicated group laws, the picture
is not as clear. In fact, there are examples of finitely generated nilpotent
groups [Pic71] and virtually cyclic groups [Bau74] that are not profinitely
rigid. Outside the world of groups satisfying group laws, harder problems
arise, such as finding profinitely rigid groups that contain non-abelian free
subgroups. The only known examples are due to Bridson–McReynolds–Reid–Spitler
[Bri20, Bri21], which are certain arithmetic lattices in
$\mathrm{PSL}_{2}(\mathbb{C})$ and $\mathrm{PSL}_{2}(\mathbb{R})$.

On the other hand, there are other classical families of groups that are
expected to be profinitely rigid, such as free groups, as conjectured by
Remeslennikov (see, for example, [Khu14]*Question 5.48 and [Nos79]*Question
12).

######  Conjecture 1.1.

Let $\Gamma$ be a finitely generated residually finite group with the same
profinite completion as a free group $F$. Then $\Gamma\cong F$.

It is also natural to pose this conjecture for surface groups. By a surface
group we will always mean the fundamental group of a closed orientable surface
$\Sigma$, denoted by $\pi_{1}\Sigma$.

######  Conjecture 1.2.

Let $\Gamma$ be a finitely generated residually finite group with the same
profinite completion as a surface group $\pi_{1}\Sigma$. Then
$\Gamma\cong\pi_{1}\Sigma$.

We refer the reader to [And212] for related conjectures regarding other
genera.

The main result of this paper solves 1.1 and 1.2 when $\Gamma$ belongs to a
family of groups ${\bf\mathcal{H}_{\bf ab}}$ (see 1.3 below), which has a
finite abelian hierarchy starting at finitely generated residually free
groups.

Let us briefly review the current status of these conjectures.
Puder–Parzanchevski [Pud15] established 1.1 when $\Gamma$ is a one-relator
group, and they also proved a related principle: if $F$ is a free group and
$H\leqslant F$ is a subgroup, then its closure $\overline{H}$ in $\widehat{F}$
is a free factor of $\widehat{F}$ if and only if $H$ was already a free factor
of $F$. An alternative proof of this fact was given by Wilton [And18] and it
was recently generalised by Garrido and Jaikin-Zapirain [Gar22] to virtually
free groups $F$. Coming back to 1.1, Wilton [Wil18] solved it when $\Gamma$ is
a limit group (defined below) by showing that these groups are either free or
contain surface subgroups. Bridson–Corner–Reid [Bri14a] had already pointed
out how the latter can be used to establish 1.1 when $\Gamma$ is residually
free. Lastly, Wilton proved 1.2 when $\Gamma$ is a limit group [Wil21]. Note
that there is a second proof of this fact by Fruchter and the author [Fru22].

A limit group $G$ is a finitely generated fully-residually free group, meaning
that for all finite subsets $X\subseteq G$, there exists a group homomorphism
$\psi\colon G\longrightarrow F$ to a free group $F$ such that the restriction
of $\psi$ to $X$ is injective. Alternatively, these are finitely generated
groups with the same universal theory as free groups (as proven by Sela
[Sel01]*Theorem 4.6). This class includes free and surface groups. A bigger
class of groups is the class of finitely generated residually free groups. For
example, $F_{2}\times F_{2}$ is residually free but not limit. The class
${\bf\mathcal{H}_{\bf ab}}$ contains all these groups.

######  Definition 1.3.

Let ${\bf\mathcal{H}_{\bf ab}}$ denote the smallest class of groups such that

  1. (1)

${\bf\mathcal{H}_{\bf ab}}$ contains all finitely generated residually free
groups.

  2. (2)

If $A,B$ are in ${\bf\mathcal{H}_{\bf ab}}$ and $C$ is a virtually abelian
subgroup of $A$ and $B$, then the amalgamated product $A*_{C}B$ belongs to
${\bf\mathcal{H}_{\bf ab}}$.

  3. (3)

If $A$ is in ${\bf\mathcal{H}_{\bf ab}}$ and $\theta\colon C\longrightarrow
C^{\prime}$ is an isomorphism between virtually abelian subgroups of $A$, then
the HNN extension $A\,*_{C,\theta}$ belongs to ${\bf\mathcal{H}_{\bf ab}}$.

  4. (4)

If $A$ is in ${\bf\mathcal{H}_{\bf ab}}$ and $A$ embeds as a finite-index
subgroup in $B$, then $B$ belongs to ${\bf\mathcal{H}_{\bf ab}}$.

  5. (5)

If $A$ and $B$ are in ${\bf\mathcal{H}_{\bf ab}}$, then $A\times B$ belongs to
${\bf\mathcal{H}_{\bf ab}}$.

Our main theorem (proven in 10.1 and 10.2) is the following.

######  Theorem A.

Let $\Gamma$ be a group belonging to the class ${\bf\mathcal{H}_{\bf ab}}$.
Then 1.1 and 1.2 hold for $\Gamma$.

We remark that the methods of Wilton in [Wil18] and [Wil21] cover A when
$\Gamma$ is a limit group or when $\Gamma$ is a hyperbolic fundamental groups
of graph of groups with free vertices and cyclic edges. Therefore, one novelty
of A is including similar fundamental groups and getting rid of the
hyperbolicity assumption. Note that many groups in ${\bf\mathcal{H}_{\bf ab}}$
are virtually compact special but not residually free, such as the relatively
hyperbolic group $(F_{2}\times F_{2})*F_{3}$ and the hyperbolic one-relator
groups $\langle a,b,c\,|\,a^{p}b^{q}c^{r}\rangle$ (see [Lyn62]).

There are several reasons why it is natural to study ${\bf\mathcal{H}_{\bf
ab}}$, other than being a family with a finite abelian hierarchy terminating
in residually free groups that naturally incorporates other groups for which
1.1 and 1.2 were solved. The consideration of hierarchies with abelian edges
has provided a setting in which combination theorems for relevant classes of
groups have been established: for coherent groups [Thu20, Corollary 5.4], for
hyperbolic groups [Bes92], for free groups [She55, Swa86, Sta91] and for
parafree groups [And21].

We say that a finitely generated group $\Gamma$ is parafree (resp.
parasurface) if it belongs to the nilpotent genus of some free group (resp.
surface group). Parafree groups were introduced by Baumslag in [Bau67] and
these are groups that resemble free groups in terms of their residual
properties. In fact, all $\Gamma$ of 1.1 (resp. 1.2) are parafree (resp.
parasurface) by [And212, Theorem 1.1]. It is not hard to see that
${\bf\mathcal{H}_{\bf ab}}$ contains all the parafree groups constructed in
[Bau67, Bau69, Bau94, Bau05, Bau06, And21] and all the parasurface groups of
[Bou10].

Our proof of A mimics the general outline of the proofs of Wilton for limit
groups. In particular, it relies on the existence of surface subgroups for
one-ended hyperbolic amalgamated products $F*_{\mathbb{Z}}F^{\prime}$ and HNN
extensions $F*_{\mathbb{Z}}$ of free groups along cyclic subgroups ([Wil18,
Theorem 6.1]). However, in order to apply these methods, one first needs to
show that the group $\Gamma$ is hyperbolic and has strong separability
properties (virtual retractions). We show that $\Gamma$ is hyperbolic using
the Bestvina–Feighn combination theorem [Bes92] and also virtually compact
special by the classification of hyperbolic groups with a malnormal quasi-
convex hierarchy ([Wis21, Theorem 11.2]), which ensures separability of quasi-
convex subgroups.

In order to apply the Bestvina–Feighn combination theorem in the first place,
we need to ensure that such $\Gamma\in{\bf\mathcal{H}_{\bf ab}}$ do not
contain Baumslag–Solitar groups. We ensure this by B, a result that also
encompasses a strong Tits alternative.

In geometric group theory, we say that a group $G$ satisfies a Tits
alternative if for every subgroup $H\leqslant G$, the following dichotomy
holds:

  * (a)

Either the group $H$ is virtually solvable; or

  * (b)

the group $H$ contains a non-abelian free group.

This principle was originally established by Tits [Tit72] for finitely
generated linear groups. This implies that linear groups satisfy the von
Neumann conjecture (i.e. non-amenable linear groups contain non-abelian free
subgroups) and it also plays a crucial role in several proofs of Gromov’s
theorem on groups of polynomial growth [Gro81, Dri84, Kle10, Sha10]. This
surprising dichotomy has been established for many other groups. For example,
for Mapping Class groups by McCarthy [Car85] and for $\mathrm{Out}(F_{n})$ by
Bestvina-Feighn-Handel [Bes05a, Bes05b]. There are also versions of the Tits
alternative in more topological settings, as given by Breuillard–Gelander
[Bre07], and in profinite or pro-$p$ settings, as studied by
Kochloukova–Zalesskii [Koc11] and Wilton–Zalesskii [Wilton17] (see also
[Zal22]).

By a standard ping-pong argument, it is known that hyperbolic groups also
satisfy a Tits alternative. In fact, the Tits alternative is improved because
solvable subgroups are virtually cyclic (and the same occurs for some of the
previously listed examples). We show this strengthened version of the Tits
alternative for groups in the genera of free or surface groups as a
consequence of the following result.

######  Theorem B.

Let $p$ be a prime and let $G$ be a group in the $p$-genus of a free or
hyperbolic surface group. Let $H$ be a subgroup of $G$. The following holds:

  * (i)

If $H$ is amenable, then $H$ is cyclic.

  * (ii)

If $H$ is two-generated, then $H$ is free.

We prove this result in 8.1. In particular, as we claimed above, the groups
$G$ of B do not contain Baumslag–Solitar subgroups and have the property that
any subgroup $H\leqslant G$ is either cyclic or contains a non-abelian free
subgroup. Continuing the previous discussion, we note that there are even
stronger forms of the Tits alternative in the literature, such as the one
proven by Button [But04] for fundamental groups of compact three-manifolds
with boundary in terms of $SQ$-universality. We refer the reader to [Ant15]
for more references and more variations of this classical dichotomy.

We remark why B can be applied to groups in the finite, solvable or nilpotent
genus of a free or surface group. The reason is that, again by [And212,
Theorem 1.1], these genera are contained in the $p$-genus of the corresponding
free or surface group, so all such groups lie under the assumptions of B.

There has been much research on relating subgroup structure and profinite
completions, specially in the context of three-manifolds. For example,
Kochloukova–Zalesskii [Koc07] show a Tits alternative for groups in the
$p$-genus of an irreducible, orientable, closed three-manifold. The topic of
connecting three-manifold topology with the profinite completion of its
fundamental group has experienced much exciting progress over the last few
years [Bri17, Wil17, And18, Wilkes18, Wilkes18b, Wilkes19, Wil19, Bri15, Ma22,
Liu22, Hug22]. A remarkable result is the profinite recognition of the
topology of geometric three-manifolds by Wilton–Zalesskii [Wilton17], where
they show that a compact orientable aspherical three-manifold $M^{3}$ is
hyperbolic if and only if $\widehat{\pi_{1}M^{3}}$ contains no
$\mathbb{Z}_{p}^{2}$. More recently, this was generalised by Zalesskii [Zal22,
Theorem 1.1], showing that a relatively hyperbolic virtually compact special
group $G$ is hyperbolic if and only if its profinite completion $\widehat{G}$
contains no $\mathbb{Z}_{p}^{2}$.

One of the difficulties of B is relating subgroup obstructions of an abstract
group to subgroup obstructions of its pro-$p$ completion in the flavour of the
previous results but in the generality of finitely generated residually-$p$
groups, where there is a priori no hierarchy at our disposal.

Using ideas from the world of $L^{2}$-invariants, more progress was made in
the previous direction, although with the apparent caveat of leading to
conclusions of more global nature. For example, Lackenby [Lac09, Lac10] proved
that largeness (the property of admitting a finite-index subgroup that
surjects onto $F_{2}$) is a profinite and pro-$p$ invariant among finitely
presented groups, and that it relates to the property of having positive first
$L^{2}$-Betti number, denoted by $b_{1}^{(2)}$. In a different flavour,
Peterson–Thom [Pet11] established a Freiheitsatz for countable abstract groups
$\Lambda$ with $b_{1}^{(2)}(\Lambda)>0$ and Bridson–Reid [Bri14, Corollary
8.3] used this to prove a Freiheitsatz for a class of groups $G$ that include
parafree groups and those of B.

We note that B is not immediate even for residually free groups. An elementary
property of residually free groups is that non-abelian two-generated subgroups
are free (see [Bau62, Section 4]). However, these can clearly contain the
group $\mathbb{Z}^{2}$ and, even in this context, one needs an additional
argument to rule it out.

The following is another consequence of B, as an application of the advances
of Louder–Wilton [Lou22, Lou22b] and Linton [Lin22] on one-relator groups.

######  Corollary C.

Let $p$ be a prime and let $G$ be a one-relator group in $p$-genus of a free
or hyperbolic surface group. Then $G$ is hyperbolic and virtually special.

This is proven in 9.5. The conclusion of this result was already ensured for
one-relator parafree groups in [Lin22, Corollary 8.4]. However, we should
stress that C extends to other genera by [And212, Theorem 1.1].

We recall the reader that, Puder–Parzanchevski [Pud15] solved 1.1 when
$\Gamma$ is a one-relator group, but this case of 1.2 is still open. That is,
we do not know whether one-relator groups $\Gamma$ in the finite genus of a
surface group $\pi_{1}\Sigma$ do necessarily satisfy
$\Gamma\cong\pi_{1}\Sigma$. Nevertheless, C ensures that such $\Gamma$ are
hyperbolic and virtually special.

We also remark that B gives a new proof of the analogous result of Baumslag
[Bau69, Theorem 4.2]) for parafree groups $G$ and we generalise it to
parasurfaces and other genera. The proof of Baumslag relies on the fact that a
parafree group can be embedded into the unit group of a ring of power series
$\mathbb{Z}\llbracket X\rrbracket$. Our proof of B relies on different
cohomological tools and the following elementary fact about the pro-$p$
topology of nilpotent groups.

######  Theorem D.

Let $G$ be a virtually polycyclic group. Then the following two statements are
equivalent:

  * •

The group $G$ is nilpotent.

  * •

For all subgroups $H\leqslant G$ and all primes $p$, the pro-$p$ topology of
$G$ induces on $H$ its full pro-$p$ topology.

The first implication is proven in 7.2 and the second in 7.7. To the best of
the author’s knowledge, this result has not appeared explicitly in the
literature before. We also think it may be of independent interest. Virtually
polycyclic groups are known to be LERF (see Section 7 for more references and
discussion). In particular, in these groups, the induced profinite topology on
subgroups is the full profinite topology. However, when it comes to the
pro-$p$ topology, D shows that this phenomenon is very restrictive among
virtually polycyclic groups.

In our last result, we use free and surface groups to give infinitely many
examples of residually free groups that are determined within residually free
groups by their finite quotients.

######  Theorem E.

Let $G$ be a finitely generated residually free group and let $S$ be a free or
surface group. Suppose that
$\widehat{G}\cong\widehat{S}\times\widehat{\mathbb{Z}}^{n}$. Then $G\cong
S\times\mathbb{Z}^{n}$.

We prove this theorem in Section 11. The first step consists on showing that
$G$ and $S\times\mathbb{Z}^{n}$ have isomorphic centre. We remark that this
does not happen in general since Lubotzky proved in [Lub14, Proposition 1.5]
that having non-trivial centre is not a profinite property.

###  1.1. Acknowledgements

The author is funded by the Oxford–Cocker Graduate Scholarship. This work has
also received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 850930). I am grateful to Andrei Jaikin-Zapirain for introducing
me to these ideas and questions on profinite rigidity and for suggesting me to
prove D. I would also like to thank Jonathan Fruchter for conversations on the
one-endedness of cyclic splittings. Lastly, special thanks are due to my
supervisors Dawid Kielak and Ric Wade for their stimulating guidance and for
carefully reading this paper, which made a very significant improvement.

###  1.2. Organisation of the paper

In Section 2, we revisit the theory of profinite and pro-$p$ groups and, in
Section 3, we review when the first $L^{2}$-Betti numbers of a residually
finite abstract group can be detected from its profinite completion. In
Section 4, we briefly introduce the cohomology theory for profinite groups,
with emphasis on pro-$p$ versions of notions such as cohomological dimension
and Poincaré duality. The main result of Section 4 is 4.14, whose application
to A is ensuring that a group that virtually retracts onto a surface subgroup
and that belongs to the finite genus of a surface group is itself a surface
group. To ensure surface subgroups and virtual retractions, we recall
separability properties of limit and special groups in Section 5 and Section
6. The Tits alternative of B is proven in 8.1 as an application of
$p$-separability principles studied in Section 7. Lastly, the proof of A is
divided into 10.1 and 10.2, and we prove E in Section 11.

##  2\. Pro-$\mathcal{C}$ groups, topologies and completions

This section continues the discussion at the beginning of the introduction,
where we defined formations $\mathcal{C}$ of finite groups and the
$\mathcal{C}$-genus of an abstract group.

We are interested in studying the finite genus of free and surface groups $S$.
For this purpose, it will be important to keep the following deep result in
mind, as it a strong structural result about these mysterious groups.

######  Theorem 2.1 (Theorem 1.1, [And212]).

Let $S$ be a finitely generated free or surface group. Suppose that $G$
belongs to the finite, soluble or nilpotent genus of $S$. Then $G$ belongs to
the $p$-genus of $S$ for all primes $p$.

We remark again that these inclusions between genera are not true in general,
since Lubotzky [Lub14] showed that the properties of being residually solvable
or residually-$p$ are not profinite invariants.

###  2.1. Pro-$\mathcal{C}$ completions

We now fix a formation $\mathcal{C}$ of finite groups. A pro-$\mathcal{C}$
group can be defined as a profinite group ${\bf G}$ such that for all its open
normal subgroups ${\bf U}\unlhd_{o}{\bf G}$, one has that ${\bf G}/{\bf
U}\in\mathcal{C}$. Alternatively, a pro-$\mathcal{C}$ group is just an inverse
limit of groups belonging to $\mathcal{C}$ that is endowed with the profinite
topology.

In order to relate residual properties of abstract groups and profinite
invariants, we recall the pro-$\mathcal{C}$ completion functor. Given an
abstract group $G$, its collection of quotients belonging to $\mathcal{C}$ is
encoded in its pro-$\mathcal{C}$ completion, which is defined to be the an
inverse limit of finite groups

$G_{\widehat{\mathcal{C}}}=\varprojlim_{U}G/U$

where $U$ ranges over finite-index normal subgroups of $G$ such that
$G/U\in\mathcal{C}$ (endowed with the profinite topology).

There is a natural and canonical group homomorphism $i_{\mathcal{C}}\colon
G\longrightarrow G_{\widehat{\mathcal{C}}}$ and, in these terms, a group $G$
is residually-$\mathcal{C}$ if and only if $\iota_{\mathcal{C}}$ is injective.
This map $\iota_{\mathcal{C}}$ enjoys the following universal property (that,
in fact, determines it up to automorphisms of $G_{\widehat{\mathcal{C}}}$, and
this can be described as follows.

######  Proposition 2.2.

Let $\Gamma$ be an abstract group. Let
$\iota_{\widehat{\mathcal{C}}}\colon\Gamma\longrightarrow\Gamma_{\widehat{\mathcal{C}}}$
be the canonical map. The data
$(\Gamma_{\widehat{\mathcal{C}}},\iota_{\widehat{\mathcal{C}}})$ is
characterised by the following properties.

  1. (1)

$\Gamma_{\widehat{\mathcal{C}}}$ is a pro-$\mathcal{C}$ group.

  2. (2)

The map $\iota_{\widehat{\mathcal{C}}}$ has dense image.

  3. (3)

For any pro-$\mathcal{C}$ group ${\bf K}$ and every group homomorphism
$f\colon\Gamma\longrightarrow{\bf K}$ with dense image, there exists a
continuous homomorphism
$f_{\widehat{\mathcal{C}}}\colon\Gamma_{\widehat{\mathcal{C}}}\longrightarrow{\bf
K}$ such that the diagram

${\Gamma}$${\Gamma_{\widehat{\mathcal{C}}}}$${{\bf
K}}$$\scriptstyle{f}$$\scriptstyle{\iota_{\widehat{\mathcal{C}}}}$$\scriptstyle{f_{\widehat{\mathcal{C}}}}$

is commutative.

When we say that the pair
$(\Gamma_{\widehat{\mathcal{C}}},\iota_{\widehat{\mathcal{C}}})$ is unique, we
mean the following: if ${\bf H}$ is a pro-$\mathcal{C}$ group and
$\iota\colon\Gamma\longrightarrow{\bf H}$ is a group homomorphism, and the
pair $({\bf H},\iota)$ verifies the three properties above, then there exists
an isomorphism of pro-$\mathcal{C}$ groups $\alpha\colon
G_{\widehat{\mathcal{C}}}\longrightarrow{\bf H}$ such that
$\alpha\circ\iota_{\widehat{\mathcal{C}}}=\iota$.

We stress the fact that the previous discussion does not rely on whether
$\iota_{\widehat{\mathcal{C}}}$ is injective (equivalently $G$ is
residually-$\mathcal{C}$) or not. A priori the pro-$\mathcal{C}$ completion
only captures information about finite quotients of $G$ in $\mathcal{C}$ and
not about how we can distinguish elements of $G$ in these quotients. There
are, for example, finitely generated residually finite groups with free
pro-$p$ completion that are not residually-$p$ for no prime $p$, such as the
trefoil knot group $\langle a,b\,|a^{2}b^{3}\rangle$.

The philosophy that finite quotients of a group are encoded in their profinite
completion is precised in the following folkloric theorem.

######  Theorem 2.3 (Dixon–Formanek–Poland–Ribes [Dix82]).

Let $\mathcal{C}$ be a formation of finite groups. Two abstract groups
$\Gamma$ and $\Lambda$ have the same class of isomorphism types of quotients
belonging to $\mathcal{C}$ if and only if
$\Gamma_{\widehat{\mathcal{C}}}\cong\Lambda_{\widehat{\mathcal{C}}}$.

So we can completely reformulate in terms of the pro-$\mathcal{C}$ completions
what it means for a group to be in the $\mathcal{C}$-genus of another one.
This is important because now we have turned a bare collection of finite
groups into a group and now we can do, say, group cohomology with this
information. Moreover, this functor “pro-$\mathcal{C}$ completion” is right-
exact.

######  Proposition 2.4 ([Rib00], Proposition 3.2.5).

Let $\mathcal{C}$ be a formation that is closed under taking normal subgroups.
Given a exact sequence of abstract groups $N\longrightarrow G\longrightarrow
Q\longrightarrow 1$, the induced sequence of maps
$N_{\widehat{\mathcal{C}}}\longrightarrow
G_{\widehat{\mathcal{C}}}\longrightarrow
Q_{\widehat{\mathcal{C}}}\longrightarrow 1$ is exact.

Given an injective map $N\longrightarrow G$, the question of whether the
induced map $N_{\widehat{\mathcal{C}}}\longrightarrow
G_{\widehat{\mathcal{C}}}$ is also injective will be a crucial topic of
discussion later in this section and during Section 7, especially with the aim
of proving the Tits alternative of 8.1. On the other hand, 2.4 also tells us
that pro-$\mathcal{C}$ completions are better understood when studying
quotients than when studying subgroups, as the following direct consequence
exhibits.

######  Proposition 2.5 (Pro-$\mathcal{C}$ completion of a quotient).

Let $G$ be an abstract group and let $N\unlhd G$ be a normal subgroup. Suppose
that $\mathcal{C}$ is a formation as in 2.4. Denote by $\iota\colon
N\longrightarrow G$ the inclusion. Then the canonical and natural map

$\left(G/N\right)_{\widehat{\mathcal{C}}}\longrightarrow
G_{\widehat{\mathcal{C}}}\Big{/}\overline{\iota_{\widehat{\mathcal{C}}}(N)}$

is an isomorphism.

###  2.2. The notion of index in profinite groups

For our purposes, given a profinite group ${\bf G}$ and a subgroup ${\bf H}$,
it will be convenient for our next discussion to define the formal meaning of
$|{\bf G}\mathrel{\mathop{\ordinarycolon}}{\bf H}|$ as a supernatural number
(as in, for example, [Rib00, Section 2.3]).

However, we recall that we say that $p^{\infty}$ divides the index $|{\bf
G}\mathrel{\mathop{\ordinarycolon}}{\bf H}|$ if and only if for all $n$ there
exists a normal open ${\bf U}\unlhd_{o}{\bf G}$ such that $p^{n}$ divides the
index of ${\bf H}{\bf U}/{\bf U}$ in the finite group ${\bf G}/{\bf U}$. If
${\bf G}$ is a pro-$p$ group, then ${\bf H}\leqslant_{c}{\bf G}$ having
infinite index is the same as $p^{\infty}$ dividing the index $|{\bf
G}\mathrel{\mathop{\ordinarycolon}}{\bf H}|$. However, for general profinite
groups, the latter is much stronger, as the following example shows.

######  Example 2.6.

The profinite group $\widehat{\mathbb{Z}}$ is canonically isomorphic to
$\prod_{p}\mathbb{Z}_{p}$, where $\mathbb{Z}_{p}$ denotes the additive part of
the ring of $p$-adic integers (isomorphic to the pro-$p$ completion of
$\mathbb{Z}$). So $\widehat{\mathbb{Z}}$ surjects in a natural way onto the
group $\prod_{p}\mathbb{Z}/p\mathbb{Z}$, where the kernel ${\bf K}$ will have
an index divisible by the supernatural number $\prod_{p}p$. In particular, no
prime $p$ will satisfy that $p^{2}$ divides the index
$|\widehat{\mathbb{Z}}\mathrel{\mathop{\ordinarycolon}}{\bf K}|$.

###  2.3. Lower central series

Given a group $G$, we will denote by $G^{\prime}=[G,G]$ its commutator
subgroup and by $G_{\operatorname{ab}}=G/G^{\prime}$ its abelianisation. The
terms of the lower central series of $G$ are defined recursively as
$\gamma_{1}G=G$ and $\gamma_{n+1}G=[\gamma_{n}G,G]$ if $n\geqslant 1$.
Similarly, the terms of the $p$-lower central series of $G$ are defined as
$\gamma_{1,p}G=G$ and $\gamma_{n+1,p}G=[\gamma_{n,p}G,G]\gamma_{n,p}G^{p}$ if
$n\geqslant 1$. The expression $\gamma_{n}\Gamma_{\widehat{p}}$ may be
ambiguous. We clarify that it means $\gamma_{n}G$ with
$G=\Gamma_{\widehat{p}}$, defined as a subgroup of $\Gamma_{\widehat{p}}$ (so
we do not mean to denote the pro-$p$ completion of $\gamma_{n}\Gamma$).

######  Proposition 2.7 ([Dix99], Proposition 1.19 and Exercise 17).

Let ${\bf G}$ be a finitely generated pro-$p$ group. Then $\gamma_{k}{\bf G}$
is a closed subgroup of ${\bf G}$.

In fact, the previous is true for finitely generated profinite groups,
although this is a much deeper result due to Nikolov and Segal [Nik07, Theorem
1.4].

######  Lemma 2.8.

Let $\Gamma$ be a finitely generated group and let $k\geqslant 0$ be an
integer. Then the canonical map
$\Gamma/\gamma_{k}\Gamma\longrightarrow\Gamma_{\widehat{p}}/\gamma_{k}\Gamma_{\widehat{p}}$
is a pro-$p$ completion map.

###### Proof.

The group $\gamma_{k}\Gamma_{\widehat{p}}$ are closed subgroups of
$\Gamma_{\widehat{p}}$ by 2.7 and contain $\gamma_{k}\Gamma$ as a dense
subgroup, so $\gamma_{k}\Gamma_{\widehat{p}}$ is the closure of
$\gamma_{k}\Gamma$. By 2.5, the conclusion follows. ∎

###  2.4. Induced profinite topologies

Given a formation $\mathcal{C}$ and a group $G$ the full pro-$\mathcal{C}$
topology of $G$ is defined to be the initial topology of the map
$G\longrightarrow G_{\widehat{\mathcal{C}}}$. In other words, it is the
smallest topology that contains among its open subsets all the cosets of
$U\unlhd G$ such that $G/U\in\mathcal{C}$. When $\mathcal{C}$ is the formation
of all finite groups (resp. finite $p$-groups), we call it the profinite
(resp. pro-$p$) topology.

######  Definition 2.9.

Let $\mathcal{C}$ be a formation of finite groups. Assume that $H\leqslant G$.
Denote by $\iota\colon H\longrightarrow G$ the injection. We say that $H$ is
pro-$\mathcal{C}$ embeddable into $G$ if the induced map

$\iota_{\widehat{\mathcal{C}}}\colon H_{\widehat{\mathcal{C}}}\longrightarrow
G_{\widehat{\mathcal{C}}}$

is an injection.

We observe that this is a transitive relation in the sense that whenever
$A\leqslant B\leqslant C$ with $A$ being pro-$\mathcal{C}$ embeddable in $B$
and $B$ being pro-$\mathcal{C}$ embeddable in $C$, then $A$ is
pro-$\mathcal{C}$ embeddable in $C$.

######  Lemma 2.10 (See [Rib00], Lemma 3.2.6).

Let $H\leqslant G$ be two groups. Then $H$ is pro-$\mathcal{C}$ embeddable
into $G$ if and only if the pro-$\mathcal{C}$ topology of $G$ induces on $H$
its full pro-$\mathcal{C}$ topology.

######  Observation 2.11.

If $H$ is finitely generated, in order to prove that the
pro-$\mathcal{C}$-topology of $G$ induces the full pro-$\mathcal{C}$ topology
of $H$, it suffices to check the following: For any characteristic subgroup
$\Lambda\unlhd H$ with $H/\Lambda\in\mathcal{C}$, there exists $\Gamma\unlhd
G$ with $G/\Gamma\in\mathcal{C}$ such that $\Gamma\cap H\subseteq\Lambda$.

######  Lemma 2.12.

Let $G$ be a group and let $H\leqslant G$ be a subgroup with the property that
every finite-index normal subgroup $K\unlhd H$ with $H/K\in\mathcal{C}$ is a
closed subset in the pro-$\mathcal{C}$ topology of $G$. Then $H$ is
pro-$\mathcal{C}$ embeddable into $G$.

###### Proof.

Let $\Lambda\leqslant H$ be a closed subgroup of $H$, where $H$ is here
endowed with its full pro-$\mathcal{C}$ topology. Then $\Lambda$ is the
intersection of standard basis open sets of $H$, that is, there are normal and
finite-index subgroups $K_{i}\unlhd H$ such that $\Lambda=\cap_{i}K_{i}$. We
want to prove that $\Lambda$ is a closed subset of $G$, but this is clear
because each $K_{i}$ is, by assumption, closed in $G$. ∎

In particular, in LERF groups $G$, every finitely generated subgroup
$H\leqslant G$ has the property that the induced
$\widehat{H}\longrightarrow\widehat{G}$ is injective.

######  Lemma 2.13.

Let $G$ be a group and assume that $H\unlhd_{o}G$ such that
$G/H\in\mathcal{C}$. Then $H$ is pro-$\mathcal{C}$ embeddable into $G$.

###### Proof.

By [Rib00, Lemma 3.1.2], $H$ is open in $G$, so the conclusion follows from
Lemma 2.10 and Lemma 2.12. ∎

We can give other natural conditions when subgroups are closed in the
pro-$\mathcal{C}$ topology, such as factors of free splittings or semidirect
products.

######  Lemma 2.14 ([Rib00], Lemma 3.1.5 and Corollary 3.1.6).

Let $G_{1}=K_{1}\rtimes H_{1}$ and let $G_{2}=K_{2}*H_{2}$. Then $H_{i}$ is
pro-$\mathcal{C}$ embeddable into $G_{i}$ for all $i\in\\{1,2\\}$.

Given a semidirect product $G=K\rtimes H$, it is also natural to study the
induced topology on the first factor. For general pro-$\mathcal{C}$
topologies, one cannot really say much (as 7.8 exhibits). However, for
profinite topologies, we have the following simple lemma.

######  Lemma 2.15 ([Bri15], Lemma 2.2).

Let $G=K\rtimes H$ be a semidirect product, with $K$ finitely generated. Then
the induced profinite topology on $K$ is its full profinite topology.

Consequently, given a semidirect product $G=K\rtimes H$, with $K$ finitely
generated, we have a natural isomorphism
$\widehat{G}\cong\widehat{K}\rtimes\widehat{H}$.

######  Proposition 2.16.

Let $G$ be a finitely generated group such that the central quotients
$G/\gamma_{k}G$ are torsion-free for all $k$. Then, for all primes $p$,
$G_{\widehat{p}}$ is residually-(torsion-free nilpotent). In fact, the
quotients $G_{\widehat{p}}/\gamma_{n}G_{\widehat{p}}$ are torsion-free for all
$n$.

###### Proof.

Since $G_{\widehat{p}}$ is residually-$p$,
$\bigcap_{k}\gamma_{k,p}(G_{\widehat{p}})=\\{1\\}$ and, in particular,
$\bigcap_{k}\gamma_{k}G_{\widehat{p}}=1$.

It remains to prove that $G_{\widehat{p}}/\gamma_{k}G_{\widehat{p}}$ is
torsion-free. We know that $\Gamma=G/\gamma_{c+1}G$ is a finitely generated
torsion-free nilpotent group. We are going to prove that this is enough to
conclude that its pro-$p$ completion $\Gamma_{\widehat{p}}$ is torsion-free.
The proof of the proposition would then be complete because the canonical map
$\Gamma_{\widehat{p}}\longrightarrow
G_{\widehat{p}}/\gamma_{k}(G_{\widehat{p}})$ is an isomorphism by 2.5.

Now let $\Gamma$ be a finitely generated torsion-free nilpotent group. Then
$\Gamma/Z(\Gamma)$ is also torsion-free (see [Cle17, Corollary 2.22]). Then,
iterating the previous observation, we deduce that the lower central series

$1=\gamma_{c+1}\Gamma\unlhd\gamma_{c}\Gamma\unlhd\cdots\gamma_{2}\Gamma\unlhd\gamma_{1}\Gamma=\Gamma$

of $\Gamma$ will be such that
$\gamma_{k}\Gamma/\gamma_{k+1}\Gamma\cong\mathbb{Z}^{n_{k}}$ for all
$1\leqslant k\leqslant c$ and some non-negative integers $n_{1},\dots,n_{c}$.

Notice that the canonical maps
$(\gamma_{k}\Gamma)_{\widehat{p}}\longrightarrow\gamma_{k}\Gamma_{\widehat{p}}$
are isomorphisms (for example, by 7.2). From this and 2.5, we derive that
$(\gamma_{k+1}\Gamma/\gamma_{k}\Gamma)_{\widehat{p}}\longrightarrow\gamma_{k+1}\Gamma_{\widehat{p}}/\gamma_{k}\Gamma_{\widehat{p}}$
are isomorphisms of pro-$p$ groups for all $1\leqslant k\leqslant c$ (recall
that these groups are, by Lemma 2.8, the pro-$p$ completions of
$\gamma_{k+1}\Gamma/\gamma_{k}\Gamma$) So there is a lower central series of
$\Gamma_{\widehat{p}}$ by closed subgroups

$1=\gamma_{c+1}\Gamma_{\widehat{p}}\unlhd\gamma_{c}\Gamma_{\widehat{p}}\unlhd\cdots\unlhd\gamma_{2}\Gamma_{\widehat{p}}\unlhd\gamma_{1}\Gamma_{\widehat{p}}=\Gamma_{\widehat{p}},$

for which each quotient
$\gamma_{k+1}\Gamma_{\widehat{p}}/\gamma_{k}\Gamma_{\widehat{p}}\cong\mathbb{Z}_{p}^{n_{k}}$
is torsion-free. So $\Gamma_{\widehat{p}}$ is torsion-free, as we claimed
before. ∎

It is not hard to see that the groups satisfying the assumptions of 2.16 is
closed under taking free products. However, the following proposition ensures
many one-ended examples as well.

######  Proposition 2.17.

Let $G$ be isomorphic to a RAAG or to a one-relator group $F/\langle\\!\langle
r\rangle\\!\rangle$ such that if $k$ is the biggest non-negative integer such
that $r\in\gamma_{k}F$, then $r\cdot\gamma_{k+1}F$ is not a proper power in
$\gamma_{k}F/\gamma_{k+1}F$. In these cases, $G$ satisfies the assumptions of
2.16. In particular, this is true if $G$ is an orientable surface group.

###### Proof.

When $G$ is a RAAG, this is [Wade15, Theorem 6.4]. When $G=F/\langle\langle
r\rangle\rangle$, this proposition is a direct consequence of [Lab70, Section
1]. The reason is the following. The main theorem of [Lab70] states that,
under our assumptions on the relator $r$, the graded algebra
$\bigoplus_{n}\gamma_{n}G/\gamma_{n+1}G$ is a free $\mathbb{Z}$-Lie algebra.
In particular, the subsequent quotients $\gamma_{n}G/\gamma_{n+1}G$ are free
abelian (the author also computes their rank). Since hyperbolic surface groups
are defined by a single relator with these properties, the last conclusion
follows. ∎

We observe that the class of one-relator groups of 2.17 includes many limit
groups. For example, those arising as doubles
$F*_{u=\overline{u}}\overline{F}$ of a free group $F$ along a relator ${u}\in
F$ satisfying the same assumptions as $r$ in 2.17.

##  3\. The first $L^{2}$-Betti number as a pro-$\mathcal{C}$ invariant

We will describe some scenarios in which the first $L^{2}$-Betti number of a
group is a profinite invariant, to discuss some consequences later on. An
standard reference about $L^{2}$-invariants is the book of Lück [Luc02] and,
in alignment with the approach taken in this section, the survey paper of
Jaikin-Zapirain [And19b].

For torsion-free groups $G$ satisfying the Strong Atiyah conjecture, an
observation of Linnel [Lin93] implies that one can define $b_{1}^{(2)}(G)$ as
follows. Suppose that the group ring $\mathbb{Q}G$ of $G$ with rational
coefficients has a universal division ring of fractions
$\mathbb{Q}G\xhookrightarrow{}\mathcal{D}_{\mathbb{Q}G}$. This provides a
notion of dimension for all $\mathbb{Q}G$-modules as follows: given a
$\mathbb{Q}G$-module $M$, we extend scalars
$\mathcal{D}_{\mathbb{Q}G}\mathbin{\mathop{\otimes}\limits_{\mathbb{Q}G}}M$
and take its linear-algebraic dimension as a vector space over the division
ring $\mathcal{D}_{\mathbb{Q}G}$. We can define the first $L^{2}$-Betti number
for these groups $G$ as

$b_{1}^{(2)}(G)=\dim_{\mathcal{D}_{\mathbb{Q}G}}H_{1}(G;\mathcal{D}_{\mathbb{Q}G}).$

Since locally indicable groups satisfy the Strong Atiyah conjecture by a
result of Jaikin-Zapirain and López-Álvarez [And192], the previous is a
definition of $b_{1}^{(2)}$ for these groups.

A general philosophy that explains why we work with first $L^{2}$-Betti
numbers instead of usual Betti numbers is that the former behave in a better
way while containing similar information. For example, they are multiplicative
in the following sense.

######  Proposition 3.1 (Theorem 1.35(9), [Luc02]).

Let $G$ be a finitely generated group and let $H$ be a finite-index subgroup.
Then $b_{1}^{(2)}(H)=b_{1}^{(2)}(G)\,|G\mathrel{\mathop{\ordinarycolon}}H|.$

###  3.1. Computations using Lück approximation

When it comes to relate first $L^{2}$-Betti numbers and profinite completions,
it is particularly helpful the characterisation of these numbers in terms of a
limit of usual Betti numbers over a filtration of $G$ by normal finite-index
subgroups. A pioneering result in this direction was given for finitely
presented groups $G$ by Lück [Luc94].

######  Theorem 3.2 (Lück’s approximation theorem).

Let $G$ be a group of type $\mathtt{FP}_{2}$ 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

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

For residually-finite groups that are not $\mathtt{FP}_{2}$, we still have a
one-sided estimate of the same type.

######  Theorem 3.3 ([Luc11]).

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

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

We move on to the observation that groups $G$ satisfying the following
inequality also have a Lück approximation theorem:

(1) $b_{1}(G_{0})\geqslant b_{1}^{(2)}(G_{0})+1\,\mbox{for every finite-index
$G_{0}\leqslant G$.}$

Examples of groups $G$ with this property are limits of left-orderable
amenable groups in the space of marked group presentations (by a result of
Osin [Osi11, Theorem 1.4]). Interestingly, the property of (1) can be proven
in greater generality.

######  Proposition 3.4 (Jaikin-Zapirain [And19]).

Let $G$ be a finitely generated residually-(locally indicable and amenable)
group. Then $b_{1}^{(2)}(G)\leqslant b_{1}(G)-1$.

###### Proof.

Since $H$ is residually-(locally indicable amenable), then there exists a
universal division ring of fractions
$\mathbb{Q}G\xhookrightarrow{}\mathcal{D}_{\mathbb{Q}G}$ by [And19, Theorem
1.1]. From previous comments, then we can compute the first $L^{2}$-Betti
number of $G$ as
$b_{1}^{(2)}(G)=\dim_{\mathcal{D}_{\mathbb{Q}G}}H_{1}(G;\mathcal{D}_{\mathbb{Q}G})=\dim_{\mathcal{D}_{\mathbb{Q}G}}\left(\mathcal{D}_{\mathbb{Q}G}\mathbin{\mathop{\otimes}\limits_{\mathbb{Z}G}}I_{G}\right)-1$.
On the other hand, the universality of the epic $\mathbb{Q}G$-ring
$\mathcal{D}_{\mathbb{Q}G}$ implies that its associated rank function is
maximal (see [And19, Corollary 1.4]). Hence, we have the inequality
$\dim_{\mathcal{D}_{\mathbb{Q}G}}\left(\mathcal{D}_{\mathbb{Q}G}\mathbin{\mathop{\otimes}\limits_{\mathbb{Z}G}}I_{G}\right)\leqslant\dim_{\mathbb{Q}}\left(\mathbb{Q}\mathbin{\mathop{\otimes}\limits_{\mathbb{Z}G}}I_{G}\right)=b_{1}(G)$.
The conclusion follows. ∎

This proposition is used in [And212, Theorem 3.3] to show that groups in the
soluble genus of a free group are parafree (similarly, these ideas are
replicated in [Fru22]). For our purposes, the interest behind 3.4 is extending
the Lück approximation 3.2 of $b_{1}^{(2)}$ to other classes of groups.

######  Corollary 3.5.

Let $G$ be a finitely generated residually-(amenable and locally indicable)
group. 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

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

###### Proof.

We know from 3.3 that

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

On the other hand, since $b_{1}(N_{m})\geqslant
b_{1}^{(2)}(N_{m})+1=b_{1}^{(2)}(G)|G\mathrel{\mathop{\ordinarycolon}}N_{m}|+1$
for all $m$ by 3.4, we have

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

Hence the given limit exists and equals $b_{1}^{(2)}(G)$. ∎

######  Definition 3.6.

We say that a finitely generated residually finite group $G$ is
$\mathrm{L\ddot{u}cky}$ if for every descending sequence
$G=N_{1}>N_{2}>\dots>N_{m}>\dots$ of finite-index normal subgroups $N_{m}$
with $\bigcap_{m}N_{m}=1$, it satisfies that

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

We review this subsection by saying that both $\mathtt{FP}_{2}$ groups and
finitely generated residually-(amenable and locally indicable) are
$\mathrm{L\ddot{u}cky}$ by 3.2 and 3.5.

###  3.2. An invariant of the genus

We are going to denote by $\mathcal{C}$ a formation of finite groups. In
particular, we will denote by $\mathcal{C}_{p}$ the formation of finite
$p$-groups.

######  Proposition 3.7.

Let $\mathcal{C}$ be a formation of finite groups containing $\mathcal{C}_{p}$
for some prime $p$. Suppose that $G$ and $H$ have the same pro-$\mathcal{C}$
completion. Then $b_{1}(G)=b_{1}(H)$.

###### Proof.

This is a direct consequence of the fact that $b_{1}(G)$ is exactly the
greatest positive integer $m$ such that $G$ maps surjectively to
$(\mathbb{Z}/p^{n}\mathbb{Z})^{m}$ for all $n$, and this is a property
detected by the pro-$\mathcal{C}$ completion because $\mathcal{C}$ contains
$\mathcal{C}_{p}$. ∎

The following is a re-statement of [Bri14, Proposition 7.5] and we reproduce
the same proof.

######  Corollary 3.8.

Let $\mathcal{C}$ be a formation of finite groups that includes
$\mathcal{C}_{p}$ for some $p$. Let $H$ be a $\mathrm{L\ddot{u}cky}$ and
residually-$\mathcal{C}$ group. If $G$ admits a dense embedding into
$H_{\widehat{\mathcal{C}}}$ then $b_{1}^{(2)}(G)\geqslant b_{1}^{(2)}(H)$.

###### Proof.

Let $H_{m}$ be the intersection of all the normal finite-index subgroups
$H_{0}\leqslant H$ such that $H/H_{0}\in\mathcal{C}$ and
$|H\\!\mathrel{\mathop{\ordinarycolon}}\\!H_{0}|\leqslant m$. It is clear that
$H_{m}$ is normal with $H/H_{m}\in\mathcal{C}$ and that the intersection of
the sequence of open subgroups $\overline{H_{m}}\subseteq H_{\mathcal{C}}$ is
trivial. Consider $G_{m}=G\cap\overline{H_{m}}$. Then $\cap_{m}G_{m}=1$ are
normal subgroups of finite-index. Then, keeping in mind that $H$ is
$\mathrm{L\ddot{u}cky}$, we can apply 3.3 to derive that

$b_{1}^{(2)}(G)\geqslant\limsup_{m\rightarrow\infty}\frac{b_{1}(G_{m})}{|G\mathrel{\mathop{\ordinarycolon}}G_{m}|}=\lim_{m\rightarrow\infty}\frac{b_{1}(H_{m})}{|H\mathrel{\mathop{\ordinarycolon}}H_{m}|}=b_{1}^{(2)}(H).\qed$

######  Corollary 3.9.

Let $G$ and $H$ be two $\mathrm{L\ddot{u}cky}$ groups. Let $\mathcal{C}$ be a
formation of finite groups that contains $\mathcal{C}_{p}$ for some prime $p$.
If $G$ and $H$ belong to the same $\mathcal{C}$-genus, then
$b_{1}^{(2)}(G)=b_{1}^{(2)}(H)$.

###### Proof.

Immediate from 3.8. ∎

##  4\. Cohomological dimension and Poincaré duality

Here we recall the finiteness properties of groups that we shall use. We
denote by $R$ a unital and associative ring (not necessarily commutative).

######  Definition 4.1.

We say that an $R$-module $M$ is of type $\mathtt{FP}_{n}(R)$ if there exists
a projective resolution

$\longrightarrow P_{n+1}\longrightarrow
P_{n}\longrightarrow\cdots\longrightarrow P_{1}\longrightarrow
P_{0}\longrightarrow M\longrightarrow 0$

with $P_{i}$ finitely generated for $0\leqslant i\leqslant n$. Moreover, we
say that an $R$-module $M$ is of type $\mathtt{FP}_{\infty}(R)$ if it is of
type $\mathtt{FP}_{n}(R)$ for all $n$. Analogously, we say that a group $G$ is
of type $\mathtt{FP}_{n}(R)$ (resp. of type $\mathtt{FP}_{\infty}(R)$) if the
trivial $RG$-module $R$ is of type $\mathtt{FP}_{n}(R)$ (resp. of type
$\mathtt{FP}_{n}(R)$ for all $n$).

###  4.1. Profinite groups

Now we quickly review the cohomology theory of profinite groups, including
their adequate notion of group ring that is used to define their (co-)homology
groups and their finiteness properties in the sense of 4.1.

Given a profinite group ${\bf G}$ and a profinite ring ${\bf R}$ given as an
inverse limit ${\bf R}=\varprojlim_{i\geqslant 1}R_{i}$ of finite rings
$R_{i}$, the complete group algebra of ${\bf G}$ over ${\bf R}$ is defined as
the inverse limit of usual group rings $R_{i}[{\bf G}/{\bf U}]$ in the
following way.

${\bf R}\llbracket{\bf G}\rrbracket=\varprojlim_{i\geqslant 1,\,{\bf
U}\unlhd_{o}{\bf G}}R_{i}[{\bf G}/{\bf U}],$

where $i\geqslant 1$ and ${\bf U}$ ranges over open normal subgroup of ${\bf
G}$. The only three profinite rings ${\bf R}$ that we consider are
$\mathbb{F}_{p}$,
$\mathbb{Z}_{p}\cong\varprojlim_{n}\mathbb{Z}/p^{n}\mathbb{Z}$ and
$\widehat{\mathbb{Z}}\cong\varprojlim_{n}\mathbb{Z}/n\mathbb{Z}$.

A group is said to be cohomologically good (or good in the sense of Serre
[Ser97, Section I. 2. 6]) if for every finite $\widehat{G}$-module $A$, the
natural homomorphism

$H^{k}(\widehat{G};A)\longrightarrow H^{k}(G;A),$

induced by $G\longrightarrow\widehat{G}$, is an isomorphism. A group $G$ is
said to be subgroup separable or LERF if every finitely generated subgroup
$H\leqslant G$ is closed in the profinite topology. This implies that the
topology induced in $H$ from the profinite topology of $G$ is the full
profinite topology of $H$. Equivalently, by Lemma 2.10, the natural map
$\widehat{H}\longrightarrow\overline{H}\subseteq\widehat{G}$ is an isomorphism
of profinite groups.

We say that a profinite group ${\bf G}$ is of type $\mathtt{FP}_{n}$ (resp. of
type $p$-$\mathtt{FP}_{n}$) if the $\llbracket\widehat{\mathbb{Z}}{\bf
G}\rrbracket$-module $\widehat{\mathbb{Z}}$ (resp. the
$\llbracket\mathbb{Z}_{p}{\bf G}\rrbracket$-module $\mathbb{Z}_{p}$) is of
type $\mathtt{FP}_{n}$.

######  Proposition 4.2 ([And18], Proposition 3.1).

If $\Gamma$ is an abstract $\mathtt{FP}_{\infty}$ and good group then its
profinite completion $\widehat{\Gamma}$ is of type $p$-$\mathtt{FP}_{\infty}$
for every prime $p$.

######  Definition 4.3.

We say that a profinite group ${\bf G}$ has $p$-cohomological dimension
$n=\mathrm{cd}_{p}({\bf G})$ if $n$ is the largest non-negative integer $m$
such that $H^{m+1}({\bf G},A)=0$ for all simple discrete
$\llbracket\widehat{\mathbb{Z}}{\bf G}\rrbracket$-modules that are annihilated
by $p$.

We recall a few alternative characterisations of this dimension from [Rib00,
Proposition 7.1.4].

######  Proposition 4.4.

Let ${\bf G}$ be a profinite group. Then the following are equivalent

  1. (1)

$\mathrm{cd}_{p}({\bf G})\leqslant n$.

  2. (2)

$H^{k}({\bf G},A)=0$ for all $k\geqslant n+1$ and all discrete
$\llbracket\widehat{\mathbb{Z}}{\bf G}\rrbracket$-modules that are annihilated
by $p$.

  3. (3)

$H^{k}({\bf G},A)$ does not have elements of order $p$ for all finite
$G$-modules.

  4. (4)

There exists a resolution of the trivial $\llbracket\mathbb{F}_{p}{\bf
G}\rrbracket$-module $\mathbb{F}_{p}$

$0\longrightarrow P_{n}\longrightarrow
P_{n-1}\longrightarrow\cdots\longrightarrow
P_{0}\longrightarrow\mathbb{F}_{p}\longrightarrow 0$

by projective profinite $\llbracket\mathbb{F}_{p}{\bf G}\rrbracket$-modules
$P_{i}$.

The $p$-cohomological dimension of profinite free (resp. profinite surface
groups) is one (resp. two). In fact, notice that good groups $G$ enjoy the
property that

$\mathrm{cd}_{p}(\widehat{G})=\mathrm{cd}_{p}(G)\,\,\,\mbox{for all primes
$p$.}$

In particular, a free profinite group has $\mathrm{cd}_{p}=1$ and profinite
surfaces have $\mathrm{cd}_{p}=2$. Beyond the ease of computations, we should
remark that focusing on $p$-primary modules in the 4.3 of the “cohomological
dimension function” $\mathrm{cd}_{p}$ also has the purpose of recovering
intuitive features that are true for abstract groups such that assigning the
value 1 to free objects. If we looked, more generally, at discrete modules and
not only finite modules, one would not avoid the pathology of
$H^{2}(\widehat{\mathbb{Z}};\mathbb{Z})$ being non-zero (as this is isomorphic
to
$H^{1}(\widehat{\mathbb{Z}},\mathbb{Q}/\mathbb{Z})\cong\mathbb{Q}/\mathbb{Z}$,
see [Ser97, Chapter 1, Section 3.2]). Furthermore, the dimension function
$\mathrm{cd}_{p}$ enjoys nice “geometric” properties.

######  Proposition 4.5.

Let ${\bf G}$ be a profinite group, ${\bf H}$ a closed subgroup of ${\bf G}$
and ${\bf N}$ a normal closed subgroup of ${\bf G}$. Then

  1. (1)

$\mathrm{cd}_{p}({\bf H})\leqslant\mathrm{cd}_{p}({\bf G})$, and

  2. (2)

$\mathrm{cd}_{p}({\bf G})\leqslant\mathrm{cd}_{p}({\bf
N})+\mathrm{cd}_{p}({\bf G}/{\bf N})$.

The first point of the proposition is proven in [Rib00, Theorem 7.3.1] and
follows from Shapiro’s lemma. The second one is more technical, as it requires
the use of spectral sequences (see [Rib00, Proposition 7.4.2]). Consequently,
$\mathrm{cd}_{p}(\widehat{F}\rtimes\widehat{\mathbb{Z}})\leqslant 2$ and
$\mathrm{cd}_{p}(\widehat{\pi_{1}(\Sigma)}\rtimes\widehat{\mathbb{Z}})\leqslant
3$. In fact, the previous are equalities in these cases, because, under
certain finiteness assumptions, $\mathrm{cd}_{p}$ can be proven to be
(virtually) additive with respect to short exact sequences (as shown in
[Wei04, Theorem 1.1]).

It is instructive to compare the properties that we are treating about
$\mathrm{cd}_{p}$ with the ones corresponding to the projective dimension in
the abstract setting (as in, for example, [Bro82, Chapter VIII, Section 2]).

A classical characteristic feature of groups of finite cohomological
dimension, both in the abstract and profinite setting, is that these are
torsion-free.

######  Lemma 4.6.

Let ${\bf G}$ be a profinite group with $\mathrm{cd}_{p}({\bf G})<\infty$ for
all primes $p$. Then ${\bf G}$ is torsion-free.

###### Proof.

By 4.5, we know that, for all closed subgroups ${\bf H}\leqslant{\bf G}$ and
all primes $p$, we have $\mathrm{cd}_{p}({\bf H})\leqslant\mathrm{cd}_{p}({\bf
G})$. In particular, since $\mathrm{cd}_{p}(\mathbb{Z}/p)=\infty$, there is no
embedding $\mathbb{Z}/p\xhookrightarrow{}{\bf G}$. Thus ${\bf G}$ is torsion-
free. ∎

The previous lemma allows to prove that the property of being torsion-free is
a property of the finite genus of good groups with finite cohomological
dimension. However, Lubotzky shows that being torsion-free is not a profinite
property ([Lub14, Proposition 1.5]).

Another crucial principle about the category of pro-$p$ groups, as opposed to
the category of profinite groups, is that we have an analogue of Stallings’
and Swan’s theorem characterising free groups by their cohomological
dimension.

######  Theorem 4.7 (Theorem 7.7.4, [Rib00]).

Let ${\bf G}$ be a pro-$p$ group of $p$-cohomological dimension at most one.
Then ${\bf G}$ is free pro-$p$.

###  4.2. Poincaré Duality and the recognition of surfaces

A hard theorem of Eckman and Müller [Eck80] gives a cohomological
characterisation of surface groups among abstract groups as being exactly the
Poincaré duality groups of dimension two (an alternative proof can be found in
[Kie21]). There is a similar principle for pro-$p$ groups but it is poorly
understood what happens for general profinite groups. In a similar fashion, it
is not known whether a finitely generated residually finite group whose
profinite completion is $\mathtt{PD}_{2}$ at all primes $p$ is itself
$\mathtt{PD}_{2}$. However, we combine a geometric result of Kerckhoff (4.8)
with an observation of Serre on profinite groups (4.11) to restate in 4.14 a
criterion of Wilton [Wil21] for detecting surface groups among abstract groups
by looking at their profinite completion. Before this, we introduce separately
these two topological and profinite ingredients.

A classical question in the theory of surfaces is the Nielsen realisation
problem, which, in its strongest formulation, asks whether any finite subgroup
of the mapping class group of a surface $\Sigma$ can be realised as a group of
isometries of $(\Sigma,g)$ for some choice of a Riemannian metric $g$ with
constant sectional curvature. Fenchel and Nielsen solved the case of cyclic
groups and the general case was solved by Kerckhoff [Ker83]. This result has
the following particular consequence, which answers a question of A. Karrass
and D. Solitar raised at the American Mathematical Society meeting in 1970.

######  Corollary 4.8 (Kerckhoff’s Nielsen realisation).

A torsion-free group that is virtually a surface group is itself a surface
group.

The previous result was used by Gardam, Kielak and Logan [Kie22] to partially
confirm some conjectural descriptions of surface groups among one-relator
groups.

In order to state the aforementioned remark of Serre in 4.11, we review the
notion of Poincaré duality in profinite groups.

######  Definition 4.9.

Let ${\bf G}$ be a profinite group of type $p$-$\mathtt{FP}_{\infty}$. We say
that ${\bf G}$ is a Poincaré duality group at $p$ of dimension $n$ (written as
$\mathtt{PD}^{n}$ at $p$) if $\mathrm{cd}_{p}({\bf G})=n$ and

$\displaystyle H^{i}({\bf G},\llbracket\mathbb{Z}_{p}{\bf
G}\rrbracket)=0\,\,\,\,\mbox{for $i\neq n$},$ $\displaystyle H^{i}({\bf
G},\llbracket\mathbb{Z}_{p}{\bf
G}\rrbracket)\cong\mathbb{Z}_{p}\,\,\,\,\mbox{(as abelian groups).}$

######  Remark 4.10.

A group ${\bf G}$ with $\mathrm{cd}_{p}({\bf G})<\infty$ is $\mathtt{PD}_{n}$
at $p$ if and only if some (equivalently all) open subgroup ${\bf
U}\leqslant_{o}{\bf G}$ is $\mathtt{PD}_{n}$ at $p$ (see [Sym00, Remark
4.2.9]).

As opposed to the abstract case, there is no classification of
$\mathtt{PD}_{2}$ profinite groups. However, these are classified for pro-$p$
groups. A pro-$p$ group ${\bf G}$ is $\mathtt{PD}_{1}$ if and only if ${\bf
G}\cong\mathbb{Z}_{p}$ (see [Sym00, Example 4.4.4]). Poincaré duality pro-$p$
groups of dimension 2 are known to coincide with the class of Demushkin groups
(see [Ser97, Section I.4.5, Example 2]), which is a class of one-relator
pro-$p$ groups that was classified by Demushkin, Serre and Labute [Dem61,
Dem63, Ser95, Lab67].

The following is a profinite variant of Strebel’s theorem [Str77] on infinite-
index subgroups of $\mathtt{PD}_{n}$ abstract groups, and it appears in
[Ser97, Page 44, exercise 5(b)] and [Neu08, Chapter 3, Section 7, Exercise 3].

######  Lemma 4.11 (Serre).

Let ${\bf G}$ be a profinite $\mathtt{PD}_{n}$ group at a prime $p$ and let
${\bf H}$ be a closed subgroup such that $p^{\infty}$ divides the index $|{\bf
G}\mathrel{\mathop{\ordinarycolon}}{\bf H}|$. Then $\mathrm{cd}_{p}({\bf
H})<n$.

The previous lemma was of fundamental importance in the work of Zalesskii and
Wilton [Wil19] on the profinite detection of prime and JSJ decompositions of
three-manifolds.

######  Remark 4.12.

When ${\bf G}$ is a pro-$p$ group, 4.11 only requires ${\bf H}$ to have
infinite index. Nevertheless, for general profinite groups ${\bf G}$, the
conclusion of the theorem would not be true by only requiring infinite index.
In fact, from 2.6, we know that $\widehat{\mathbb{Z}}$ contains closed
subgroups isomorphic to $\widehat{\mathbb{Z}}$ of index $\prod_{p}p$.

The most natural source of examples of profinite Poincaré duality groups
consists of profinite completions of abstract Poincaré duality groups, as the
following result from [Koc08, Theorem 4.1] ensures.

######  Theorem 4.13.

Let $\Gamma$ be an abstract cohomologically good $\mathtt{PD}_{n}$ group. Then
$\widehat{\Gamma}$ is a profinite $\mathtt{PD}_{n}$ group at every prime and
$\Gamma_{\widehat{p}}$ is $\mathtt{PD}_{n}$ at $p$.

Finally, we are ready to give a more refined criterion to detect surfaces,
used by Wilton [Wil21] to establish 1.2 when $G$ is a limit group.

######  Proposition 4.14.

Let $p$ be a prime and let $G$ be a torsion-free residually finite group with
separable cyclic subgroups such that its profinite completion $\widehat{G}$ is
$\mathtt{PD}_{2}$ at $p$. Suppose that $G$ contains a surface subgroup as a
virtual retract. Then $G$ is a surface group.

###### Proof.

By assumption, there exists a finite-index subgroup $L$ of $G$ containing a
surface group $\pi_{1}\Sigma$ as a retract. Let $K$ be the kernel of the
retraction $L\longrightarrow\pi_{1}\Sigma$. Our aim is to prove that $K=1$.
This way, we would have that $L\cong\pi_{1}\Sigma$ and that the torsion-free
group $G$ would be virtually a surface group. So the conclusion would follow
from 4.8.

Hence, it remains to show that $K=1$. We proceed by contradiction. Suppose
that $K\neq 1$. The retraction
$\pi_{1}\Sigma\xhookrightarrow{}L\longrightarrow\pi_{1}\Sigma$ induces a
retraction
$\widehat{\pi_{1}\Sigma}\xhookrightarrow{}\widehat{L}\longrightarrow\widehat{\pi_{1}\Sigma}$
at the level of the profinite completions. Take $1\neq k\in K$. Since $G$ is
torsion-free and cyclic subgroups are separable, then $\langle
k\rangle\cong\mathbb{Z}$ and the supernatural number $p^{\infty}$ divides the
order of $\overline{\langle k\rangle}\cong\widehat{\mathbb{Z}}$. Since $k$ is
contained in the kernel ${\bf K}$ of the retraction
$\widehat{L}\longrightarrow\widehat{\pi_{1}\Sigma}$, then $|{\bf
K}|=|\overline{\langle k\rangle}||{\bf
K}\mathrel{\mathop{\ordinarycolon}}\overline{\langle k\rangle}|$ is also
divisible by $p^{\infty}$. We now claim that $p^{\infty}$ also divides the
index $|\widehat{L}\mathrel{\mathop{\ordinarycolon}}\widehat{\pi_{1}\Sigma}|$.
This way, it would also divide the index
$|\widehat{G}\mathrel{\mathop{\ordinarycolon}}\widehat{\pi_{1}\Sigma}|$ and,
by 4.11, the $p$-cohomological dimension of $\widehat{\pi_{1}\Sigma}$ should
be at most $\mathrm{cd}_{p}(\widehat{G})-1=1$, which is a contradiction.

Thus, it simply remains to prove that $p^{\infty}$ divides
$|\widehat{L}\\!\mathrel{\mathop{\ordinarycolon}}\\!\widehat{\pi_{1}\Sigma}|$.
From comments above, we know that, for every $n\geqslant 1$, there exists a
characteristic open subgroup ${\bf U}\unlhd_{o}{\bf K}$ such that $p^{n}$
divides $|{\bf K}\\!\mathrel{\mathop{\ordinarycolon}}\\!{\bf U}|$. We observe
that ${\bf H}={\bf U}\cdot\widehat{\pi_{1}\Sigma}$ is an open subgroup of
$\widehat{L}$. After taking a normal open subgroup ${\bf
H}_{0}\unlhd_{o}\widehat{L}$ contained in ${\bf H}$ (for example, its core),
we are going to check that $p^{n}$ divides
$|\widehat{L}\mathrel{\mathop{\ordinarycolon}}{\bf
H}_{0}\cdot\widehat{\pi_{1}\Sigma}|$. For this, is suffices to notice that the
previous index is divisible by

$|\widehat{L}\mathrel{\mathop{\ordinarycolon}}{\bf
H}\cdot\widehat{\pi_{1}\Sigma}|=|{\bf
K}\cdot\widehat{\pi_{1}\Sigma}\mathrel{\mathop{\ordinarycolon}}{\bf
U}\cdot\widehat{\pi_{1}\Sigma}|=|{\bf K}\mathrel{\mathop{\ordinarycolon}}{\bf
U}|,$

since ${\bf U}\unlhd_{o}{\bf K}$ and ${\bf
K}\cap\widehat{\pi_{1}\Sigma}=\\{1\\}$. We chose ${\bf U}$ so $p^{n}$ divides
$|{\bf K}\mathrel{\mathop{\ordinarycolon}}{\bf U}|$, so this proves that
$p^{\infty}$ divides
$|\widehat{L}\mathrel{\mathop{\ordinarycolon}}\widehat{\pi_{1}\Sigma}|$, as we
wanted. ∎

For the purpose of proving A using 4.14, we remark that the profinite
completion of a surface $\widehat{\pi_{1}\Sigma}$ is $\mathtt{PD}_{2}$ at $p$
for all primes $p$.

##  5\. Residually free and limit groups

The purpose of this section is reviewing the separability properties of
residually free groups and criteria to detect hyperbolic and limit groups
among them.

It is straightforward that these groups satisfy a Tits alternative.

######  Proposition 5.1 (Section 4, [Bau62]).

Let $G$ be a residually free group and let $H\leqslant G$ be a two-generated
subgroup. Then $H$ is isomorphic to one of the groups
$\\{1,\mathbb{Z},\mathbb{Z}^{2},F_{2}\\}$.

In particular, a non-amenable residually free group must contain a non-abelian
free group. The following result allows to relate the structure of finitely
generated residually free groups to limit groups.

######  Proposition 5.2 ([Bau99], Corollary 19; and [Sel01], Claim 7.5).

Let $G$ be a finitely generated residually free group. Then $G$ is a subdirect
product of finitely many limit groups.

A classical question attributed to Gromov asks whether one-ended hyperbolic
groups must contain surface subgroups, and Wilton answers this question
positively for many cyclic splittings in the following way.

######  Theorem 5.3 ([Wil18], Theorem 6.1).

Let $G$ be the fundamental group of a graph of virtually free groups and
virtually cyclic edge subgroups. Suppose that $G$ is hyperbolic and one-ended.
Then $G$ contains a surface subgroup.

As proven in [Wil18, Corollary C], the theorem above can be applied, in
combination with Sela’s hierarchy on limit groups [Sel01], to find surface
subgroups in non-free limit groups.

###  5.1. Criterion for hyperbolicity of residually free groups

######  Proposition 5.4.

A limit group $G$ is hyperbolic if and only if it contains no
$\mathbb{Z}^{2}$.

###### Proof.

One direction is clear since we know that hyperbolic groups cannot contain
$\mathbb{Z}^{2}$. For the converse, if $G$ contains no $\mathbb{Z}^{2}$ then
it contains no non-cyclic abelian subgroup. It is proven in [Sel01, Corollary
4.4. (iii)] that this is enough to ensure that $G$ is hyperbolic. ∎

We remark that currently we know much more than this. For example, Alibegovic
(in [Ali05, Corollary 3.4]) and Dahmani (in [Dah03, Theorem 0.3])
independently proved that limit groups are hyperbolic relative to their
maximal abelian subgroups. Furthermore, in the case a limit group contains no
$\mathbb{Z}^{2}$, we also conclude using Haglund–Wise’s hierarchies, as we
will see in 6.2, that it is hyperbolic virtually compact special.

######  Proposition 5.5 ([Bri14a], Lemma 4.15).

Let $G$ be a finitely generated residually free group. Then $G$ is a
hyperbolic limit group if and only if it contains no $\mathbb{Z}^{2}$.

###### Proof.

One direction is clear. For the other, it suffices to prove, by 5.4, that $G$
is limit if it contains no $\mathbb{Z}^{2}$. By 5.2, $G$ must be a subgroup of
a direct product $G_{1}\times\cdots\times G_{n}$ of limit groups
$G_{1},\dots,G_{n}$. We can assume that each $G_{i}$ intersects $G$ non-
trivially. Otherwise, if some $G_{i}$ satisfies that $G\cap G_{i}=\\{1\\}$,
then the composition
$G\xhookrightarrow{}\oplus_{j}G_{j}\longrightarrow\oplus_{j\neq i}G_{j}$ is
still injective. If $G$ intersected at least two of the factors, then $G$
would contain a $\mathbb{Z}^{2}$ (since limit groups are torsion-free). So
this implies that $G\xhookrightarrow{}G_{1}$ must be a finitely generated
subgroup of a limit group, and so it is also limit. ∎

###  5.2. Separability properties of residually free groups

Wilton [Wil08] proved that finitely generated subgroups of limit groups are
virtual retracts (implying that, in particular, these are LERF). Since limit
groups are of type $\mathtt{FP}$, the following separability principle is a
generalisation.

######  Theorem 5.6 ([Bri08], Theorems A and B).

Let $G$ be a finitely generated residually free group and let $H\leqslant G$
be a finitely presented subgroup. Then $H$ is closed in the profinite topology
of $G$. Moreover, if $H$ is of type $\mathtt{FP}$, then $H$ is a virtual
retract of $G$.

##  6\. Hyperbolic and special groups

We start by stating a criterion that appears in the introduction of [Bes92]
for the hyperbolicity of extensions of hyperbolic groups over cyclic groups.

######  Theorem 6.1 (A combination theorem for negatively curved groups).

Let $(\mathcal{G},Y)$ be a finite graph of groups with hyperbolic vertex
groups and virtually cyclic edge subgroups. Then its fundamental group $\pi$
is hyperbolic if and only if it contains no Baumslag–Solitar subgroups.

######  Proposition 6.2.

Let $G$ be a group belonging to ${\bf\mathcal{H}_{\bf ab}}$ that

  1. (1)

is torsion-free,

  2. (2)

contains no Baumslag–Solitar subgroups,

  3. (3)

and contains no non-cyclic abelian subgroups.

Then $G$ is a hyperbolic virtually compact special group.

Before moving on to the proof, it is important to keep in mind that the
assumptions 1–3 of 6.2 pass to subgroups.

###### Proof.

On the first place, we show that $G$ is hyperbolic by an inductive argument on
the level of the hierarchy of ${\bf\mathcal{H}_{\bf ab}}$. In the base of the
induction, that is, at the level 0 of the hierarchy, $G$ is a finitely
generated residually free group. So $G$ is hyperbolic by 5.5. For the
induction step, we distinguish three cases:

  * •

If $G=A*_{C}B$ or $G=A_{C,\theta}$ for some injection
$\theta\mathrel{\mathop{\ordinarycolon}}C\longrightarrow A$), where $A$ and
$B$ are hyperbolic in ${\bf\mathcal{H}_{\bf ab}}$, and $C$ is virtually
abelian, then $C$ is cyclic and hence $G$ is hyperbolic by 6.1.

  * •

If $H\leqslant G$ is a finite-index subgroup in ${\bf\mathcal{H}_{\bf ab}}$
for which the statement is already true, then $G$ is clearly hyperbolic.

  * •

If $G=A\times B$, where the statement is true for each factor and each factor
is in ${\bf\mathcal{H}_{\bf ab}}$, we notice that at least one factor is
trivial (since $G$ is torsion-free and has no $\mathbb{Z}^{2}$), so it holds
for $G$, too.

The induction is complete and this proves that such $G$ are hyperbolic.
Secondly, we prove, again inductively, that $G$ is virtually compact special.
Let us start with the base case. When $G$ is finitely generated residually
free, it follows from 5.5 that $G$ is a hyperbolic limit group. By [Sel01],
hyperbolic limit groups have a finite cyclic hierarchy terminating in free
groups, so $G$ is virtually compact special by [Wis21, Theorem 11.2].
Moreover, from this, we notice that any hyperbolic $G\in{\bf\mathcal{H}_{\bf
ab}}$ that is torsion-free and has no non-cyclic abelian subgroup obviously
admits a finite cyclic hierarchy terminating in free groups. So, again by
[Wis21, Theorem 11.2], this proves that $G$ is virtually compact special. ∎

######  Definition 6.3.

Let $A$ be a subgroup of $B$. We say that $A$ is a virtual retract of $B$ if
there exists a finite-index subgroup $B_{1}$ of $B$ containing $A$ and a group
homomorphism $r\mathrel{\mathop{\ordinarycolon}}B_{1}\longrightarrow A$ such
that $r$ restricts to the identity function on $A$.

From the proof of Theorem 7.3 of [Hag08], one knows that quasi-convex
subgroups of hyperbolic compact special groups are separable in the following
strong way.

######  Theorem 6.4.

Let $G$ be a hyperbolic virtually compact special group and let $H\leqslant G$
be a quasi-convex subgroup. Then there exists a finite-index subgroup
$H^{\prime}$ of $H$ such that $H^{\prime}$ is a virtual retract of $G$.

######  Proposition 6.5.

Let $G$ be as in 6.2. Suppose that $G$ is not free. Then $G$ contains a
surface subgroup as a virtual retract.

###### Proof.

By 6.4, it is enough to ensure that $G$ contains a quasi-convex surface
subgroup. As an additional note, we know that for finitely generated subgroups
of a hyperbolic group, being quasi-convex and quasi-isometrically embedded is
equivalent. Hence it suffices to ensure a quasi-isometrically embedded surface
subgroup in $G$.

As in the proof of 6.2, we will proceed by induction on the level of the
hierarchy. Let us start with the level 0, that is, when $G$ is a finitely
generated residually free group. If $G$ contains $\mathbb{Z}^{2}$, then this
$\mathbb{Z}^{2}$ will be quasi-convex by 5.6. On the other hand, if $G$
contains no $\mathbb{Z}^{2}$, then $G$ is a hyperbolic limit group and $G$
contains a quasi-convex surface subgroup by 5.3.

For the induction step, we distinguish two cases:

  * •

Suppose that $G=A*_{C}B$ (resp. $G=A_{C,\theta}$ for some injection
$\theta\mathrel{\mathop{\ordinarycolon}}C\longrightarrow A$), where $A$ and
$B$ (resp. $A$) are groups in ${\bf\mathcal{H}_{\bf ab}}$ for which the
statement holds and $C$ is cyclic. Since both $A$ and $B$ are quasi-
isometrically embedded in $G$ (for example, by [Kap01, Theorem 1.2]), then $G$
will contain a quasi-isometrically embedded surface if at least one of $A$ or
$B$ does. So it remains to ensure the conclusion in the case when $A$ and $B$
(resp. $A$) are free. Viewing $G$ as the fundamental group of a graph of two
free groups (resp. one free group) with infinite cyclic edge groups, we apply
[Wil12, Theorem 18] to deduce that, if $G$ is not one-ended, then some vertex
will split freely relative to its incident edge groups. In particular,
applying several times this result (and taking out an infinite cyclic free
factor of $G$), we deduce that $G$ is isomorphic to the free product
$F*G^{\prime}$ for a free group $F$ and a one-ended group
$G^{\prime}=A^{\prime}*_{C}B^{\prime}$ (resp. $G=A^{\prime}_{C,\theta}$ for
the injection $\theta\mathrel{\mathop{\ordinarycolon}}C\longrightarrow
A^{\prime}$), where $A^{\prime}$ and $B^{\prime}$ are free factors of $A$ and
$B$ (resp. $A^{\prime}$ is a free factor of $A$). Since $G$ was hyperbolic,
and $G^{\prime}$ is a quasi-convex subgroup, then $G^{\prime}$ is also
hyperbolic. So $G^{\prime}$ is a hyperbolic one-ended fundamental group of a
graph of free groups with cyclic edges. By 5.3, the group $G^{\prime}$ (and
hence also $G$) contains a quasi-convex surface subgroup.

  * •

Suppose that $G$ has a finite-index subgroup in ${\bf\mathcal{H}_{\bf ab}}$
for which the statement is already true. Since $G$ is torsion-free and not
free, then $H$ not free. So $H$ (and hence also $G$) must contain a surface
subgroup that is a virtual retract. ∎

##  7\. Profinite and pro-$p$ topologies in polycyclic groups

It was proven by Hirsch [Hir46, Theorem 3.25] that polycyclic groups are
residually finite, and Malcev later in 1948 proved that they are, in fact,
LERF (see [Seg83, Exercise 11, Chapter 1] for a proof). The former result of
Hirsch has a pro-$p$ variant for nilpotent groups, given by Gruenberg (7.1),
while Malcev’s result does not. However, we recall that it follows from
Malcev’s theorem the weaker property that the induced topology on subgroups of
polycyclic groups is the full profinite topology. We study in 7.2 a pro-$p$
version of this statement. Before stating it, we first introduce Gruenberg’s
result and review some definitions.

Given a set of primes $P$ and a non-negative integer $n$, we say that $n$ is a
$P$-number if all the primes dividing $n$ are contained in $P$. We say that a
group $G$ is residually-$P$ if for all $1\neq g\in G$, there exists a prime
$p\in P$ and a finite-index normal subgroup $U\unlhd G$, where
$|G\mathrel{\mathop{\ordinarycolon}}U|$ is a power of $p$, such that $g\notin
U$.

We observe that the set of primes $P$ appearing as the orders of elements of a
group $G$ is exactly the smallest set of primes $P$ such that every element
with finite order in $G$ has a $P$-number order.

######  Theorem 7.1 ([Gru57], Theorem 2.1).

Let $G$ be a finitely generated nilpotent group.

  * (a)

If $G$ is torsion-free, then $G$ is residually-$p$ for all primes $p$.

  * (b)

Otherwise, let $P$ be the (non-empty) collection of primes $p$ that appear as
orders of elements in $G$. Then $G$ is residually-$P$.

###  7.1. Induced pro-$p$ topology in nilpotent groups

Continuing the discussion at the beginning of the section, one may also try to
extend the result of Malcev to the pro-$p$ setting by, on the first place,
figuring out a definition of $p$-LERF groups. A tentative definition is to say
that a group $G$ is $p$-LERF if all its finitely generated subgroups are
closed in the pro-$p$ topology. However, this does not even hold in
$\mathbb{Z}$ (since $p\mathbb{Z}$ is not a closed subgroup in the pro-$q$
topology for any prime $q\neq p$). In fact, a virtually polycyclic group is
$p$-LERF if and only if it is a finite $p$-group. So this is not an
interesting notion for infinite groups. In any case, many infinite groups have
the weaker property that the induced pro-$p$ topology on subgroups is their
full pro-$p$ topology. We confirm this in the next theorem.

######  Theorem 7.2.

Let $G$ be a finitely generated nilpotent group and let $H$ be any subgroup.
Then the pro-$p$ topology of $G$ induces in $H$ its full pro-$p$ topology.

We recall 2.9, where we say under the previous setting that $H$ is pro-$p$
embeddable into $G$. This result says that the functor “pro-$p$ completion” is
exact in the category of finitely generated nilpotent groups (since it is
always right-exact by 2.4).

Before the proof of 7.2, we will review how one can extract roots of subgroups
in nilpotent groups in 7.4.

######  Definition 7.3.

Let $G$ be a group, let $P$ be a collection of primes and assume $H\leqslant
G$. We define the $P$-radical of $H$ in $G$ as the set

$R_{P}(H)=\\{g\in G\,\,\mbox{such that there exists a $P$-number $n$ with
$g^{n}\in H$}\\}.$

######  Proposition 7.4.

Let $G$ be a finitely generated nilpotent group and let $H\leqslant G$. Then
$R_{P}(H)$ is a subgroup of $G$ and the finite index
$|R_{P}(H)\mathrel{\mathop{\ordinarycolon}}H|$ is a $P$-number.

###### Proof.

Let $K$ be the subgroup generated by $R_{P}(H)$. Since $G$ is noetherian, then
$K$ is finitely generated. So $K$ is generated by a finite subset of
$R_{P}(H)$. Then, by [Cle17, Theorem 2.24], it follows that $K=R_{P}(H)$ and
that $H$ has a finite index in $R_{P}(H)$ which is a $P$-number. ∎

###### Proof of 7.2.

We divide the proof in two steps. First, we prove in 7.5 that the conclusion
follows if we additionally suppose that $H$ is normal, and then we prove the
general case when $H$ is any subgroup.

######  Claim 7.5.

The conclusion follows if $H$ is normal in $G$.

###### Proof of 7.5.

We follow the strategy depicted in 2.11. Let $\Lambda\unlhd H$ be a
characteristic subgroup of $H$ whose index
$|H\mathrel{\mathop{\ordinarycolon}}\Lambda|$ is finite and a $p$-power. In
particular, $\Lambda$ is normal in $G$. Let $P$ be the collection of primes
different from $p$. The subgroup $N=R_{P}(\Lambda)$ is a normal in $G$ and it
contains $\Lambda$ with finite-index. It is immediate to see that $N\cap
H=\Lambda$ and that the finite group $G/N$ does not contain elements of order
coprime with $p$. The latter implies, by 7.1, that the nilpotent group $G/N$
is residually-$p$. We have an injection $HN/N\xhookrightarrow{}G/N$, where
$HN/N$ is a finite subgroup. There exists a finite $p$-power index normal
subgroup $K\unlhd G$ containing $N$ such that $K/N$ intersects $HN/N$
trivially (or, equivalently, $K\cap HN=N$). Lastly, we check that $K\cap
H=\Lambda$, which would finish the proof. For this, observe that $K\cap
H=(K\cap HN)\cap H=N\cap H=\Lambda.$ ∎

We can now finish the proof of the theorem. Consider any subgroup $H\leqslant
G$. Let $c$ denote the nilpotency class of $G$. By [Hal76, Theorem 10.3.3],
there is an ascending chain $H=H_{0}\unlhd H_{1}\unlhd\cdots\unlhd H_{c}=G$.
By 7.5, each $H_{i}$ is pro-$p$ embeddable into $H_{i+1}$. Finally, by
transitivity, $H$ is pro-$p$ embeddable into $G$. ∎

######  Remark 7.6.

Gareth Wilkes pointed out to the author another proof of one of the directions
of 7.2, that is, that if $G$ is finitely generated nilpotent and $H\leqslant
G$ is a subgroup, then $H$ is pro-$p$ embeddable into $G$. The argument can be
sketched as follows. Since pro-nilpotent groups are direct products of their
$p$-Sylows (see [Rib00, Proposition 2.3.8]), then there is a commutative
diagram

(2)
${H}$${G}$${H_{\widehat{p}}}$${G_{\widehat{p}}}$${\widehat{H}}$${\widehat{G}}$$\scriptstyle{\iota}$$\scriptstyle{\iota_{\widehat{p}}}$$\scriptstyle{\widehat{\iota}}$

where we denoted by $\iota\colon H\longrightarrow G$ the injection map and by
$\iota_{\widehat{p}}$ and $\widehat{\iota}$ the natural induced maps between
profinite and pro-$p$ completions, respectively. Since $G$ is LERF, then
$\widehat{\iota}$ is injective and, consequently, $\iota_{\widehat{p}}$ is
injective, as we wanted to prove.

###  7.2. Induced pro-$p$ topology on polycyclic groups

As we anticipated, although the profinite version of 7.2 holds for virtually
polycyclic groups, we prove a converse that shows the limitations of this
phenomenon outside of the world of nilpotent groups.

######  Theorem 7.7.

Let $G$ be a virtually polycyclic group. Suppose that for every prime $p$ and
every subgroup $H\leqslant G$, the induced $H_{\widehat{p}}\longrightarrow
G_{\widehat{p}}$ is injective. Then $G$ is nilpotent.

We recall that this theorem in conjunction to 7.2 constitute D from the
introduction.

######  Lemma 7.8.

Let $P$ be a finite $p$-group and let $Q$ be a finite $q$-group, with $p$ and
$q$ being different primes. Let $\theta\colon
Q\longrightarrow\operatorname{Aut}(P)$ be a group homomorphism and let
$G=P\rtimes Q$ be the corresponding semi-direct product. Then the following
are equivalent:

  1. (1)

The map $\theta$ is trivial.

  2. (2)

$G$ is isomorphic to $P\times Q$.

  3. (3)

$G$ is nilpotent.

  4. (4)

The subgroup $P$ is pro-$p$ embeddable into $G$.

###### Proof.

The equivalence between (1) and (2) is well-known and follows from the
definition of semi-direct product (since $Q$ is normal if and only if it is
central). By [Hal76, Theorem 10.3.4], a finite group is nilpotent if and only
it is a direct product of finite $t$-groups with $t$ ranging over prime
numbers. So the equivalence between (2) and (3) is also clear. We notice that
if (1) is given, then (4) is obviously ensured. So it suffices to prove that
(4) implies (1). We consider the trivial subgroup $\\{1\\}$, which has
$p$-power index in $P$ and it is normal. By assumption, there must exist a
normal subgroup $N\unlhd G$ of $p$-power index such that $N\cap
P\subseteq\\{1\\}$, so $N\cap P=\\{1\\}$. Since the index is coprime to $q$,
then $N$ is a $q$-Sylow subgroup of $G$, as well as $Q$. So $N$ and $Q$ are
conjugated to each other by the second Sylow’s theorem, which shows that $Q$
is also normal and (1) follows. ∎

######  Lemma 7.9.

Let $p$ be a prime. Suppose that $N\unlhd G$ and $N\leqslant H\leqslant G$
have the property that the induced maps $N_{\widehat{p}}\longrightarrow
H_{\widehat{p}}$ and $N_{\widehat{p}}\longrightarrow G_{\widehat{p}}$ are
injective. Then, the induced map $H_{\widehat{p}}\longrightarrow
G_{\widehat{p}}$ is injective if and only if the induced map
$\left(H/N\right)_{\widehat{p}}\longrightarrow\left(G/N\right)_{\widehat{p}}$
is injective.

###### Proof.

If the induced maps $N_{\widehat{p}}\longrightarrow H_{\widehat{p}}$ and
$N_{\widehat{p}}\longrightarrow G_{\widehat{p}}$ are injective, we have, by
2.4, a commutative diagram of the form

(3)
${1}$${N_{\widehat{p}}}$${H_{\widehat{p}}}$${\left(H/N\right)_{\widehat{p}}}$${1}$${1}$${N_{\widehat{p}}}$${G_{\widehat{p}}}$${\left(G/N\right)_{\widehat{p}}}$${1,}$

where all the arrows are the natural ones and the rows are exact. A simple
diagram chase gives the desired conclusion. ∎

######  Lemma 7.10.

Let $P$ be a direct product $P_{1}\times\cdots\times P_{m}$, where each
$P_{i}$ is a finite-$p_{i}$ group and $p_{i}\neq p_{j}$ if $i\neq j$. Suppose
that $q$ is a prime not belonging to the set $\\{p_{1},\dots,p_{n}\\}$ and let
$Q$ be a finite $q$-group. Let $\theta\colon
Q\longrightarrow\operatorname{Aut}(P)$ be a group homomorphism and let
$G=P\rtimes Q$ be the corresponding semi-direct product. Then the following
are equivalent:

  1. (1)

The map $\theta$ is trivial.

  2. (2)

$G$ is isomorphic to $P\times Q$.

  3. (3)

$G$ is nilpotent.

  4. (4)

The subgroup $P$ is pro-$p_{i}$ embeddable into $G$ for all $i$.

###### Proof.

The equivalence between the first three items is established in the exact same
way as in 7.8. Again, it is clear that $(2)$ implies (4). We suppose now that
(4) holds. For every $1\leqslant i\leqslant n$, we consider
$N_{i}=\prod_{j\neq i}P_{j}$, which is a characteristic subgroup of $G$ that
is pro-$t$ embeddable into $P$ for all primes $t$. So $N_{i}$ is normal in $G$
and, by 7.9, $P/N_{i}\cong P_{i}$ is pro-$p_{i}$ embeddable into $G/N_{i}\cong
P_{i}\rtimes Q$. By 7.8, the action of $Q$ on $P_{i}$ is trivial. We have
proven this for all $i$, so the action of $Q$ on $P$ is trivial and hence we
have (2). ∎

######  Lemma 7.11.

Let $G$ be a finite group such that for every subgroup $H\leqslant G$, the
induced map $H_{\widehat{p}}\longrightarrow G_{\widehat{p}}$ is injective.
Then $G$ is nilpotent.

###### Proof.

We can prove this by induction on the order of $G$. When $|G|=1$ or, in fact,
when it is a finite $p$-group for some $p$, the statement is trivial. Suppose
then that $|G|$ is divisible by at least two primes $p$ and $q$. We consider a
$p$-Sylow $P\leqslant G$ (which exists by the first Sylow theorem). Since
$\\{1\\}$ is open in $P$, there must exist a normal subgroup $N\unlhd G$ of
$p$-power index such that $N\cap P=\\{1\\}$. Since $p$-subgroups of $G$ can be
conjugated into $P$ (by the second Sylow theorem), then the order of $N$ is
coprime with $p$. Consequently, $|P|=|G\mathrel{\mathop{\ordinarycolon}}N|$
and then $G=\langle N,P\rangle$ is isomorphic to a semi-direct product
$N\rtimes P$. We know that $|N|<|G|$ and then by the induction hypothesis, $N$
is nilpotent and so a direct product of its Sylow subgroups. By 7.10, we
conclude that $G$ is nilpotent. ∎

###### Proof of 7.7.

By 7.9, any finite quotient $Q$ of $G$ will still have the property that for
every subgroup $H\leqslant Q$, the induced map $H_{\widehat{p}}\longrightarrow
Q_{\widehat{p}}$ is injective. So, by 7.11, every finite quotient of $G$ is
nilpotent. Finally, by a result of Hirsch (see [Seg83, Theorem 2, Chapter 1]),
a virtually polycyclic group has the property that all its finite quotients
are nilpotent if and only if it is nilpotent itself. So $G$ is nilpotent. ∎

##  8\. A Tits Alternative for the $p$-genus of free or surface groups

In this section we prove B from the introduction.

######  Theorem 8.1.

Let $p$ be a prime and let $G$ be a group in the $p$-genus of a free or
hyperbolic surface group. Assume that $H\leqslant G$. The following holds.

  * (i)

If $H$ is amenable, then it is cyclic.

  * (ii)

If $H$ is two-generated, then $H$ is free.

The property (i) of this theorem will follow from (ii) by the following lemma.

######  Lemma 8.2.

Let $G$ be a finitely generated residually-(torsion-free polycyclic) group and
let $H\leqslant G$ be locally-cyclic. Then $H$ is either trivial or infinite-
cyclic.

###### Proof.

If $H$ is finitely generated, then the conclusion follows immediately. We
proceed by contradiction. Suppose, otherwise, that $H$ is not finitely
generated. Take $1\neq h_{1}\in H$. Since $H$ is locally cyclic but not
cyclic, there exists $a_{2}\notin\langle h_{1}\rangle$ and also $h_{2}\in H$
such that $\langle h_{2}\rangle=\langle h_{1},a_{2}\rangle$. So there exists
$n_{1}\in\mathbb{Z}$ with $h_{2}^{n_{1}}=h_{1}$ and, due to the fact that
$a_{2}\notin\langle h_{1}\rangle$, we derive that $|n_{1}|>1$. Iterating this
process, we show that then there must exist a sequence $(h_{i})_{i\geqslant
1}$ in $H$ of non-trivial elements and a sequence of integers
$(n_{i})_{i\geqslant 1}$ with $|n_{i}|>1$ such that $h_{i+1}^{n_{i}}=h_{i}$
for all $i\geqslant 1$. Let $\phi\colon G\longrightarrow Q$ be a surjective
map to a torsion-free polycyclic group $Q$ such that $\phi(h_{1})\neq 1$ (and
hence $\phi(h_{i})\neq 1$ for all $i$). Now consider the ascending chain of
subgroups

$\langle\phi(h_{1})\rangle\subseteq\langle\phi(h_{2})\rangle\subseteq\cdots\subseteq\langle\phi(h_{n})\rangle\subseteq\cdots$

of $Q$. Since $Q$ is noetherian, this sequence must stabilise. So there exists
$k\geqslant 1$ such that
$\phi(h_{k+1})\in\langle\phi(h_{k})\rangle=\langle\phi(h_{k+1})^{n_{k}}\rangle.$
Since $|n_{k}|>1$, this implies that $\phi(h_{k+1})$ has finite order in $Q$,
and so $\phi(h_{k+1})=1$, which is a contradiction. ∎

Before proving 8.1, we state the following well-known principle.

######  Lemma 8.3.

Let $S$ be a non-abelian free or surface group. Suppose that ${\bf H}$ is a
topologically 2-generated closed subgroup of $S_{\widehat{p}}$. Then ${\bf H}$
is a free pro-$p$ group.

###### Proof.

If $S$ is free, this follows from 4.7. If $S$ is a hyperbolic surface group of
genus $g$, then the abelianisation of $S$ has rank $2g\geqslant 4$. This
implies that the image of ${\bf H}$ in the abelianisation of $S_{\widehat{p}}$
has infinite index and, in particular, ${\bf H}$ already had infinite-index in
$S_{\widehat{p}}$. By 4.11, $\mathrm{cd}_{p}({\bf H})\leqslant 1$. Lastly,
again by 4.7, ${\bf H}$ is a free pro-$p$ group. ∎

###### Proof of 8.1.

We first explain how the property (i) follows from (ii). Remark that amenable
groups cannot contain non-abelian free subgroups. So, if $H$ is amenable,
then, by property (ii), $H$ is locally cyclic. In addition, $G$ is a subgroup
of $G_{\widehat{p}}$, which is residually-(torsion-free nilpotent) by 2.17.
Finally, by Lemma 8.2, $H$ is cyclic.

Now we move on to prove (ii). Let $S$ be a free or surface group and suppose
that $G$ belongs to the $p$-genus of $S$. Te statement is clear when
$S\cong\mathbb{Z}$ because, in this case, $G\cong\mathbb{Z}$. Let us suppose
that $S$ a non-abelian free or a hyperbolic surface group. Let $H\leqslant G$
be a two-generated subgroup. We analyse separately two cases, depending on
whether $H$ is abelian or not.

  * (i)

Firstly, consider the case when $H$ is non-abelian. We consider
$\overline{H}\subseteq G_{\widehat{p}}$. If $G$ By 4.11, then
$\mathrm{cd}_{p}(\overline{H})\leqslant 1$. So by 4.7, $\overline{H}$ is a
free pro-$p$ group. Since $H$ is two-generated, then $\overline{H}$ has rank
at most two. We notice that $H\leqslant\overline{H}$ cannot be an abelian
group, so the rank of $\overline{H}$ has to be exactly equal to two. So
$\overline{H}$ is the free pro-$p$ group of rank 2. Let $x$ and $y$ be two
elements that generate $H$, then $x$ and $y$ are topological generators of
$\overline{H}$ and hence they are also free topological generators by the Hopf
property. In particular, the group $H$ generated by $x$ and $y$ is free of
rank two, as we wanted.

  * (ii)

Lastly, suppose that $H$ is abelian. It has to be torsion-free (in fact, we
already know that $H$ is residually-(torsion-free nilpotent)). Since $H$ is
two-generated, it suffices to show that $H\not\cong\mathbb{Z}^{2}$ in order to
conclude that it is cyclic. We proceed by contradiction. Assume that
$H\cong\mathbb{Z}^{2}$ and let $a$ and $b$ be free-abelian generators of $H$.
By 8.3, $\overline{H}\cong\mathbb{Z}_{p}$, where this group is topologically
generated by some $c\in G_{\widehat{p}}$, so there exist non-trivial
$m_{a},m_{b}\in\mathbb{Z}_{p}$ such that $a=c^{m_{a}}$ and $b=c^{m_{b}}$. Let
$n\geqslant 1$ be such that $c\notin\gamma_{n}G_{\widehat{p}}$. We consider
the canonical projection $p\colon G_{\widehat{p}}\longrightarrow
G_{\widehat{p}}/\gamma_{n}G_{\widehat{p}}$. The pro-$p$ group
$G_{\widehat{p}}/\gamma_{n}G_{\widehat{p}}$ is torsion-free by 2.16. In
particular, since $p(c)\neq 1$, it follows that the restriction of
$p_{\widehat{p}}$ to $\overline{H}$ is injective. If we denote by $\iota\colon
H\longrightarrow G$ the inclusion, we have a commutative diagram of the form

(4)
${H}$${H_{\widehat{p}}}$${G}$${G_{\widehat{p}}}$${G/\gamma_{n}G}$${G_{\widehat{p}}/\gamma_{n}G_{\widehat{p}}}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\scriptstyle{f_{\widehat{p}}}$$\scriptstyle{\iota_{\widehat{p}}}$$\scriptstyle{p}$$\scriptstyle{p_{\widehat{p}}}$

The horizontal arrows are the canonical pro-$p$ completion maps (the fact that
the lowest arrow represents a pro-$p$ completion map follows from Lemma 2.8).
The map $f=p\circ\iota\colon H\longrightarrow G/\gamma_{n}G$ is injective by
previous comments. Since $G/\gamma_{n}G$ is finitely generated and nilpotent,
it follows by 7.2 that the induced $f_{\widehat{p}}$ is injective, too. In
particular, $\iota_{\widehat{p}}$ must be injective. But this is absurd,
because we derived that $H_{\widehat{p}}\cong\mathbb{Z}_{p}^{2}$ and
$\iota_{\widehat{p}}(H_{\widehat{p}})=\overline{H}\cong\mathbb{Z}_{p}$. ∎

##  9\. Pro-$p$ detection of algebraic and topological properties

We recall that by 2.1, groups in the finite, soluble or nilpotent genus of a
free or surface group $S$, do belong to the $p$-genus of $S$ for all primes
$p$. So all the results here about any $p$-genus of $S$ include all these
genera.

###  9.1. Lücky groups

It is generally not well understood if being Lücky (3.6) is a profinite
invariant (see 9.2 below). However, Jaikin-Zapirain [And212] can answer
affirmatively to this question for the genera of free and surface groups as a
consequence of the following.

######  Proposition 9.1.

Let $S$ be a free or surface group and let $G$ be an abstract subgroup of
$S_{\widehat{p}}$. Then $G$ is $\mathrm{L\ddot{u}cky}$.

###### Proof.

By 2.16 and 2.17, $G$ is residually-(torsion-free nilpotent). By 3.5, $G$ is
$\mathrm{L\ddot{u}cky}$. ∎

In fact, it is not difficult to see that the previous $G$ have to be RFRS. As
a consequence, if $G$ is a group in the finite genus of a free or surface
group $S$, then $b_{1}^{(2)}(G)=b_{1}^{(2)}(S)$.

######  Question 9.2.

Let $\Gamma$ and $\Lambda$ be finitely generated residually finite groups with
$\widehat{\Gamma}\cong\widehat{\Lambda}$. If $\Gamma$ is
$\mathrm{L\ddot{u}cky}$, does it follow that $\Lambda$ is
$\mathrm{L\ddot{u}cky}$?

This relate the question of whether being RFRS is a profinite invariance.

######  Question 9.3.

Let $\Gamma$ and $\Lambda$ be finitely generated residually finite groups with
$\widehat{\Gamma}\cong\widehat{\Lambda}$. If $\Gamma$ is RFRS, does it follow
that $\Lambda$ is RFRS?

###  9.2. Hyperbolic and special groups

We also study when groups from ${\bf\mathcal{H}_{\bf ab}}$ (1.3) or one-
relator groups can be proven to be hyperbolic and virtually special by looking
at the pro-$p$ completion.

######  Corollary 9.4.

Let $p$ be a prime and let $G$ be a group in the class ${\bf\mathcal{H}_{\bf
ab}}$ that belongs to the $p$-genus of a free or hyperbolic surface group.
Then $G$ is hyperbolic and virtually compact special.

###### Proof.

By the Tits alternative of 8.1, $G$ is torsion-free, contains no
Baumslag–Solitar groups and contains no non-cyclic abelian subgroups. Hence,
by 6.2, $G$ is hyperbolic virtually compact special. ∎

Recently, Linton [Lin22] established that all one-relator groups with no
negative immersions are hyperbolic and virtually special, and uses this result
to prove that one-relator parafree groups are hyperbolic. We will use the Tits
alternative of 8.1 to extend this argument to parasurfaces and to other
genera.

######  Corollary 9.5.

Let $G$ be a one-relator group that belongs to the $p$-genus of a free group
or a hyperbolic surface group. Then $G$ is hyperbolic and virtually special.

###### Proof.

Let $G=\langle S|\omega\rangle$ be presented by a finite set of generators $S$
and a relator $\omega\in F(S)$. By 8.1, all 2-generated subgroups are free and
hence there are no freely indecomposable one-relator subgroups of rank 2 in
$G$. Hence, by [Lou22, Theorem 1.3], the primitivity rank $\pi(\omega)$ is at
least 3; and then, by [Lou22, Theorem 1.3], $G=\langle S|\omega\rangle$ has a
presentation complex with no negative immersions. Finally, the conclusion
follows immediately from [Lin22, Theorem 8.2]. ∎

##  10\. On the profinite rigidity of free and surface groups

We are now ready to establish the profinite rigidity of free and surface
groups within the family ${\bf\mathcal{H}_{\bf ab}}$. The proof of A is
divided in 10.1 and 10.2.

######  Theorem 10.1.

Let $G$ be a group belonging to the class ${\bf\mathcal{H}_{\bf ab}}$ and let
$F$ be a finitely generated free group. If $G$ is residually finite and
$\widehat{G}\cong\widehat{F}$, then $G\cong F$.

###### Proof.

Let $G$ be a group in ${\bf\mathcal{H}_{\bf ab}}$ that belongs to the finite
genus of a free group $F$. By [And212, Theorem 1.1], $G$ also belongs to the
$p$-genus of $F$. Consequently, by 9.4, $G$ is hyperbolic virtually compact
special. We want to prove that $G$ is free and we proceed by contradiction.
Suppose that is $G$ is not free. Then, by 6.5, there exists a surface subgroup
$\pi_{1}S$ that it is a virtual retract of $G$. In particular,
$\widehat{\pi_{1}S}\xhookrightarrow{}\widehat{G}\cong\widehat{F}$. This would
imply by the goodness of the surface $\pi_{1}S$ and the monotonicity of
$\mathrm{cd}_{p}$ (4.5), the following contradiction:

$2=\mathrm{cd}_{p}(\pi_{1}S)=\mathrm{cd}_{p}(\widehat{\pi_{1}S})\leqslant\mathrm{cd}_{p}(\widehat{F})=1.$

Thus, $G$ is free and has the same abelianisation as $F$, so $G\cong F$. ∎

######  Theorem 10.2.

Let $G$ be a group belonging to the class ${\bf\mathcal{H}_{\bf ab}}$ and let
$\pi_{1}\Sigma$ be the fundamental group of a closed orientable surface. If
$G$ is residually finite and $\widehat{G}\cong\widehat{\pi_{1}\Sigma}$, then
$G\cong\pi_{1}\Sigma$.

###### Proof.

Let $G$ be a group in ${\bf\mathcal{H}_{\bf ab}}$ that belongs to the finite
genus of a free group $\pi_{1}\Sigma$. By [And212, Theorem 1.1], $G$ also
belongs to the $p$-genus of $\pi_{1}\Sigma$. Consequently, by 9.4, $G$ is
hyperbolic virtually compact special. Since $G$ cannot be free, it follows
from 6.5 that there exists a surface subgroup $\pi_{1}S$ that it is a virtual
retract of $G$. Lastly, by 4.14, $G$ will be isomorphic to a surface group.
Since the abelianisation of a group is a profinite invariant and surface
groups are distinguished by their abelianisation, then we have an isomorphism
$G\cong\pi_{1}\Sigma$. ∎

##  11\. Proof of E

We will list two lemmas that we will use during the proof of E, such as the
following elementary fact about residually free groups.

######  Lemma 11.1 ([Bau67a], Lemma 4).

Let $G$ be a residually free group. Then $G/Z(G)$ is a residually free group
with trivial centre.

We will also need a criterion that detects when short exact sequences split by
looking at profinite completions.

######  Lemma 11.2.

Let $S$ be a surface group and let

(5) ${1}$${\mathbb{Z}^{n}}$${G}$${S}$${1}$

be a central extension of $S$ by the group $\mathbb{Z}^{n}$. Suppose that the
induced exact sequence in profinite completions

(6) ${1}$${\widehat{\mathbb{Z}}^{n}}$${\widehat{G}}$${\widehat{S}}$${1}$

is also exact and that it splits. Then the sequence (5) splits, too.

The previous lemma is taken from [Wilton17, Lemma 8.3], where a similar
version is used to distinguish circle bundles over the surface by looking at
finite quotients of the fundamental group (later on used in dimension 4 in
[Ma22, Proposition 25]). We remark that the exactness of (6) is always ensured
if (5) is exact, by the goodness of $S$ (as observed by Serre [Ser97, Chapter
1, Section 2, Exercise 2(b)]). However, we stated this as an assumption for
clearness, as it is also enough for our application. Regarding this lemma of
Serre, we also refer the reader to [Lor08, Proposition 2.4] for a precise
proof and a converse.

###### Proof of E.

Let $S$ be a free or surface group and let $G$ be a finitely generated
residually free group in the finite genus of $S\times\mathbb{Z}^{n}$. If $S$
is abelian, then it is clear that $G\cong S\times\mathbb{Z}^{n}$. So we can
suppose that $S$ is hyperbolic and non-abelian. In particular,
$b_{1}^{(2)}(S)>0$ and $\widehat{S}$ has trivial centre.

We denote by $\iota\colon Z(G)\xhookrightarrow{}G$ the canonical injection.

######  Claim 11.3.

The centre $Z(G)$ is finitely generated and the induced map
$\widehat{\iota}\mathrel{\mathop{\ordinarycolon}}\widehat{Z(G)}\longrightarrow\widehat{G}$
is also injective.

To prove the claim, first recall that
$Z(\widehat{G})=\widehat{\mathbb{Z}}^{n}$ is a retract of $\widehat{G}$.
Denoting such retraction by $r\colon\widehat{G}\longrightarrow
Z(\widehat{G})$, we have the following natural commutative diagram:

(7)
${Z(G)}$${Z(\widehat{G})}$${G}$${\widehat{G}}$${G/[G,G]}$${Z(\widehat{G}),}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{p}$$\scriptstyle{r}$$\scriptstyle{g}$

for some naturally defined $f$ and $g$ from the canonical maps $\iota,p$ and
$r$. We can read off from the diagram that $g\circ f$ is injective, so $f$ is
injective, too. This implies that $Z(G)$ is a subgroup of the finitely
generated abelian group $G/[G,G]$, so it is finitely generated, too. Now we
consider a second natural commutative diagram:

(8)
${Z(G)}$${\widehat{Z(G)}}$${G}$${\widehat{G}}$${G/[G,G]}$${\widehat{G/[G,G]}.}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\scriptstyle{\widehat{f}}$$\scriptstyle{\widehat{\iota}}$$\scriptstyle{p}$$\scriptstyle{\widehat{p}}$

We know that $f$ is injective, so $\widehat{f}$ is also injective (since
finitely generated abelian groups are LERF). Thus $\widehat{\iota}$ is
injective and 11.3 is proven.

######  Claim 11.4.

The image of $\widehat{\iota}$ is the whole $Z(\widehat{G})$.

The image of $\widehat{\iota}$ can be alternatively described as the closure
of $Z(G)\xhookrightarrow{}\widehat{G}$ and it is easy to see that it is a
normal subgroup of $\widehat{G}$ that is contained in
$Z(\widehat{G})=\widehat{\mathbb{Z}}^{n}$. We consider the group $H=G/Z(G)$,
which is a finitely generated residually free group with trivial centre by
11.1. By 2.5, $\widehat{H}$ is naturally isomorphic to
$\widehat{G}/\overline{Z(G)}\cong\widehat{S}\times\left(Z(\widehat{G})/\overline{Z(G)}\right)$.
Let us denote by $p_{1}$ and $p_{2}$ the projections of $\widehat{H}$ onto its
first and second direct factor according to the decomposition
$\widehat{H}\cong\widehat{S}\times Z(\widehat{G})/\overline{Z(G)}$. 11.4
follows if the second factor is trivial. Let us suppose that it is not trivial
to reach a contradiction. We study two separate cases.

  * (i)

If $Z(\widehat{G})/\overline{Z(G)}$ is infinite, then we claim that it is
isomorphic to the profinite completion of a finitely generated abelian group
$A$. This is immediate from the fact that this profinite group appear as the
direct factor of
$\left(\widehat{S}/[\widehat{S},\widehat{S}]\right)\times\left(Z(\widehat{G})/\overline{Z(G)}\right)$,
which is the profinite completion of $H/[H,H]$. As a consequence,
$\widehat{H}$ is the profinite completion of the group $K=S\times A$. It
follows that $K$ is $\mathrm{L\ddot{u}cky}$ and $b_{1}^{(2)}(K)=0$. Since $H$
is residually free, then it is also $\mathrm{L\ddot{u}cky}$ by 3.5, so it
follows from 3.9 that $b_{1}^{(2)}(H)=b_{1}^{(2)}(K)=0$. From this, we deduce
that the restriction of $p_{1}$ to $H$ cannot be injective (otherwise, we
would see $H$ as a dense subgroup of $\widehat{S}$ and it would follow from
3.8 that $b_{1}^{(2)}(H)\geqslant b_{1}^{(2)}(S)>0$). So $H$ intersects $\ker
p_{1}=Z(\widehat{G})/\overline{Z(G)}=Z(\widehat{H})$ non-trivially and then
$Z(H)\neq 1$, which contradicts 11.1.

  * (ii)

If $Z(\widehat{G})/\overline{Z(G)}$ is finite, then we can still prove that
$H$ intersects non-trivially this factor to lead to the same contradiction as
before. To prove this, let us denote $H_{1}=H\cap\widehat{S}$, which is a
finite-index subgroup of $H$ with $\widehat{H}_{1}\cong\widehat{S}$. So
$H_{1}$ is also $\mathrm{L\ddot{u}cky}$ and
$b_{1}^{(2)}(H_{1})=b_{1}^{(2)}(S)$ by 3.9. Observe that $p_{1}(H_{1})$ is
isomorphic to $H_{1}$, that this is finite-index in $p_{1}(H)$, and that both
$p_{1}(H_{1})$ and $p_{1}(H)$ are dense in $\widehat{S}$. Let us denote the
index $k=|p_{1}(H)\mathrel{\mathop{\ordinarycolon}}p_{1}(H_{1})|$. Using the
multiplicativity of $b_{1}^{(2)}$ and 3.8, it follows that

$b_{1}^{(2)}(p_{1}(H))\geqslant
b_{1}^{(2)}(S)=b_{1}^{(2)}(H_{1})=b_{1}^{(2)}(p_{1}(H_{1}))=k\,b_{1}^{(2)}(p_{1}(H)).$

Since $b_{1}^{(2)}(S)$ is positive, we derive from the previous inequalities
that $k=1$ and hence $p_{1}(H)=p_{1}(H_{1})$. From this, it is immediate to
see that $Z(\widehat{G})/\overline{Z(G)}=p_{2}(H)=p_{2}(H\cap
Z(\widehat{G})/\overline{Z(G)})$, and then $H\cap
Z(\widehat{G})/\overline{Z(G)}\neq 1$, as we wanted.

We have completed the proof of 11.4. Since $H=G/Z(G)$ is finitely generated
residually free and $\widehat{H}\cong\widehat{S}$, then it follows by 10.1 and
10.2 that $H\cong S$.

######  Claim 11.5.

The short exact sequence

(9) ${1}$${Z(G)}$${G}$${G/Z(G)}$${1}$

splits. In particular, $G\cong Z(G)\times G/Z(G)\cong\mathbb{Z}^{n}\times S$.

The proof of the claim would complete the proof of E. This claim is obvious
when $S$ is free and it is not hard to see that, in the case when $S$ is a
surface group, it is a direct consequence of 11.2. ∎

## References