$\displaystyle=\begin{cases}\prod_{i\in\mathbb{Z}}\exp(y_{-i}\operatorname{ad}_{X_{i}})\tau_{0},&\text{if
}\lambda=0,\\\
\prod_{i\in\mathbb{Z}}\exp((-1)^{\delta_{i}^{0}}y_{-i}\operatorname{ad}_{X_{i}})\tau_{-1},&\text{if
}\lambda=-1,\end{cases}$

Therefore, if we multiply two arbitrary elements of
$\operatorname{Aut}(\mathcal{W}_{X}(\lambda))$, we get

$\left(\prod_{i\in\mathbb{Z}}\exp(y_{i}\operatorname{ad}_{X_{i}})\Phi_{k;a,b;\alpha,\xi}^{X(\lambda)}\right)\left(\prod_{i\in\mathbb{Z}}\exp(z_{i}\operatorname{ad}_{X_{i}})\Phi_{\ell;c,d;\beta,\zeta}^{X(\lambda)}\right)\\\
=\begin{cases}\prod_{i\in\mathbb{Z}}\exp((y_{i}+\alpha^{\varepsilon_{1}i}\xi
z_{\varepsilon_{1}i})\operatorname{ad}_{X_{i}})\Phi_{k+\ell;\varepsilon_{2}a+\xi
c,b+\xi d;\alpha^{\varepsilon_{2}}\beta,\xi\zeta}^{X(\lambda)},&\text{if
}\lambda=0,\\\
\prod_{i\in\mathbb{Z}}\exp((y_{i}+\varepsilon_{1}^{\delta_{i}^{0}}\alpha^{\varepsilon_{1}i}\xi
z_{\varepsilon_{1}i})\operatorname{ad}_{X_{i}})\Phi_{k+\ell;\varepsilon_{2}a+\xi
c,b+\xi d;\alpha^{\varepsilon_{2}}\beta,\xi\zeta}^{X(\lambda)},&\text{if
}\lambda=-1,\end{cases}$

where $k,\ell\in\mathbb{Z}/2\mathbb{Z}$ and $\varepsilon_{1}=(-1)^{k}$,
$\varepsilon_{2}=(-1)^{\ell}$. This concludes the proof. ∎

##  6\. Derivations

We finish the paper by computing derivations of $\mathcal{W}_{X}(\lambda)$. We
begin by recalling the notion of a derivation of a Lie algebra.

######  Definition 6.1.

A _derivation_ of a Lie algebra $\mathfrak{g}$ with values in an
$\mathfrak{g}$-module $\mathfrak{M}$ is a linear map $d\colon\mathfrak{g}\to
M$ such that

$d([x,y])=x\cdot d(y)-y\cdot d(x)$

for $x,y\in\mathfrak{g}$. We write
$\operatorname{Der}(\mathfrak{g},\mathfrak{M})$ for the set of derivations
$\mathfrak{g}$ with values in $\mathfrak{M}$, and simply write
$\operatorname{Der}(\mathfrak{g})$ instead of
$\operatorname{Der}(\mathfrak{g},\mathfrak{g})$.

A derivation $d\in\operatorname{Der}(\mathfrak{g},\mathfrak{M})$ is _inner_ if
there exists $m\in\mathfrak{M}$ such that $d(x)=x\cdot m$. We write
$\operatorname{Inn}(\mathfrak{g},\mathfrak{M})$ for the set of inner
derivations of $\mathfrak{g}$ with values in $\mathfrak{M}$. As before, we
write $\operatorname{Inn}(\mathfrak{g})$ instead of
$\operatorname{Inn}(\mathfrak{g},\mathfrak{g})$.

If $\mathfrak{g}=\bigoplus_{n\in\mathbb{Z}}\mathfrak{g}_{n}$ is a
$\mathbb{Z}$-graded Lie algebra and
$\mathfrak{M}=\bigoplus_{n\in\mathbb{Z}}\mathfrak{M}_{n}$ is a
$\mathbb{Z}$-graded $\mathfrak{g}$-module, then for $k\in\mathbb{Z}$ we write

$\operatorname{Der}(\mathfrak{g},\mathfrak{M})_{k}=\\{d\in\operatorname{Der}(\mathfrak{g},\mathfrak{M})\mid
d(\mathfrak{g}_{n})\subseteq\mathfrak{M}_{n+k}\text{ for all
}n\in\mathbb{Z}\\}.$

If $d\in\operatorname{Der}(\mathfrak{g},\mathfrak{M})_{k}$, we say that $d$ is
a _graded derivation of degree $k$_.

######  Remark 6.2.

If $\mathfrak{g}$ is a Lie algebra and $\mathfrak{M}$ is a
$\mathfrak{g}$-module, then

$\operatorname{H}^{1}(\mathfrak{g};\mathfrak{M})\cong\operatorname{Der}(\mathfrak{g},\mathfrak{M})/\operatorname{Inn}(\mathfrak{g},\mathfrak{M}).$

Therefore, computing derivations of $\mathfrak{g}$ is equivalent to computing
$\operatorname{H}^{1}(\mathfrak{g};\mathfrak{g})$.

There is a close relationship between the automorphisms of a Lie algebra
$\mathfrak{g}$ and its derivations: if we have a (smooth) one-parameter
subgroup of $\operatorname{Aut}(\mathfrak{g})$, differentiating it at $0$
gives a derivation of $\mathfrak{g}$. In other words, derivations of
$\mathfrak{g}$ can be seen as “infinitesimal automorphisms”.

We now introduce notation for the derivations that arise from one-parameter
subgroups of $\operatorname{Aut}(\mathcal{W}_{X}(\lambda))$.

######  Notation 6.3.

Let $X\in\\{A,B\\}$ and $\lambda\in\mathbb{P}^{1}$. Recall the automorphisms
defined in Notation 5.1. For both $X=A$ and $X=B$, define

$d_{\operatorname{Ab}}=\left.\frac{d}{dt}\right|_{t=0}\mu_{e^{t}}\in\operatorname{Der}(\mathcal{W}_{X}(\lambda)).$

For $X=A$, define

$\delta_{\lambda}^{A}=\left.\frac{d}{dt}\right|_{t=0}\varphi_{t}^{A(\lambda)},\quad\partial_{\lambda}^{A}=\left.\frac{d}{dt}\right|_{t=0}\psi_{t}^{A(\lambda)}\in\operatorname{Der}(\mathcal{W}_{A}(\lambda)).$

For $X=B$, define

$d_{\lambda}^{B}=\left.\frac{d}{dt}\right|_{t=0}\varphi_{t}^{B(\lambda)}\in\operatorname{Der}(\mathcal{W}_{B}(\lambda)).$

For $X=B$ and $\lambda=0$, define

$\partial^{B}_{0}=\left.\frac{d}{dt}\right|_{t=0}\psi_{t}^{B(0)}\in\operatorname{Der}(\mathcal{W}_{B}(0)).$

######  Remark 6.4.

Certainly, there is another one-parameter subgroup $\sigma_{e^{t}}$ of
$\operatorname{Aut}(\mathcal{W}_{X}(\lambda))$, giving rise to another
derivation. However, the derivation we obtain from this one-parameter subgroup
is inner: we have

$\left.\frac{d}{dt}\right|_{t=0}\sigma_{e^{t}}=\operatorname{ad}_{L_{0}}\in\operatorname{Inn}(\mathcal{W}_{X}(\lambda)).$

Indeed, we can immediately see this from the observations of Remark 5.3.

Having defined these derivations that arise from automorphisms, we now compute
their actions on $\mathcal{W}_{X}(\lambda)$.

######  Lemma 6.5.

Let $X\in\\{A,B\\}$ and $\lambda\in\mathbb{P}^{1}$. Then

$\begin{gathered}d_{\operatorname{Ab}}(L_{n})=0,\quad
d_{\operatorname{Ab}}(X_{n})=X_{n};\\\
\delta_{\lambda}^{A}(L_{n})=nA_{n},\quad\delta_{\lambda}^{A}(A_{n})=0;\quad\partial_{\lambda}^{A}(L_{n})=n^{2}A_{n},\quad\partial_{\lambda}^{A}(A_{n})=0;\\\
d_{\lambda}^{B}(L_{n})=\begin{cases}(\lambda+1)B_{0},&\text{if }n=0\text{ and
}\lambda\neq\infty,\\\ B_{n},&\text{otherwise},\end{cases}\quad
d_{\lambda}^{B}(B_{n})=0;\\\
\partial^{B}_{0}(L_{n})=nB_{n},\quad\partial^{B}_{0}(B_{n})=0,\end{gathered}$

for all $n\in\mathbb{Z}$.

###### Proof.

Follows immediately from the definitions of the automorphisms in Notation 5.1.
∎

We now consider the question of when the derivations defined in Notation 6.3
are inner.

######  Lemma 6.6.

Let $\lambda\in\mathbb{P}^{1}$. If $\lambda\neq\infty$, then
$(\lambda+1)\delta_{\lambda}^{A}+\lambda\partial_{\lambda}^{A}\in\operatorname{Inn}(\mathcal{W}_{A}(\lambda))$,
while if $\lambda=\infty$, then
$\delta_{\infty}^{A}+\partial_{\infty}^{A}=\operatorname{ad}_{-A_{0}}\in\operatorname{Inn}(\mathcal{W}_{A}(\infty))$.
Defining

$d_{\lambda}^{A}=\begin{cases}\delta_{\lambda}^{A},&\text{if }\lambda\neq
0,\\\ \partial_{0}^{A},&\text{if }\lambda=0,\end{cases}$

then $d_{\lambda}^{A}\notin\operatorname{Inn}(\mathcal{W}_{A}(\lambda))$ for
all $\lambda\in\mathbb{P}^{1}$. Certainly, we have

$d_{\lambda}^{A}=\left.\frac{d}{dt}\right|_{t=0}\varpi_{t}^{A(\lambda)},$

where $\varpi_{t}^{A(\lambda)}$ is defined in (5.4).

Furthermore,
$d_{\lambda}^{B}\notin\operatorname{Inn}(\mathcal{W}_{B}(\lambda))$, and there
is no nonzero linear combination of $d_{0}^{B}$ and $\partial^{B}_{0}$ which
produces an inner derivation of $\mathcal{W}_{B}(0)$.

###### Proof.

By Lemma 6.5, we can easily see that
$(\lambda+1)\delta_{\lambda}^{A}+\lambda\partial_{\lambda}^{A}=\operatorname{ad}_{-A_{0}}$
if $\lambda\neq\infty$, while
$\delta_{\infty}^{A}+\partial_{\infty}^{A}=\operatorname{ad}_{-A_{0}}$ if
$\lambda=\infty$. Since $\operatorname{ad}_{A_{0}}$ and
$\operatorname{ad}_{L_{0}}$ span the space of inner derivations of
$\mathcal{W}_{A}(\lambda)$ of degree zero, it follows immediately that
$d_{\lambda}^{A}$ is not inner for all $\lambda\in\mathbb{P}^{1}$.

The only nonzero (up to multiplication by a scalar) inner derivation of
$\mathcal{W}_{B}(\lambda)$ of degree zero is $\operatorname{ad}_{L_{0}}$. This
is because $B_{0}$ is central in $\mathcal{W}_{B}(\lambda)$, so
$\operatorname{ad}_{B_{0}}=0$. The result follows. ∎

The main goal of this section is to prove that, together with inner
derivations, the derivations defined in Notation 6.3 span the entire space of
derivations of $\mathcal{W}_{X}(\lambda)$.

######  Theorem 6.7.

Let $\lambda\in\mathbb{P}^{1}$. Then

$\displaystyle\operatorname{Der}(\mathcal{W}_{A}(\lambda))$
$\displaystyle=\operatorname{Inn}(\mathcal{W}_{A}(\lambda))\oplus\mathbb{C}d_{\operatorname{Ab}}\oplus\mathbb{C}d_{\lambda}^{A};$
$\displaystyle\operatorname{Der}(\mathcal{W}_{B}(\lambda))$
$\displaystyle=\begin{cases}\operatorname{Inn}(\mathcal{W}_{B}(\lambda))\oplus\mathbb{C}d_{\operatorname{Ab}}\oplus\mathbb{C}d_{0}^{B}\oplus\mathbb{C}\partial_{0}^{B},&\text{if
}\lambda=0,\\\
\operatorname{Inn}(\mathcal{W}_{B}(\lambda))\oplus\mathbb{C}d_{\operatorname{Ab}}\oplus\mathbb{C}d_{\lambda}^{B},&\text{if
}\lambda\neq 0,\end{cases}$

where the derivations are defined in Notation 6.3 and Lemma 6.6. Consequently,

$\dim(\operatorname{H}^{1}(\mathcal{W}_{A}(\lambda);\mathcal{W}_{A}(\lambda)))=2,\quad\dim(\operatorname{H}^{1}(\mathcal{W}_{B}(\lambda);\mathcal{W}_{B}(\lambda)))=\begin{cases}3,&\text{if
}\lambda=0,\\\ 2,&\text{if }\lambda\neq 0.\end{cases}$

######  Remark 6.8.

Theorem 6.7 says that all derivations of $\mathcal{W}(0,1)$ extend to the one-
parameter family $\mathcal{W}_{A}(\lambda)$. On the other hand, all
derivations in
$\operatorname{Inn}(\mathcal{W}(0,0))\oplus\mathbb{C}d_{\operatorname{Ab}}\oplus\mathbb{C}d_{0}^{B}\subseteq\operatorname{Der}(\mathcal{W}(0,0))$
extend to the one-parameter family $\mathcal{W}_{B}(\lambda)$, while
$\partial_{0}^{B}$ does not.

It follows immediately from Theorem 3.8 that
$H^{1}(\mathfrak{g};\mathfrak{g})$ is generated by derivations of degree 0. In
other words, the internally graded structure of $\mathcal{W}_{X}(\lambda)$
implies the next result.

######  Corollary 6.9.

Let $X\in\\{A,B\\}$ and $\lambda\in\mathbb{P}^{1}$. Then

$\operatorname{Der}(\mathcal{W}_{X}(\lambda))=\operatorname{Inn}(\mathcal{W}_{X}(\lambda))+\operatorname{Der}(\mathcal{W}_{X}(\lambda))_{0}.\qed$

Given Corollary 6.9, we only need to focus on computing derivations of
$\mathcal{W}_{X}(\lambda)$ of degree zero to prove Theorem 6.7.

######  Lemma 6.10.

Let $X\in\\{A,B\\}$ and $\lambda\in\mathbb{P}^{1}$. Let
$d\colon\mathcal{W}_{X}(\lambda)\to\mathcal{W}_{X}(\lambda)$ be a linear map
of degree zero. In other words, for all $n\in\mathbb{Z}$, we have

$d(L_{n})=\alpha_{n}L_{n}+\beta_{n}X_{n},\quad
d(X_{n})=\eta_{n}L_{n}+\theta_{n}X_{n},$

for some $\alpha_{n},\beta_{n},\eta_{n},\theta_{n}\in\mathbb{C}$. Then $d$ is
a derivation of $\mathcal{W}_{X}(\lambda)$ if and only if

(6.1) $\begin{gathered}\alpha_{n+m}=\alpha_{n}+\alpha_{m},\quad\text{if }n\neq
m,\\\
(m-n)\beta_{n+m}=\omega_{X(\lambda)}(n,m)\beta_{m}-\omega_{X(\lambda)}(m,n)\beta_{n},\\\
\omega_{X(\lambda)}(n,m)\theta_{n+m}=\omega_{X(\lambda)}(n,m)(\alpha_{n}+\theta_{m}),\\\
\omega_{X(\lambda)}(n,m)\eta_{n+m}=(m-n)\eta_{m},\\\
\omega_{X(\lambda)}(n,m)\eta_{n}=\omega_{X(\lambda)}(m,n)\eta_{m},\end{gathered}$

for all $n,m\in\mathbb{Z}$.

###### Proof.

The linear map $d$ is a derivation if and only if the following equations are
satisfied for all $n,m\in\mathbb{Z}$:

(6.2) $\displaystyle d([L_{n},L_{m}])$
$\displaystyle=[L_{n},d(L_{m})]+[d(L_{n}),L_{m}],$ (6.3) $\displaystyle
d([L_{n},X_{m}])$ $\displaystyle=[L_{n},d(X_{m})]+[d(L_{n}),X_{m}],$ (6.4)
$\displaystyle 0=d([X_{n},X_{m}])$
$\displaystyle=[X_{n},d(X_{m})]+[d(X_{n}),X_{m}].$

Expanding the above equations and grouping like terms, we deduce the result.
Note that the first two equations of (6.1) come from (6.2), the third and
fourth equations come from (6.3), and the final equation comes from (6.4). ∎

We now show that degree zero derivations of $\mathcal{W}_{X}(\lambda)$
preserve $X(\lambda)$.

######  Lemma 6.11.

Let $\lambda\in\mathbb{P}^{1}$ and $X\in\\{A,B\\}$. Let
$d\in\operatorname{Der}(\mathcal{W}_{X}(\lambda))$ such that
$d(L_{n})=\alpha_{n}L_{n}+\beta_{n}X_{n}$ and
$d(X_{n})=\eta_{n}L_{n}+\theta_{n}X_{n}$ for all $n\in\mathbb{Z}$, where
$\alpha_{n},\beta_{n},\eta_{n},\theta_{n}\in\mathbb{C}$. Then $\eta_{n}=0$ for
all $n\in\mathbb{Z}$.

###### Proof.

We split the proof into the cases $X=A$ and $X=B$.

Case 1: $X=A$.

By Lemma 6.10, we have

(6.5) $(m-n)\eta_{m}=(n+m+n(n+1)\lambda\delta_{m}^{0})\eta_{n+m}$

for all $n,m\in\mathbb{Z}$. Letting $m\in\mathbb{Z}\setminus\\{0\\}$ and
$n=-m$ in (6.5), we get $\eta_{m}=0$ for all $m\in\mathbb{Z}\setminus\\{0\\}$.
If we set $m=0,n\in\mathbb{Z}\setminus\\{0\\}$ in (6.5), we also deduce that
$\eta_{0}=0$. This proves the case $X=A$.

Case 2: $X=B$.

By Lemma 6.10, we have

(6.6) $(m-n)\eta_{m}=(m-n(n+1)\lambda\delta_{n+m}^{0})\eta_{n+m}$

for all $n,m\in\mathbb{Z}$. Setting $m=0$ and $n\in\mathbb{Z}\setminus\\{0\\}$
(6.6), we get $\eta_{0}=0$. If we let $m\in\mathbb{Z}\setminus\\{0\\}$ and
$n=-m$ in (6.6), we also deduce that $\eta_{m}=0$ for $m\neq 0$. This
concludes the proof. ∎

We are now ready to prove Theorem 6.7.

###### Proof of Theorem 6.7.

Let $d\in\operatorname{Der}(\mathcal{W}_{X}(\lambda))_{0}$. Then, for
$n\in\mathbb{Z}$ we have

$d(L_{n})=\alpha_{n}L_{n}+\beta_{n}X_{n},\quad
d(X_{n})=\eta_{n}L_{n}+\theta_{n}X_{n},$

for some $\alpha_{n},\beta_{n},\eta_{n},\theta_{n}\in\mathbb{C}$. By Lemma
6.11, $\eta_{n}=0$ for all $n\in\mathbb{Z}$. Furthermore, replacing $d$ with
$d-\theta_{0}d_{\operatorname{Ab}}$, where $d_{\operatorname{Ab}}$ is as in
Notation 6.3, we may assume that $\theta_{0}=0$.

By Lemma 6.10, we have $\alpha_{n}+\alpha_{m}=\alpha_{n+m}$ if $n\neq m$. Now,
$\alpha_{0}+\alpha_{1}=\alpha_{1}$, so $\alpha_{0}=0$. Furthermore,
$\alpha_{-n}+\alpha_{n}=\alpha_{0}=0$, so $\alpha_{-n}=-\alpha_{n}$ for all
$n\in\mathbb{Z}$. Now consider $2\alpha_{1}+\alpha_{3}$:

$2\alpha_{1}+\alpha_{3}=\alpha_{1}+\alpha_{4}=\alpha_{5}=\alpha_{2}+\alpha_{3}.$

Hence, $\alpha_{2}=2\alpha_{1}$. Inductively, we deduce that
$n\alpha_{1}=\alpha_{n}$ for all $n\in\mathbb{N}$. By a symmetric argument, we
can also deduce that $\alpha_{-n}=n\alpha_{-1}=-n\alpha_{1}$ for all
$n\in\mathbb{N}$. Therefore, $\alpha_{n}=n\alpha_{1}$ for all
$n\in\mathbb{Z}$. Note that
$\alpha_{1}\operatorname{ad}_{L_{0}}(L_{n})=n\alpha_{1}L_{n}=\alpha_{n}L_{n}$.
Therefore, replacing $d$ with
$d-\alpha_{1}\operatorname{ad}_{L_{0}}\in\operatorname{Der}(\mathcal{W}_{X}(\lambda))_{0}$,
we may assume that $\alpha_{n}=0$ for all $n\in\mathbb{Z}$.

We now analyse the cases $X=A$ and $X=B$ separately.

Case 1: $X=A$.

Replacing $d$ with
$d-\beta_{1}(2\delta_{\lambda}^{A}-\partial_{\lambda}^{A})+\frac{\beta_{2}}{2}(\delta_{\lambda}^{A}-\partial_{\lambda}^{A})$,
we may assume that $\beta_{1}=\beta_{2}=0$. We claim that $d=0$.

By Lemma 6.10, we have

(6.7)
$(n+m+n(n+1)\lambda\delta_{m}^{0})\beta_{m}-(n+m+m(m+1)\lambda\delta_{n}^{0})\beta_{n}=(m-n)\beta_{n+m}$

for all $n,m\in\mathbb{Z}$. Notice that this equation is the same as (3.18).
Since we assumed that $\beta_{1}=\beta_{2}=0$, we see that $\beta_{n}=0$ for
all $n\in\mathbb{Z}$, by the proof of Proposition 3.23.

We have now reduced to the case where $d(L_{n})=0$, while
$d(A_{n})=\theta_{n}A_{n}$ for all $n\in\mathbb{Z}$, where $\theta_{0}=0$. By
Lemma 6.10, we have

$(n+m+n(n+1)\lambda\delta_{m}^{0})\theta_{m}=(n+m+n(n+1)\lambda\delta_{m}^{0})\theta_{n+m}$

for all $n,m\in\mathbb{Z}$. Therefore, $\theta_{n}=\theta_{0}=0$ for all
$n\in\mathbb{Z}$.

This completes the proof in the case $X=A$. Indeed, what we have shown is that
if $d\in\operatorname{Der}(\mathcal{W}_{A}(\lambda))_{0}$ with

$d(L_{n})=\alpha_{n}L_{n}+\beta_{n}A_{n},\quad
d(A_{n})=\eta_{n}L_{n}+\theta_{n}A_{n},$

where $\alpha_{n},\beta_{n},\eta_{n},\theta_{n}\in\mathbb{C}$, then

$d=\alpha_{1}\operatorname{ad}_{L_{0}}+\theta_{0}d_{\operatorname{Ab}}+\beta_{1}(2\delta_{\lambda}^{A}-\partial_{\lambda}^{A})-\frac{\beta_{2}}{2}(\delta_{\lambda}^{A}-\partial_{\lambda}^{A})\in\operatorname{Inn}(\mathcal{W}_{A}(\lambda))+\mathbb{C}d_{\operatorname{Ab}}+\mathbb{C}\delta_{\lambda}^{A}+\mathbb{C}\partial_{\lambda}^{A}.$

Recall that, by Lemma 6.6, there is a linear combination of
$\delta_{\lambda}^{A}$ and $\partial_{\lambda}^{A}$ which gives an inner
derivation, so the above sum of vector spaces is not a direct sum. Indeed, by
Lemma 6.6, it follows that

$\operatorname{Inn}(\mathcal{W}_{A}(\lambda))+\mathbb{C}d_{\operatorname{Ab}}+\mathbb{C}\delta_{\lambda}^{A}+\mathbb{C}\partial_{\lambda}^{A}=\operatorname{Inn}(\mathcal{W}_{A}(\lambda))\oplus\mathbb{C}d_{\operatorname{Ab}}\oplus\mathbb{C}d_{\lambda}^{A}.$

The result for $X=A$ now follows by Corollary 6.9.

Case 2: $X=B$.

If $\lambda=0$ then $\mathcal{W}_{B}(0)\cong\mathcal{W}(0,0)$, so the result
follows by [GJP11, Theorem 4.7]. Therefore, we assume that $\lambda\neq 0$.

Replacing $d$ with $d-\beta_{1}d_{\lambda}^{B}$, we may assume that
$\beta_{1}=0$. By Lemma 6.10, we have

$(-n+m(m+1)\lambda\delta_{n+m}^{0})\beta_{n}+(m-n(n+1)\lambda\delta_{n+m}^{0})\beta_{m}=(m-n)\beta_{n+m}$

for all $n,m\in\mathbb{Z}$. Notice that this equation is the same as (3.16).
By the proof of Proposition 3.19, it follows that $\beta_{n}=0$ for all
$n\in\mathbb{Z}$.

We have now reduced to the case where $d(L_{n})=0$, while
$d(B_{n})=\theta_{n}B_{n}$ for all $n\in\mathbb{Z}$, where $\theta_{0}=0$. By
Lemma 6.10, we have

$(m-n(n+1)\lambda\delta_{n+m}^{0})\theta_{m}=(m-n(n+1)\lambda\delta_{n+m}^{0})\theta_{n+m}$

for all $n,m\in\mathbb{Z}$. Therefore, $\theta_{n}=\theta_{0}=0$ for all
$n\in\mathbb{Z}$.

In this case, we showed that if $\lambda\neq 0$ and
$d\in\operatorname{Der}(\mathcal{W}_{B}(\lambda))_{0}$ with

$d(L_{n})=\alpha_{n}L_{n}+\beta_{n}X_{n},\quad
d(X_{n})=\eta_{n}L_{n}+\theta_{n}X_{n},$

where $\alpha_{n},\beta_{n},\eta_{n},\theta_{n}\in\mathbb{C}$, then

$d=\alpha_{1}\operatorname{ad}_{L_{0}}+\theta_{0}d_{\operatorname{Ab}}+\beta_{1}d_{\lambda}^{B}\in\operatorname{Inn}(\mathcal{W}_{B}(\lambda))\oplus\mathbb{C}d_{\operatorname{Ab}}\oplus\mathbb{C}d_{\lambda}^{B},$

where we used Lemma 6.6 to conclude that the sum is direct. Using [GJP11,
Theorem 4.7] for the case $\lambda=0$ and applying Corollary 6.9, this
completes the proof. ∎

## References

  * [Aiz12] N. Aizawa, _Some representations of planar Galilean conformal algebra_ , 7th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, 2012, arXiv:1212.6288.
  * [AR16] Dražen Adamović and Gordan Radobolja, _On free field realizations of $W(2,2)$-modules_, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 113, 13 pp.
  * [Bav16] Vladimir Bavula, _The groups of automorphisms of the Witt $W_{n}$ and Virasoro Lie algebras_, Czechoslovak Math. J. 66 (141) (2016), no. 4, 1129–1141.
  * [BBC+20] Arjun Bagchi, Aritra Banerjee, Shankhadeep Chakrabortty, Sudipta Dutta, and Pulastya Parekh, _A tale of three—tensionless strings and vacuum structure_ , J. High Energy Phys. (2020), no. 4, Paper No. 061, 52 pp.
  * [BBGR15] Arjun Bagchi, Rudranil Basu, Daniel Grumiller, and Max Riegler, _Entanglement entropy in Galilean conformal field theories and flat holography_ , Phys. Rev. Lett. 114 (2015), Paper No. 111602, 6 pp.
  * [BCG20] Carles Batlle, Víctor Campello, and Joaquim Gomis, _A canonical realization of the Weyl BMS symmetry_ , Phys. Lett. B 811 (2020), Paper No. 135920, 5 pp.
  * [BG09] Arjun Bagchi and Rajesh Gopakumar, _Galilean conformal algebras and AdS/CFT_ , J. High Energy Phys. (2009), no. 7, Paper No. 037, 22 pp.
  * [BH86] J. D. Brown and Marc Henneaux, _Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity_ , Comm. Math. Phys. 104 (1986), no. 2, 207–226.
  * [BvdBM62] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, _Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems_ , Proc. Roy. Soc. London Ser. A 269 (1962), 21–52.
  * [CT16] Eduardo Casali and Piotr Tourkine, _On the null origin of the ambitwistor string_ , J. High Energy Phys. (2016), no. 11, Paper No. 036, front matter + 22 pp.
  * [DBFK24] Patrícia Damas Beites, Bruno Leonardo Macedo Ferreira, and Ivan Kaygorodov, _Transposed Poisson structures_ , Results Math. 79 (2024), no. 2, Paper No. 93, 29 pp.
  * [DCL24] Munayim Dilxat, Liangyun Chen, and Dong Liu, _Classification of simple Harish-Chandra modules over the Ovsienko-Roger superalgebra_ , Proc. Roy. Soc. Edinburgh Sect. A 154 (2024), no. 2, 483–493.
  * [FGP+21] Oscar Fuentealba, Hernán A. González, Alfredo Pérez, David Tempo, and Ricardo Troncoso, _Superconformal Bondi-Metzner-Sachs algebra in three dimensions_ , Phys. Rev. Lett. 126 (2021), no. 9, Paper No. 091602, 9 pp.
  * [FKL21] Bruno Leonardo Macedo Ferreira, Ivan Kaygorodov, and Viktor Lopatkin, _$\frac{1}{2}$ -derivations of Lie algebras and transposed Poisson algebras_, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021), no. 3, Paper No. 142, 19 pp.
  * [FMP+18] Oscar Fuentealba, Javier Matulich, Alfredo Pérez, Miguel Pino, Pablo Rodríguez, David Tempo, and Ricardo Troncoso, _Integrable systems with $\rm BMS_{3}$ Poisson structure and the dynamics of locally flat spacetimes_, J. High Energy Phys. (2018), no. 1, Paper No. 148, front matter \+ 36 pp.
  * [FOPS21] Laurent Freidel, Roberto Oliveri, Daniele Pranzetti, and Simone Speziale, _The Weyl BMS group and Einstein’s equations_ , J. High Energy Phys. (2021), no. 7, Paper No. 170, 63 pp.
  * [FOV24] José Figueroa-O’Farrill and Girish S Vishwa, _The BRST quantisation of chiral BMS-like field theories_ , arXiv:2407.12778.
  * [Fuk86] D. B. Fuks, _Cohomology of infinite-dimensional Lie algebras_ , Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986, Translated from the Russian by A. B. Sosinskiĭ.
  * [FW21] Jörg Feldvoss and Friedrich Wagemann, _On Leibniz cohomology_ , J. Algebra 569 (2021), 276–317.
  * [GJP11] Shoulan Gao, Cuipo Jiang, and Yufeng Pei, _Low-dimensional cohomology groups of the Lie algebras $W(a,b)$_, Comm. Algebra 39 (2011), no. 2, 397–423.
  * [GLP16] Shoulan Gao, Dong Liu, and Yufeng Pei, _Structure of the planar Galilean conformal algebra_ , Rep. Math. Phys. 78 (2016), no. 1, 107–122.
  * [GMZ22] Dongfang Gao, Yao Ma, and Kaiming Zhao, _Non-weight modules over the mirror Heisenberg-Virasoro algebra_ , Sci. China Math. 65 (2022), no. 11, 2243–2254.
  * [GR23] Daniel Grumiller and Max Riegler, _Carrollian $c$ functions and flat space holographic RG flows in ${\rm BMS}_{3}/{\rm CCFT}_{2}$_, Phys. Rev. D 108 (2023), no. 12, Paper No. 126008, 11 pp.
  * [HCS17] Jianzhi Han, Qiufan Chen, and Yucai Su, _Modules over the algebra $\mathcal{Vir}(a,b)$_, Linear Algebra Appl. 515 (2017), 11–23.
  * [HPL08] Naihong Hu, Yufeng Pei, and Dong Liu, _A cohomological characterization of Leibniz central extensions of Lie algebras_ , Proc. Amer. Math. Soc. 136 (2008), no. 2, 437–447.
  * [HPS16] Stephen W. Hawking, Malcolm J. Perry, and Andrew Strominger, _Soft hair on black holes_ , Phys. Rev. Lett. 116 (2016), no. 23, Paper No. 231301.
  * [HPS17] by same author, _Superrotation charge and supertranslation hair on black holes_ , J. High Energy Phys. (2017), no. 5, Paper No. 161, front matter + 32 pp.
  * [JPZ18] Wei Jiang, Yufeng Pei, and Wei Zhang, _Determinant formula and a realization for the Lie algebra $W(2,2)$_, Sci. China Math. 61 (2018), no. 4, 685–694.
  * [JZ15] Wei Jiang and Wei Zhang, _Verma modules over the $W(2,2)$ algebras_, J. Geom. Phys. 98 (2015), 118–127.
  * [KS85] Irving Kaplansky and L. J. Santharoubane, _Harish-Chandra modules over the Virasoro algebra_ , Infinite-dimensional groups with applications (Berkeley, Calif., 1984), Math. Sci. Res. Inst. Publ., vol. 4, Springer, New York, 1985, pp. 217–231.
  * [Lod98] Jean-Louis Loday, _Cyclic homology_ , second ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili.
  * [LP93] Jean-Louis Loday and Teimuraz Pirashvili, _Universal enveloping algebras of Leibniz algebras and (co)homology_ , Math. Ann. 296 (1993), no. 1, 139–158.
  * [LPXZ24] Dong Liu, Yufeng Pei, Limeng Xia, and Kaiming Zhao, _Smooth modules over the $n=1$ Bondi–Metzner–Sachs superalgebra_, Commun. Contemp. Math. electronic (2024), Paper No. 2450021.
  * [MS14] Lionel Mason and David Skinner, _Ambitwistor strings and the scattering equations_ , J. High Energy Phys. 2014 (2014), no. 7, Paper No. 048, front matter + 33 pp.
  * [OR96] V. Yu. Ovsienko and K. Rozhe, _Extensions of the Virasoro group and the Virasoro algebra by means of modules of tensor densities on $S^{1}$_, Funktsional. Anal. i Prilozhen. 30 (1996), no. 4, 86–88.
  * [Rad13] Gordan Radobolja, _Subsingular vectors in Verma modules, and tensor product of weight modules over the twisted Heisenberg-Virasoro algebra and $W(2,2)$ algebra_, J. Math. Phys. 54 (2013), no. 7, Paper No. 071701, 24 pp.
  * [RR17] Jørgen Rasmussen and Christopher Raymond, _Galilean contractions of $W$-algebras_, Nuclear Phys. B 922 (2017), 435–479.
  * [RR19] by same author, _Higher-order Galilean contractions_ , Nuclear Phys. B 945 (2019), Paper No. 114680, 15 pp.
  * [Sch08] M. Schottenloher, _A mathematical introduction to conformal field theory_ , second ed., Lecture Notes in Physics, vol. 759, Springer-Verlag, Berlin, 2008\.
  * [SŠ17] Susan J. Sierra and Špela Špenko, _Generalised Witt algebras and idealizers_ , J. Algebra 483 (2017), 415–428.
  * [Su94] Yu Cai Su, _Harish-Chandra modules of the intermediate series over the high rank Virasoro algebras and high rank super-Virasoro algebras_ , J. Math. Phys. 35 (1994), no. 4, 2013–2023.
  * [SZ16] Andrew Strominger and Alexander Zhiboedov, _Gravitational memory, BMS supertranslations and soft theorems_ , J. High Energy Phys. (2016), no. 1, Paper No. 086, front matter + 14 pp.
  * [TYZ23] Haijun Tan, Yufeng Yao, and Kaiming Zhao, _Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules_ , 2023, arXiv:2110.05714.
  * [Wei94] Charles A. Weibel, _An introduction to homological algebra_ , Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994.
  * [ZD09] Wei Zhang and Chongying Dong, _$W$ -algebra $W(2,2)$ and the vertex operator algebra $L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)$_, Comm. Math. Phys. 285 (2009), no. 3, 991–1004.


# Strong conciseness of coprime commutators in profinite groups

Iker de las Heras  Iker de las Heras: Mathematisches Institut, Heinrich-Heine-
Universität, 40225 Düsseldorf, Germany; Department of Mathematics, Euskal
Herriko Unibertsitatea UPV/EHU, 48940 Leioa, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Matteo Pintonello  Matteo Pintonello: Department of
Mathematics, Euskal Herriko Unibertsitatea UPV/EHU, 48940 Leioa, Spain
<EMAIL_ADDRESS>and  Pavel Shumyatsky  Pavel Shumyatsky: Department
of Mathematics, University of Brasilia, Brasilia DF, Brazil<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a profinite group. The coprime commutators $\gamma_{j}^{*}$ and
$\delta_{j}^{*}$ are defined as follows. Every element of $G$ is both a
$\gamma_{1}^{*}$-value and a $\delta_{0}^{*}$-value. For $j\geq 2$, let $X$ be
the set of all elements of $G$ that are powers of $\gamma_{j-1}^{*}$-values.
An element $a$ is a $\gamma_{j}^{*}$-value if there exist $x\in X$ and $g\in
G$ such that $a=[x,g]$ and $(|x|,|g|)=1$. For $j\geq 1$, let $Y$ be the set of
all elements of $G$ that are powers of $\delta_{j-1}^{*}$-values. The element
$a$ is a $\delta_{j}^{*}$-value if there exist $x,y\in Y$ such that $a=[x,y]$
and $(|x|,|y|)=1$.

In this paper we establish the following results.

A profinite group $G$ is finite-by-pronilpotent if and only if there is $k$
such that the set of $\gamma_{k}^{*}$-values in $G$ has cardinality less than
$2^{\aleph_{0}}$ (Theorem 1.1).

A profinite group $G$ is finite-by-(prosoluble of Fitting height at most $k$)
if and only if there is $k$ such that the set of $\delta_{k}^{*}$-values in
$G$ has cardinality less than $2^{\aleph_{0}}$ (Theorem 1.2).

###### Key words and phrases:

Conciseness, strong conciseness, profinite groups, word problems

###### 2010 Mathematics Subject Classification:

Primary 20E18; Secondary 28A78

The first and second authors are supported by the Spanish Government project
PID2020-117281GB-I00, partially by FEDER funds and by the Basque Government
project IT483-22. The first author has also received funding from the European
Union’s Horizon 2021 research and innovation programme under the Marie
Sklodowska-Curie grant agreement, project 101067088. The second author is also
supported by a grant FPI-2018 of the Spanish Government. The third author was
supported by FAPDF and CNPq.

##  1\. Introduction

A group word $w$ is called concise in the class of groups $\mathcal{C}$ if the
verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values
in a group $G\in\mathcal{C}$. In the sixties Hall raised the problem whether
every word is concise in the class of all groups but in 1989 S. Ivanov [17]
solved the problem in the negative (see also [23, p. 439]). On the other hand,
the problem for residually finite groups remains open (cf. Segal [24, p. 15]
or Jaikin-Zapirain [18]). In recent years several new positive results with
respect to this problem were obtained (see [2, 15, 14, 6, 8, 7, 11]).

A natural variation of the notion of conciseness for profinite groups was
introduced in [9]: the word $w$ is strongly concise in a class of profinite
groups $\mathcal{C}$ if the verbal subgroup $w(G)$ is finite in any group
$G\in\mathcal{C}$ in which $w$ takes less than $2^{\aleph_{0}}$ values. Here
and throughout the paper, whenever $G$ is a profinite group we write $w(G)$ to
denote the closed subgroup generated by $w$-values. A number of new results on
strong conciseness of group words can be found in [9, 13, 3, 20].

It was noted in [10] that the concept of (strong) conciseness can be applied
in a wider context. Suppose $\mathcal{C}$ is a class of profinite groups and
$\phi(G)$ is a subset of $G$ for every $G\in\mathcal{C}$. Is the subgroup
generated by $\phi(G)$ finite whenever $|\phi(G)|<2^{\aleph_{0}}$? This
question is interesting whenever $\phi(G)$ is defined in some natural way
and/or properties of the subgroup $\langle\phi(G)\rangle$ have strong impact
on the structure of $G$. It was shown in [10] that the set of coprime
commutators in a profinite group is strongly concise.

Given a profinite group $G$ and an element $x\in G$, we denote by $|x|$
(respectively $|G|$) the order of $x$ (respectively $G$) as a supernatural
number (or Stenitz number), and $\pi(x)$ (respectively $\pi(G)$) will stand
for the set of prime numbers dividing $|x|$ (respectively $|G|$). An element
$a\in G$ is a coprime commutator if there are $x,y\in G$ such that $a=[x,y]$
and $(|x|,|y|)=1$. Here, as usual, $[x,y]$ stands for $x^{-1}y^{-1}x,y$. It is
well-known that the set of coprime commutators in a profinite group $G$
generates the nilpotent residual $\gamma_{\infty}(G)$, that is, the smallest
normal subgroup $N$ such that $G/N$ is pronilpotent. Of course,
$\gamma_{\infty}(G)=\cap_{i}\gamma_{i}(G)$ is the intersection of the lower
central series of $G$.

Coprime commutators of higher order in finite groups were defined in [25]. The
definition naturally extends to the profinite case. Let $G$ be a profinite
group. Every element of $G$ is both a $\gamma_{1}^{*}$-value and a
$\delta_{0}^{*}$-value. For $j\geq 2$, let $X$ be the set of all elements of
$G$ that are powers (in the profinite sense) of $\gamma_{j-1}^{*}$-values. An
element $a$ is a $\gamma_{j}^{*}$-value if there exist $x\in X$ and $g\in G$
such that $a=[x,g]$ and $(|x|,|g|)=1$. For $j\geq 1$, let $Y$ be the set of
all elements of $G$ that are powers (in the profinite sense) of
$\delta_{j-1}^{*}$-values. The element $a$ is a $\delta_{j}^{*}$-value if
there exist $x,y\in Y$ such that $a=[x,y]$ and $(|x|,|y|)=1$.

Set $D_{1}(G)=\gamma_{\infty}(G)$ and $D_{i}(G)=\gamma_{\infty}(D_{i-1})$ for
$i\geq 2$. It was shown in [25] that for any $j$ we have the equalities
$\gamma_{j}^{*}(G)=D_{1}(G)$ and $\delta_{j}^{*}(G)=D_{j}(G)$. Thus,
$\gamma_{j}^{*}(G)=1$ if and only if $G$ is pronilpotent while
$\delta_{j}^{*}(G)=1$ if and only if $G$ is prosoluble and has Fitting height
at most $j$ (we say that a profinite group has Fitting height at most $h$ if
it has a normal series of length $h$ all of whose factors are pronilpotent).

The main results in this paper are as follows.

######  Theorem 1.1.

A profinite group $G$ is finite-by-pronilpotent if and only if there is $k$
such that the set of $\gamma_{k}^{*}$-values in $G$ has cardinality less than
$2^{\aleph_{0}}$.

######  Theorem 1.2.

A profinite group $G$ is finite-by-(prosoluble of Fitting height at most $k$)
if and only if there is $k$ such that the set of $\delta_{k}^{*}$-values in
$G$ has cardinality less than $2^{\aleph_{0}}$.

In the case where the set of $\gamma_{k}^{*}$-values (respectively,
$\delta_{j}^{*}$-values) in a profinite group is finite of size $m$ the
subgroup generated by the set is finite of order bounded in terms of $m$ only.
This was established for finite groups in [1] and can be extended to profinite
groups in a straightforward manner.

##  2\. Preliminaries

For a subgroup $H$ of a profinite group $G$, we denote by $H^{G}$ the normal
closure of $H$ in $G$, that is, the minimal normal subgroup of $G$ containing
$H$. For an element $g\in G$ we denote by $g^{G}$ the conjugacy class of $g$
in $G$. We use the left-normed notation for commutators; for example,
$[x,y,z]=[[x,y],z]$. If $X$ and $Y$ are subsets of $G$, then $[X,Y]$ stands
for the subgroup generated by commutators $[x,y]$, where $x\in X$ and $y\in
Y$. If $X=\\{x\\}$, then we just write $[x,Y]$. If $U$ is an open normal
subgroup of $G$, we write $U\trianglelefteq_{o}G$. All subgroups of profinite
groups are assumed to be closed.

One can easily see that if $N$ is a normal subgroup of $G$ and $x$ an element
whose image in $G/N$ is a $\gamma_{j}^{*}$-value (respectively, a
$\delta_{j}^{*}$-value), then there exists a $\gamma_{j}^{*}$-value $y$ in $G$
(respectively, a $\delta_{j}^{*}$-value) such that $x\in yN$. This observation
will be used throughout the paper without explicit references.

We begin by quoting a theorem that was already mentioned in the introduction.
It was established in [25] for finite groups. The variant for profinite groups
is an obvious modification of the finite case.

######  Theorem 2.1 ([25] Theorems 2.1, 2.7).

Let $G$ be a profinite group. The subgroup $\gamma_{k}^{*}(G)$ is trivial if
and only if $G$ is pronilpotent. The subgroup $\delta_{k}^{*}(G)$ is trivial
if and only if $G$ is prosoluble of Fitting height at most $k$.

The following results will be helpful.

######  Proposition 2.2 ([9] Lemma 2.1).

Let $\varphi:X\to Y$ be a continuous map between two non-empty profinite
spaces that is nowhere locally constant (i.e. there is no non-empty open
subset $U\subseteq_{o}X$ where $\varphi|_{U}$ is constant). Then
$|\varphi(X)|\geq 2^{\aleph_{0}}$.

######  Lemma 2.3 ([9] Lemma 2.2).

Let $G$ be a profinite group and $g\in G$ be an element whose conjugacy class
$g^{G}$ contains less than $2^{\aleph_{0}}$ elements. Then $g^{G}$ is finite.

It is well-known that if $A$ is a group of automorphisms of a finite group $G$
with $(|A|,|G|)=1$, then $[G,A]=[G,A,A]$. The following lemma is a stronger
version of this result for the case where $G$ is a pronilpotent group.

######  Lemma 2.4 ([19] Lemma 4.6).

Let $\varphi$ be a coprime automorphism of a pronilpotent group $G$. Then the
restriction of the mapping

$\theta:x\to[x,\varphi]$

to the set $S=\\{[g,\varphi]|g\in G\\}$ is bijective.

The original version of Lemma 2.4 states that the continuous map $\theta$ is
injective when restricted to the set $S$. However, since $S$ is closed and
$\theta(S)\subseteq S$, the surjectivity of $\theta$ follows immediately.

The following is a profinite version of Lemma 2.4 in [25]. It will be used
together with Lemma 2.4 in order to show that certain elements of a group $G$
are $\delta_{k}^{*}$-values.

######  Lemma 2.5.

Let $G$ be a profinite group and $g_{1},\ldots,g_{k}$ be
$\delta^{*}_{k-1}$-values in $G$. Suppose that $g_{1},\ldots,g_{k}\in
N_{G}(H)$ for a subgroup $H\leq G$ with $(|H|,|g_{i}|)=1$ for every
$i\in\\{1,\ldots,k\\}$. Then, for every $h\in H$, the element
$[h,g_{1},\ldots,g_{k}]$ is a $\delta_{k}^{*}$-value.

The next result is a profinite version of Lemma 2.4 in [1]. By a meta-
pronilpotent group we mean a profinite group $G$ having a normal pronilpotent
subgroup $N$ such that $G/N$ is pronilpotent.

######  Lemma 2.6.

Let $G$ be a meta-pronilpotent group. Then
$\gamma_{\infty}(G)=\prod_{p}[K_{p},H_{p^{\prime}}]$, where $K_{p}$ is a Sylow
$p$-subgroup of $\gamma_{\infty}(G)$ and $H_{p^{\prime}}$ is a Hall
$p^{\prime}$-subgroup of $G$.

For a general group word $w$, the set $G_{w}$ of $w$-values of a profinite
group $G$ is always closed in $G$. We will show that the same is true for the
sets of $\gamma_{k}^{*}$ and $\delta_{k}^{*}$-values.

######  Proposition 2.7.

Let $S_{1},\ldots,S_{t}$ be closed subsets of a profinite group $G$. Then the
set

$C=\\{(g_{1},\ldots,g_{t})\in S_{1}\times\cdots\times
S_{t}\mid(|g_{i}|,|g_{i+1}|)=1\text{ for all }i=1,\ldots,t-1\\}$

is closed in $S_{1}\times\cdots\times S_{t}$. Furthermore, the sets
$G_{\gamma_{k}^{*}}$ and $G_{\delta_{k}^{*}}$ are closed in $G$.

###### Proof.

Let $\mathcal{P}$ be the set of all primes and $p\in\mathcal{P}$. First notice
that for every closed subset $S$ of $G$ the set

$S_{p^{\prime}}=\\{g\in S\mid p\notin\pi(g)\\}$

is closed since $S_{p^{\prime}}=\cap_{N\trianglelefteq_{o}G}S_{p^{\prime}}N$.
Also, the set $S^{\widehat{\mathbb{Z}}}=\\{g^{\lambda}\mid g\in
S,\lambda\in\widehat{\mathbb{Z}}\\}$ is the image under the continuous map
$f(g,\lambda)=g^{\lambda}$ of the compact set $S\times\widehat{\mathbb{Z}}$,
so it is closed too.

Let now $A,B$ be subsets of $G$. We claim that the set

$R_{A,B}=\bigcap_{p\in\mathcal{P}}\big{(}(A\times
B_{p^{\prime}})\cup(A_{p^{\prime}}\times B)\big{)}$ (2.1)

is exactly the set of elements $(a,b)\in A\times B$ with $|a|$ and $|b|$
coprime. On the one hand, if $|a|$ and $|b|$ are coprime then
$(a,b)\in(A\times B_{p^{\prime}})\cup(A_{p^{\prime}}\times B)$ for every
$p\in\mathcal{P}$, because, if $b\in B\smallsetminus B_{p^{\prime}}$, then
$a\in A_{p^{\prime}}$ necessarily. On the other hand, if $(a,b)\in R_{A,B}$
and a prime $p$ divides $|a|$, then $(a,b)\in(A,B_{p^{\prime}})$ so $p$ does
not divide $|b|$, and the claim follows. Notice now that if $A$ and $B$ are
closed, the set $R_{A,B}$ is an intersection of closed subsets of $G\times G$
so it is closed too.

It is now easy to prove by induction on $k$ that the sets
$G_{\gamma_{k}^{*}}$, $G_{\delta_{k}^{*}}$ are closed: just note that
$G_{\gamma_{k}^{*}}$ is exactly the set $R_{A,B}$ in (2.1) with
$A=G_{\gamma_{k-1}^{*}}^{\widehat{\mathbb{Z}}}$, $B=G$, whereas
$G_{\delta_{k}^{*}}$ is the set $R_{A,B}$ in (2.1) with
$A=B=G_{\delta_{k-1}^{*}}^{\widehat{\mathbb{Z}}}$.

To prove that the set $C$ is closed in $S_{1}\times\cdots\times S_{t}$, it
suffices to notice that by the above arguments the set

$C_{i}=S_{1}\times\cdots\times S_{i-1}\times R_{S_{i},S_{i+1}}\times
S_{i+2}\times\cdots\times S_{t}$

is closed for every $i\in\\{1,\ldots,t-1\\}$ and $C=\bigcap_{i=1}^{t-1}C_{i}$.
∎

##  3\. Coprime commutators in prosoluble groups

In this section we will prove some particular cases of Theorems 1.1 and 1.2.

First we will prove Theorem 1.1 in the case when the profinite group $G$ is
meta-pronilpotent. The second part of the section will be devoted to the proof
of Theorem 1.2 in the case where $G$ is poly-pronilpotent.

We first start with a general reduction.

######  Lemma 3.1.

Let $\phi$ be a map that associates to every group $G$ a normal subset
$\phi(G)\subseteq G$. Let $G$ be a profinite group with
$|\phi(G)|<2^{\aleph_{0}}$ and let $K$ be a pronilpotent subgroup of
$\langle\phi(G)\rangle$ generated by a subset of $\phi(G)$. If $K/K^{\prime}$
is finite, then $K$ is finite.

###### Proof.

Since $K$ is pronilpotent, $K^{\prime}\leq\Phi(K)$, where $\Phi(K)$ stands for
the Frattini subgroup of $K$. Thus $K/\Phi(K)$ is finite, and hence we can
find a finite subset $S$ of $\phi(G)$ generating $K$. Since $\phi(G)$ is a
normal subset of $G$, by Lemma 2.3 each of these generators has finitely many
conjugates in $G$, so in particular $|G:C_{G}(s)|\leq\infty$ for every $s\in
S$. Since $C_{G}(K)=\cap_{s\in S}C_{G}(s)$, this implies that $Z(K)=K\cap
C_{G}(K)$ has finite index in $K$, and by Schur’s theorem $K^{\prime}$ is
finite. ∎

We will use Lemma 3.1 for $\phi=\gamma_{k}^{*}$ or $\phi=\delta_{k}^{*}$, but
it could be applied to other cases, such as group word maps or uniform (anti-
coprime) commutators (see [10] or [12]).

###  3.1. The meta-pronilpotent case for $\gamma_{k}^{*}$

The next lemma is given without a proof since this is an obvious modification
of Lemma 3.1 in [10].

######  Lemma 3.2.

Let $G$ be a profinite group that is a product of a subgroup $H$ and a normal
pronilpotent subgroup $Q$ with $(|H|,|Q|)=1$. Suppose that

$|\\{[h,q]\mid h\in H,q\in Q\\}|<2^{\aleph_{0}}.$

Then $[H,Q]$ is finite.

Similarly, by using Lemma 3.2 in place of Lemma 3.1 of [10], we can refine the
proof of Lemma 3.4 of [10] and obtain the following version.

######  Lemma 3.3.

Let $G$ be a meta-pronilpotent group with

$|\\{[g,h]\mid g\in G,h\in\gamma_{\infty}(G),(|g|,|h|)=1\\}|<2^{\aleph_{0}}.$

Then $\gamma_{\infty}(G)$ is finite.

We are now ready to prove the strong conciseness of $\gamma_{k}^{*}$ in meta-
pronilpotent groups.

######  Proposition 3.4.

Let $G$ be a meta-pronilpotent group with
$|G_{\gamma_{k}^{*}}|<2^{\aleph_{0}}$. Then $\gamma_{\infty}(G)$ is finite.

###### Proof.

Let $g\in G$ and $h\in\gamma_{\infty}(G)$ such that $(|g|,|h|)=1$, and let $H$
be the minimal Hall subgroup of $\gamma_{\infty}(G)$ containing $h$. Since $H$
is pronilpotent, Lemma 2.4 shows that there exists $h^{\prime}\in H$ such that
$[h,g]=[h^{\prime},g,\overset{k-1}{\ldots},g]$, and therefore $[h,g]\in
G_{\gamma_{k}^{*}(G)}$. Hence, we have

$|\\{[g,h]\mid g\in G,h\in\gamma_{\infty}(G),(|g|,|h|)=1\\}|<2^{\aleph_{0}},$

and we conclude by Lemma 3.3. ∎

###  3.2. The poly-pronilpotent case for $\delta_{k}^{*}$

Recall that a Sylow basis of a profinite group $G$ is a family $\\{P_{i}\\}$
of Sylow subgroups of $G$, one for each prime in $\pi(G)$, such that
$P_{i}P_{j}=P_{j}P_{i}$ for every $i,j$. The normalizer of a Sylow basis,
namely $T=\bigcap_{i}N_{G}(P_{i})$, is pronilpotent and satisfies
$G=T\gamma_{\infty}(G)$. Basic results on Sylow bases can be found in Section
9.2 of [21]. It is well-known that any prosoluble group admits a Sylow basis
(see Proposition 2.3.9 of [22]). Moreover, if $G$ is meta-pronilpotent, it is
easy to see that $\gamma_{\infty}(G)=[T,\gamma_{\infty}(G)]$.

For every profinite group $G$, the subgroup $\delta^{*}_{k}(G)$ coincides with
the nilpotent residual $\gamma_{\infty}(\delta_{k-1}^{*}(G))$ of
$\delta_{k-1}^{*}(G)$ (see [25]). Let $\\{P_{i}\\}$ be a Sylow basis of $G$
and observe that $\\{P_{i}\cap\delta_{j}^{*}(G)\\}$ is a Sylow basis of
$\delta_{j}^{*}(G)$ for every $j\geq 0$. Let $T_{j}$ be the normalizer in
$\delta_{j}^{*}(G)$ of the Sylow basis $\\{P_{i}\cap\delta_{j}^{*}(G)\\}$, so
that $G=T_{0}T_{1}\cdots T_{k}\delta^{*}_{k+1}(G)$ for every $k\geq 1$ and
$T_{j}\leq N_{G}(T_{i})$ for every $j\leq i$. In particular, if $G$ is a
prosoluble group of Fitting height $k$, then $\delta_{k+1}^{*}(G)=1$, and
therefore $G=T_{0}\cdots T_{k}$. In this case, we have
$T_{k}=\delta_{k}^{*}(G)$ and
$T_{i}=[T_{0},\ldots,T_{i}]\pmod{\delta_{i+1}^{*}(G)}$ for all
$i\in\\{0,\ldots,k\\}$.

For this reason, we will often work with subgroups $G_{1},\ldots,G_{t}$, for a
positive integer $t$, of a profinite group $G$ such that $G_{j}\leq
N_{G}(G_{i})$ for every $j\leq i$; in this setting, we will write

$\begin{array}[]{cccc}\varphi:&G_{1}\times\cdots\times
G_{t}&\longrightarrow&G_{t}\\\
&(g_{1},\ldots,g_{t})&\longmapsto&[g_{1},\ldots,g_{t}].\end{array}$

Let

$\mathcal{C}=\\{(g_{1},...,g_{t})\in G_{1}\times\cdots\times
G_{t}\mid(|g_{i}|,|g_{i+1}|)=1\\};$ (3.1)

for $S_{i}\subseteq G_{i}$, $i\in\\{1,\ldots,t\\}$, we define the set

$\varphi^{*}(S_{1},\ldots,S_{t})=\varphi((S_{1}\times\cdots\times
S_{t})\cap\mathcal{C}).$

The subgroups $G_{1},\ldots,G_{t}$ for which the definitions of $\varphi$ and
$\varphi^{*}$ apply will be clear from the context.

For $i\in\\{1,\ldots,t\\}$, let $X_{i},Y_{i}\subseteq G_{i}$. Similar to [9],
for $J\subseteq\\{1,\ldots,t\\}$ we can define the set

$\varphi_{J}(X_{i};Y_{i})=\varphi(Z_{1},\ldots,Z_{t})\quad\text{ with }\quad
Z_{i}=\left\\{\begin{array}[]{ll}X_{i}&\text{if }\ i\in J,\\\ Y_{i}&\text{if
}\ i\not\in J.\end{array}\right.$

Notice that in order for $\varphi_{J}$ to be well-defined, we just need the
subsets $X_{i}$ where $i\in J$ and the subsets $Y_{i}$ where $i\not\in J$. In
a similar way, define the set

$\varphi_{J}^{*}(X_{i};Y_{i})=\varphi((Z_{1}\times\cdots\times
Z_{t})\cap\mathcal{C}).$

If $J=\\{1,\ldots,t\\}$, by abuse of notation, we will just write
$\varphi_{J}(X_{i};Y_{i})=\varphi(X_{i})$ and
$\varphi_{J}^{*}(X_{i};Y_{i})=\varphi^{*}(X_{i})$.

######  Remark 3.5.

Notice that whenever we have subgroups $G_{1},\ldots,G_{\ell}$ of a profinite
group $G$ with $G_{j}\leq N_{G}(G_{i})$ for every $j\leq i$, and we take an
open subgroup $U\trianglelefteq_{o}G_{\ell}$, there exists an open normal
subgroup $V\trianglelefteq_{o}G$ such that $V\cap G_{\ell}\leq U$. This
implies that $V\cap G_{\ell}\trianglelefteq G_{1}\cdots G_{\ell}$ and
$V\trianglelefteq_{o}G_{\ell}$.

The next two lemmas are useful applications of basic commutator calculus. The
first one follows the ideas of Lemma 2.8 of [5] while Lemma 3.7 is an
application of Lemma 3.6 to coprime commutators.

######  Lemma 3.6.

Let $G_{1},\ldots,G_{t}$ be subgroups of a profinite group $G$ such that
$G_{j}\leq N_{G}(G_{i})$ for every $j\leq i$. For every $i\in\\{1,\ldots,t\\}$
let $g_{i}\in G_{i}$, and for a fixed $\ell\in\\{1,\ldots,t\\}$, let
$g_{\ell}^{\prime}\in G_{\ell}$. Then

$\varphi_{\\{\ell\\}}(g^{\prime}_{\ell}g_{\ell};g_{i})=[g_{1},\ldots,g_{\ell-1},g_{\ell}^{\prime},g_{\ell+1}^{h_{\ell}},\ldots,g_{t}^{h_{t-1}}]^{h_{t}}\varphi(g_{i}),$

where $h_{i}\in G_{\ell}\cdots G_{i}$ for $i\in\\{\ell,\ldots,t\\}$. In
particular $g_{i+1}^{h_{i}}\in G_{i+1}$ for all $i\in\\{\ell,\ldots,t-1\\}$

###### Proof.

Assume first that $\ell\neq 1$, and proceed by induction on $t-\ell$. If
$t-\ell=0$, then

$[g_{1},\ldots,g_{t-1},g^{\prime}_{t}g_{t}]=[g_{1},\ldots,g^{\prime}_{t}]^{{g_{t}}[g_{1},\ldots,g_{t}]^{-1}}[g_{1},\ldots,g_{t}],$

and the result follows. Assume $t-\ell>0$, and we write, for the sake of
brevity, $y=[g_{1},\ldots,g_{\ell-1}]$. By induction, we have

$[y,g^{\prime}_{\ell}g_{\ell},g_{\ell+1},\ldots,g_{t}]=[[y,g_{\ell}^{\prime},g_{\ell+1}^{h_{\ell}},\ldots,g_{t-1}^{h_{t-2}}]^{h_{t-1}}[g_{1},\ldots,g_{t-1}],g_{t}]$

with $h_{i}\in G_{\ell}\cdots G_{i}$ for $i\in\\{\ell,\ldots,t-1\\}$. Now,

$\displaystyle[[y,g_{\ell}^{\prime},g_{\ell+1}^{h_{\ell}},\ldots,g_{t-1}^{h_{t-2}}]$
${}^{h_{t-1}}[g_{1},\ldots,g_{t-1}],g_{t}]$ $\displaystyle\
=[[y,g_{\ell}^{\prime},g_{\ell+1}^{h_{\ell}},\ldots,g_{t-1}^{h_{t-2}}]^{h_{t-1}},g_{t}]^{[g_{1},\ldots,g_{t-1}]}[g_{1},\ldots,g_{t}]$
$\displaystyle\
=[y,g_{\ell}^{\prime},g_{\ell+1}^{h_{\ell}},\ldots,g_{t-1}^{h_{t-2}},g_{t}^{(h_{t-1})^{-1}}]^{h_{t-1}[g_{1},\ldots,g_{t-1}]}[g_{1},\ldots,g_{t}],$

and the lemma follows. If $\ell=1$, a similar argument applies. ∎

######  Lemma 3.7.

Let $G_{1},\ldots,G_{t}$ be subgroups of a profinite group $G$ such that
$G_{j}\leq N_{G}(G_{i})$ for every $j\leq i$. Let $\ell\in\\{1,\ldots,t\\}$
and $H_{1},H_{2}\subseteq G_{\ell}$ be such that
$\pi(h_{1}),\pi(h_{2})\subseteq\pi(h_{1}h_{2})$ for every $h_{1}\in H_{1}$,
$h_{2}\in H_{2}$. Let $X_{i}\subseteq G_{i}$ for $i\in\\{1,\ldots,\ell-1\\}$,
and for $i\in\\{\ell+1,\ldots,t\\}$ denote $X_{i}=G_{i}$. Then:

  1. (i)

If
$\varphi^{*}_{\\{\ell\\}}(H_{1};X_{i})=\varphi^{*}_{\\{\ell\\}}(H_{2};X_{i})=1$,
then $\varphi^{*}_{\\{\ell\\}}(H_{1}H_{2};X_{i})=1$.

  2. (ii)

If $\varphi^{*}_{\\{\ell\\}}(H_{j};X_{i})=\varnothing$ for some
$j\in\\{1,2\\}$, then
$\varphi^{*}_{\\{\ell\\}}(H_{1}H_{2};X_{i})=\varnothing$.

###### Proof.

Since $\pi(h_{1}),\pi(h_{2})\subseteq\pi(h_{1}h_{2})$ for every $h_{1}\in
H_{1}$, $h_{2}\in H_{2}$, the second statement is straightforward. Moreover,
if
$\varphi_{\\{\ell\\}}(h_{1}h_{2};g_{i})\in\varphi_{\\{\ell\\}}^{*}(H_{1}H_{2};X_{i})$,
then for $j\in\\{1,2\\}$ we have
$\varphi_{\\{\ell\\}}(h_{j};g_{i})\in\varphi_{\\{\ell\\}}^{*}(H_{j};X_{i})$.
The result follows now directly from Lemma 3.6. ∎

In view of the preceding lemma, we now introduce a convenient way to choose
coset representatives of normal subgroups. These will play an important role
throughout the paper.

######  Definition 3.8.

Let $G$ be a profinite group and $U\trianglelefteq G$. An element $g\in G$ is
a _good representative_ of the coset $gU$ if $\pi(g),\pi(u)\subseteq\pi(gu)$
for every $u\in U$.

######  Remark 3.9.

In Definition 3.8, if $G$ is pronilpotent, then any element $g\in G$ can be
written in an unique way as $g=\prod_{p\in\pi(G)}g_{p}$ with $g_{p}$ a
$p$-element of $G$. Hence, $g$ is a good representative of the coset $gU$ if
$g_{p}=1$ whenever $g_{p}\in U$ for $p\in\pi(G)$, or equivalently, if $\pi(g)$
is minimal among all representatives of the coset $gU$.

The following lemma is an application of Proposition 2.2 to a special type of
coprime commutators.

######  Lemma 3.10.

Let $G_{1},\ldots,G_{t}$ be pronilpotent subgroups of a profinite group $G$
such that $G_{j}\leq N_{G}(G_{i})$ for all $j\leq i$, and
$|G_{\delta^{*}_{t-1}}|<2^{\aleph_{0}}$. For every $i\in\\{1,\ldots,t\\}$, let
$S_{i}$ be a closed subset of $G_{i}$. If $\varphi^{*}(S_{i})\neq\varnothing$,
then, there exist elements $x_{i}\in G_{i}$ and open subgroups
$U_{i}\trianglelefteq_{o}G_{i}$ such that $|\varphi^{*}(x_{i}U_{i}\cap
S_{i})|=1$.

###### Proof.

Let

$\mathcal{C}=\Big{\\{}(x_{1},\ldots,x_{t})\in S_{1}\times\cdots\times
S_{t}\Bigm{|}(|x_{i}|,|x_{i+1}|)=1\text{ for all }i=1,\ldots,t\Big{\\}}.$

As $\varphi(\mathcal{C})=\varphi^{*}(S_{i})$, we have
$\mathcal{C}\neq\varnothing$. Note that $\mathcal{C}$ is closed in
$G_{1}\times\cdots\times G_{t}$ by Lemma 2.7, and define the map

$\displaystyle\varphi:\mathcal{C}$ $\displaystyle\longrightarrow G_{t}$
$\displaystyle(x_{1},\ldots,x_{t})$
$\displaystyle\longmapsto[x_{1},\ldots,x_{t}].$

Fix $(x_{1},\ldots,x_{t})\in\mathcal{C}$. We first prove by induction on $i$
that $g_{i}:=[x_{1},\ldots,x_{i}]\in G_{\delta^{*}_{i-1}}$ for every
$i\in\\{1,\ldots,t\\}$ and that $\pi(g_{i})\subseteq\pi(x_{i})$. If $i=1$ the
result is obvious, so assume $i>1$ and that $g_{i-1}$ is a
$\delta_{i-2}^{*}$-value with $\pi(g_{i-1})\subseteq\pi(x_{i-1})$, so in
particular $(|g_{i-1}|,|x_{i}|)=1$. If $H$ is the minimal Hall subgroup of the
pronilpotent group $G_{i}$ containing $x_{i}$, then $g_{i-1}$ acts as a
coprime automorphism of $H$. By Lemma 2.4, there exists $y_{i}\in H$ such that

$[x_{i},g_{i-1}]=[y_{i},g_{i-1},\overset{i-1}{\ldots},g_{i-1}],$

and Lemma 2.5 shows that $g_{i}=[x_{i},g_{i-1}]$ is a
$\delta_{i-1}^{*}$-value, as desired. Hence,
$|\operatorname{Im}(\varphi)|<2^{\aleph_{0}}$, and by Proposition 2.2, it
follows that there exist elements $x_{i}\in G_{i}$ and open normal subgroups
$U_{i}\trianglelefteq G_{i}$ such that

$\mathcal{C}\cap(x_{1}U_{1}\times\cdots\times x_{t}U_{t})\neq\varnothing$

and $|\varphi^{*}(x_{i}U_{i}\cap S_{i})|=1$. ∎

Lemma 3.10 will often provide some cosets of open subgroups of $G$ in which
coprime commutators are trivial. Lemmas 3.11 and 3.13 below will allow us to
relate coprime commutators of these cosets with coprime commutators of the
open subgroups themselves.

######  Lemma 3.11.

Let $G_{1},\ldots,G_{t}$ be subgroups of a profinite group $G$ such that
$G_{j}\leq N_{G}(G_{i})$ for every $j\leq i$, and for every
$i\in\\{1,\ldots,t\\}$, let $x_{i}\in G_{i}$ and $U_{i}\trianglelefteq G_{i}$.
Assume also that $G_{j}\leq N_{G}(U_{i})$ for every $j\leq i$. Fix
$j\in\\{1,\ldots,t\\}$ and write $J=\\{1,\ldots,j-1\\}$, then:

  1. (i)

If $\varphi(x_{i}U_{i})=1$ then $\varphi_{J}(x_{i}U_{i};U_{i})=1$.

  2. (ii)

If $\varphi_{J}(x_{i}U_{i};U_{i})=1$ then

$\varphi_{J\cup\\{j\\}}(x_{i}U_{i};U_{i})=\varphi(x_{1}U_{1},\ldots,x_{j-1}U_{j-1},x_{j},U_{j+1},\ldots,U_{t}).$

###### Proof.

(i) We will proceed by reverse induction on $j\in\\{1,\ldots,t+1\\}$, where
the base case $j=t+1$ translates to $\varphi(x_{i}U_{i})=1$, which is true by
hypothesis. Let thus $j<t+1$ and assume that
$\varphi_{J\cup\\{j\\}}(x_{i}U_{i};U_{i})=1$.

Let $C_{t}=1$ and for every $i\in\\{j+1,\ldots,t-1\\}$ define
$C_{i}=C_{U_{i}}(U_{i+1}/C_{i+1})$. Note that $C_{i}$ is well-defined, since
using that for every $\ell$ the subgroup $U_{\ell}$ is normal in $G_{1}\cdots
G_{\ell}$, one can easily show by induction that $C_{\ell}\trianglelefteq
G_{1}\cdots G_{\ell}$.

If $j\geq 2$, let

$Y=\\{[x_{1}u_{1},\ldots,x_{j-1}u_{j-1}]\mid u_{i}\in
U_{i},\,i=1,\ldots,j-1\\}.$

Then, we can rewrite $\varphi_{J\cup\\{j\\}}(x_{i}U_{i};U_{i})=1$ as

$[Y,x_{j}U_{j}]\subseteq C_{G_{j}}(U_{j+1}/C_{j+1}).$

For every $i\in\\{1,\ldots,j\\}$, fix $u_{i}\in U_{i}$ and shorten
$y=[x_{1}u_{1},\ldots,x_{j-1}u_{j-1}]$. Then we have
$[y,x_{j}u_{j}]=[y,u_{j}][y,x_{j}]^{u_{j}}$, and since
$C_{G_{j}}(U_{j+1}/C_{j+1})$ is a normal subgroup of $G_{j}$ containing
$[y,x_{j}u_{j}]$ and $[y,x_{j}]$, it follows that $[y,u_{j}]\in
C_{G_{j}}(U_{j+1}/C_{j+1})$. This shows that
$\varphi(x_{1}U_{1},\ldots,x_{j-1}U_{j-1},U_{j},U_{j+1},\ldots,U_{t})=1$, as
we wanted.

For the case $j=1$, note that both $x_{1}$ and $x_{1}U_{1}$ lay in
$C_{G_{1}}(U_{2}/C_{2})$, so that $U_{1}\leq C_{G_{1}}(U_{2}/C_{2})$.

(ii) For every $i\in\\{j+1,\ldots,t\\}$ we define $C_{i}$ as in (i). For
$i\in\\{1,\ldots,t\\}$, let $u_{i}\in U_{i}$ and shorten
$y=[x_{1}u_{1},\ldots,x_{j-1}u_{j-1}]$. Then,

$[y,x_{j}u_{j}]=[y,u^{\prime}x_{j}]=[y,x_{j}][y,u^{\prime}]^{x_{j}}=[y,u^{\prime}]^{x_{i}[x_{j},y]}[y,x_{j}]$

for some $u^{\prime}\in U_{j}$, and note that
$z:=[y,u^{\prime}]^{x_{j}[x_{j},y]}\in C_{G_{j}}(U_{j+1}/C_{j+1})$. Then
$[z,u^{\prime}_{j+1},\ldots,u^{\prime}_{t}]=1$ for every $u^{\prime}_{i}\in
U_{i}$, $i\in\\{j+1,\ldots,t\\}$, so that

$[y,x_{j}u_{j},u_{j+1},\ldots,u_{t}]=[z[y,x_{j}],u_{j+1},\ldots,u_{t}]=[y,x_{j},u_{j+1},\ldots,u_{t}],$

where the last equality follows from Lemma 3.6. The lemma follows. ∎

######  Definition 3.12.

Let $G_{1},\ldots,G_{t}$ be pronilpotent subgroups of a profinite group $G$
such that $G_{j}\leq N_{G}(G_{i})$ for all $j\leq i$. Let $\sigma$ be a finite
set of primes. We define the normal subgroup

$N_{\sigma}=\langle\varphi^{*}_{\\{j\\}}(H_{i};G_{i})\mid j\text{ is such that
}|\pi(G_{j})|=\infty\rangle^{G},$

where $H_{i}$ is the Hall $\sigma$-subgroup of $G_{i}$ for every $i$. If
$|\pi(G_{i})|<\infty$ for all $i$, then
$N_{\sigma}=\langle\varnothing\rangle^{G}=1$ for every $\sigma$.

The subgroups $G_{1},\ldots,G_{t}$ of $G$ for which the definition of
$N_{\sigma}$ applies will be clear from the context. Notice that for any
finite sets of primes $\sigma_{1}$ and $\sigma_{2}$ such that
$\sigma_{1}\subseteq\sigma_{2}$ we have

$N_{\sigma_{1}}\leq N_{\sigma_{2}}.$ (3.2)

######  Lemma 3.13.

Let $G_{1},\ldots,G_{t}$ be pronilpotent subgroups of a profinite group $G$
such that $G_{j}\leq N_{G}(G_{i})$ for all $j\leq i$. Fix
$\ell\in\\{1,\ldots,t\\}$. For $i\in\\{1,\ldots,\ell-1\\}$, let
$X_{i}\subseteq G_{i}$, and for $i\in\\{\ell,\dots,t\\}$ let $x_{i}\in G_{i}$
and $U_{i}\trianglelefteq_{o}G_{i}$ be such that $G_{j}\leq N_{G}(U_{i})$ for
$j\leq i$. Suppose that $(|x_{\ell}|,|x_{\ell-1}|)=1$ for every $x_{\ell-1}\in
X_{\ell-1}$. Then:

  1. (i)

If $(|x_{\ell}|,|U_{\ell+1}|)=1$ and
$\varphi^{*}(X_{1},\ldots,X_{\ell-1},x_{\ell}U_{\ell},U_{\ell+1},\ldots,U_{t})=1$,
then $\varphi^{*}(X_{1},\ldots,X_{\ell-1},U_{\ell},\ldots,U_{t})=1$.

  2. (ii)

For $i\in\\{\ell,\ldots,t\\}$, suppose that either $|\pi(G_{i})|=\infty$, in
which case we write $Y_{i}=G_{i}$, or $|\pi(G_{i})|=1$, in which case we write
$Y_{i}=U_{i}$. If
$\varphi^{*}(X_{1},\ldots,X_{\ell-1},x_{\ell}U_{\ell},\ldots,x_{t}U_{t})=1,$
then there exists a finite set of primes $\sigma$ such that
$\varphi^{*}(X_{1},\ldots,X_{\ell-1},Y_{\ell},\ldots,Y_{t})\subseteq
N_{\sigma}$ (cf. Definition 3.12). Moreover, if the $x_{i}$ are good
representatives of the cosets $x_{i}U_{i}$ (cf. Definition 3.8) such that
$(|x_{i}|,|x_{i+1}|)=1$ for all $i\in\\{\ell,\ldots,t-1\\}$, then if
$\varphi^{*}(X_{1},\ldots,X_{\ell-1},x_{\ell}U_{\ell},\ldots,x_{t}U_{t})=\varnothing$,
also $\varphi^{*}(X_{1},\ldots,X_{\ell-1},U_{\ell},\ldots,U_{t})=\varnothing$.

###### Proof.

(i) Since
$\varphi^{*}(X_{1},\ldots,X_{\ell-1},x_{\ell}U_{\ell},U_{\ell+1},\ldots,U_{t})\neq\varnothing$,
there are $x_{1},\ldots,x_{\ell-1}$ such that $x_{i}\in X_{i}$ and

$(|x_{j}|,|x_{j+1}|)=1$ (3.3)

for all $j\in\\{1,\ldots,\ell-2\\}$. For $i\in\\{\ell,\ldots,t\\}$, let
$H_{i}$ be a Hall $\pi_{i}$-subgroup of $U_{i}$, where we choose $\pi_{i}$
such that

$(|x_{\ell-1}|,|H_{\ell}|)=(|H_{j}|,|H_{j+1}|)=1$ (3.4)

for all $j\in\\{\ell,\ldots,t\\}$. Since $G_{\ell}$ is pronilpotent, we have
$\pi(x_{\ell}h)\subseteq\pi(x_{\ell})\cup\pi(h)$ for all $h\in H_{\ell}$, and
hence, as $(|x_{\ell}|,|U_{\ell+1}|)=1$, the set
$\varphi(x_{1},\ldots,x_{\ell-1},x_{\ell}H_{\ell},H_{\ell+1},\ldots,H_{t})$ is
contained in
$\varphi^{*}(X_{1},\ldots,X_{\ell-1},x_{\ell}U_{\ell},U_{\ell+1},\ldots,U_{t})$,
and so it is equal to the trivial subgroup. Lemma 3.11 now gives
$\varphi(x_{1},\ldots,x_{\ell-1},H_{\ell},\ldots,H_{t})=1$. Since this holds
for all $x_{1},\ldots,x_{\ell-i}$ and for all Hall subgroups of
$U_{i},i\geq\ell$ satisfying (3.3) and (3.4) respectively, the claim follows.

(ii) Write $L=\\{\ell,\ldots,t\\}$, and for $i\in L$, define

$\sigma_{i}=\left\\{\begin{array}[]{ll}\pi(G_{i}/U_{i})&\text{if
}|\pi(G_{i})|=\infty,\\\ \pi(G_{i})&\text{if
}|\pi(G_{i})|=1.\end{array}\right.$

Let $\sigma=\sigma_{\ell}\cup\cdots\cup\sigma_{t}$, and assume that the
$x_{i}$ are all good representatives, and in particular that they are all
$\sigma$-elements (cf. Remark 3.9). Furthermore, since
$\varphi^{*}_{L}(x_{i}U_{i};X_{i})\neq\varnothing,$ we may also assume that
$(|x_{i}|,|x_{i+1}|)=1$ for all $i\in\\{\ell,\ldots,t-1\\}$. For $i\in L$ with
$|\pi(G_{i})|=\infty$, let $V_{i}$ be the Hall $\sigma^{\prime}$-subgroups of
$G_{i}$, and for $i\in L$ with $|\pi(G)|=1$, set $V_{i}=U_{i}$ (notice that
$V_{i}\leq U_{i}$ if $|\pi(G_{i})|=\infty$). We can apply (i) $t-\ell+1$ times
to obtain $\varphi^{*}_{L}(V_{i};X_{i})=1$, where in each step the condition
$(|x_{j}|,|V_{j+1}|)=1$ is guaranteed either by the hypothesis that
$|\pi(G_{j})|=1$ or $\pi(V_{j})\subseteq\sigma^{\prime}$. Now, from Lemma
3.7(i), we obtain $\varphi^{*}_{L}(Y_{i};X_{i})\subseteq N_{\sigma}$, as
desired.

Regarding the last claim, if
$\varphi^{*}(X_{1},\ldots,X_{\ell-1},x_{\ell}U_{\ell},\ldots,x_{t}U_{t})=\varnothing$
then we also have
$\varphi^{*}(X_{1},\ldots,X_{\ell-1},x_{\ell},\ldots,x_{t})=\varnothing$, so
there exists an index $j\in\\{1,\ldots,l-2\\}$ such that
$(|x_{j}|,|x_{j+1}|)\neq 1$ for all $x_{j}\in X_{j}$, $x_{j+1}\in X_{j+1}$,
and the lemma follows. ∎

The following lemma is the focal point of the proof of Proposition 3.16, as it
will allow us to funnel some values of certain coprime commutators into an
accurately chosen subgroup.

######  Lemma 3.14.

Let $G_{1},\ldots,G_{t}$ be pronilpotent subgroups of a profinite group $G$
such that $G_{j}\leq N_{G}(G_{i})$ for all $j\leq i$, and
$|G_{\delta_{t-1}^{*}}|<2^{\aleph_{0}}$. Then, there exist a finite set
$W\subseteq\varphi^{*}(G_{i})$ and a finite set $\sigma$ of primes such that
$\varphi^{*}(G_{i})\subseteq N_{\sigma}\langle W\rangle^{G}$.

###### Proof.

Let

$\mathcal{I}=\\{i\in\\{1,\ldots,t\\}\mid|\pi(G_{i})|=\infty\\}.$

It suffices to prove the theorem in the case when $|\pi(G_{i})|=1$ for all
$G_{i}$ with $i\notin\mathcal{I}$. The general case, where each $G_{i}$,
$i\notin\mathcal{I}$, is the product of its Sylow subgroups follows by
applying Lemma 3.7.

For $i\notin\mathcal{I}$, let $p_{i}$ be a prime such that
$\pi(G_{i})=\\{p_{i}\\}$. Then we have

$\varphi^{*}(G_{i})=\varphi^{*}(G_{1},\ldots,G_{i-2},H_{i-1},G_{i},H_{i+1},G_{i+2},\ldots,G_{t}),$

where $H_{i-1}$ and $H_{i+1}$ are the Hall $p_{i}^{\prime}$-subgroups of
$G_{i-1}$ and $G_{i+1}$, respectively. We can therefore assume, again by Lemma
3.7(i), that for all $i\notin\mathcal{I}$ we have

$\pi(G_{i})\not\subseteq\pi(G_{i-1})\cup\pi(G_{i+1}).$ (3.5)

We claim that that for every
$J\subseteq\\{1,\ldots,t\\}\smallsetminus\mathcal{I}$ there exist a finite set
$W_{J}\subseteq\varphi^{*}(G_{i})$, a finite set of primes $\sigma(J)$ and
subgroups $U_{i}\trianglelefteq_{o}G_{i}$ with $i\notin\mathcal{I}\cup J$ such
that $\varphi^{*}_{\mathcal{I}\cup J}(G_{i};U_{i})\subseteq
N_{\sigma(J)}\langle W_{J}\rangle^{G}$.

We proceed by induction on $|J|$. Assume first $J=\varnothing$. By Lemma 3.10,
for every $i\in\\{1,\ldots,t\\}$ there exist elements $x_{i}\in G_{i}$ and
subgroups $U_{i}\trianglelefteq_{o}G_{i}$ such that
$\varphi^{*}(x_{i}U_{i})=\\{w_{\varnothing}\\}$ for a suitable
$w_{\varnothing}\in G$. Moreover, by Remark 3.5, we may assume that $G_{j}\leq
N_{G}(U_{i})$ for every $j\leq i$. Hence, Lemma 3.13(ii) produces a finite set
$\sigma(\varnothing)$ of primes such that
$\varphi^{*}_{\mathcal{I}}(G_{i};U_{i})\subseteq
N_{\sigma(\varnothing)}\langle w_{\varnothing}\rangle^{G}$, so the claim
follows for $|J|=0$.

Assume now that $|J|\geq 1$ and that for every $J^{-}\subsetneq J$ there exist
a finite set $W_{J^{-}}\subseteq\varphi^{*}(G_{i})$, a finite set of primes
$\rho(J^{-})$ and subgroups $U^{J^{-}}_{i}\trianglelefteq_{o}G_{i}$,
$i\notin\mathcal{I}$, such that $\varphi^{*}_{\mathcal{I}\cup
J^{-}}(G_{i};U^{J^{-}}_{i})\subseteq N_{\rho(J^{-})}\langle
W_{J^{-}}\rangle^{G}$ (here we are taking $U_{i}^{J^{-}}=G_{i}$ if $i\in
J^{-}$). Let $W_{J}=\bigcup_{J^{-}}W_{J^{-}}$,
$\rho=\bigcup_{J^{-}}\rho(J^{-})$ and $V_{i}=\bigcap_{J^{-}}U^{J^{-}}_{i}$, so
that, by (3.2), we have $\varphi^{*}_{\mathcal{I}\cup
J^{-}}(G_{i};V_{i})\subseteq N_{\rho}\langle W_{J}\rangle^{G}$ for every
$J^{-}\subsetneq J$. Furthermore, by factoring out $N_{\rho}\langle
W_{J}\rangle^{G}$, we may assume that

$\varphi^{*}_{\mathcal{I}\cup J^{-}}(G_{i};V_{i})=1$ (3.6)

for every $J^{-}\subsetneq J$. Moreover, taking into account Remark 3.5 we may
further assume that $V_{i}$ is invariant under the conjugacy action of $G_{j}$
for every $j\leq i$.

Write $J=\\{j_{1},\ldots,j_{n}\\}$ with $j_{1}<\cdots<j_{n}$, and for every
$i\in J$, fix a set $S_{i}$ of coset representatives for $V_{i}$ in $G_{i}$
containing the identity. Write

$S_{j_{1}}\times\cdots\times
S_{j_{n}}=\\{\textbf{s}_{1},\ldots,\textbf{s}_{m}\\}$

with $\textbf{s}_{\ell}=(s_{\ell,j_{1}},\ldots,s_{\ell,j_{n}})$ for
$\ell\in\\{1,\ldots,m\\}$. Denote $V_{i}=G_{i}$ for $i\in\mathcal{I}$. Since
$1\in S_{i}$ for every $i$, we have
$\varphi^{*}_{J}(S_{i};V_{i})\neq\varnothing$, so applying Lemma 3.10 we
obtain elements $x_{i}\in V_{i}$ and subgroups $U_{i}\trianglelefteq_{o}V_{i}$
such that $\varphi^{*}_{J}(S_{i};x_{i}U_{i})$ takes a single value. Actually,
since
$1=\varphi^{*}_{J}(1;x_{i}U_{i})\subseteq\varphi^{*}_{J}(S_{i};x_{i}U_{i})$,
we have $\varphi^{*}_{J}(S_{i};x_{i}U_{i})=1$. Also, we may assume the $x_{i}$
to be good representatives of the $U_{i}$ (cf. Definition 3.8) and therefore,
if $J$ does not contain neither $i$ nor $i+1$, then $(|x_{i}|,|x_{i+1}|)=1$.
Thus, for every $\ell\in\\{1,\ldots,m\\}$, we either have

$\varphi^{*}_{J}(s_{\ell,i};x_{i}U_{i})=\varnothing\quad\text{ or
}\quad\varphi^{*}_{J}(s_{\ell,i};x_{i}U_{i})=1.$ (3.7)

Again, by Remark 3.5 we may further assume that $U_{i}$ is invariant under the
conjugacy action of $G_{j}$ for every $j\leq i$.

Let $J_{0}=\varnothing$, and for $r\in\\{1,\ldots,n\\}$, let
$J_{r}=\\{j_{1},\ldots,j_{r}\\}$. We also write $j_{0}=0$ for convenience. We
will show that for every $r\in\\{0,\ldots,n\\}$, there exists a finite set of
primes $\sigma(r)$ such that
$\varphi^{*}_{J_{r}}(s_{\ell,i};Y_{i}^{(r)})\subseteq N_{\sigma(r)}$ for every
$\ell\in\\{1,\ldots,m\\}$, where

$Y_{i}^{(r)}=\left\\{\begin{array}[]{ll}G_{i}&\text{if }\ i\geq
j_{r},\,i\in\mathcal{I}\cup J,\\\ U_{i}&\text{if }\
i>j_{r},\,i\not\in\mathcal{I}\cup J,\\\ x_{i}U_{i}&\text{if }\
i<j_{r}.\end{array}\right.$

We argue by reverse induction on $r\in\\{0,\ldots,n\\}$; assume first $r=n$.
Since $j_{r}\not\in\mathcal{I}$, we deduce from (3.5) that
$(|s_{\ell,j_{r}}|,|G_{j_{r}-1}|)=(|s_{\ell,j_{r}}|,|x_{j_{r}+1}|)=1$. Thus,
for all $\ell\in\\{1,\ldots,m\\}$, we obtain from (3.7) and Lemma 3.13(ii) a
finite set of primes $\sigma(r,\ell)$ such that
$\varphi^{*}_{J_{r}}(s_{\ell,i};Y_{i}^{(r)})\subseteq N_{\sigma(r,\ell)}$.
Defining $\sigma(r)=\bigcup_{\ell=1}^{m}\sigma(r,\ell)$, we obtain
$\varphi^{*}_{J_{r}}(s_{\ell,i};Y_{i}^{(r)})\subseteq N_{\sigma(r)}$ for every
$\ell\in\\{1,\ldots,m\\}$.

Hence, we assume $r\leq n-1$. By induction, we know that there exists a finite
set of primes $\sigma(r+1)$ such that

$\varphi^{*}_{J_{r+1}}(s_{\ell,i};Y_{i}^{(r+1)})\subseteq N_{\sigma(r+1)}$
(3.8)

for every $\ell\in\\{1,\ldots,m\\}$. We will first show that
$\varphi^{*}_{J_{r}}(s_{\ell,i};Y_{i}^{(r+1)})\subseteq N_{\sigma(r+1)}$. Note
that $Y_{i}^{(r+1)}\leq V_{i}$ for every $i\not\in\mathcal{I}\cup J$ and that
$U_{j_{r+1}}\leq V_{j_{r+1}}$, so (3.6) yields

$\varphi^{*}_{J_{r}}(s_{\ell,i};\widetilde{Y}_{i})\subseteq N_{\sigma(r+1)},$
(3.9)

where $\widetilde{Y}_{i}=Y_{i}^{(r+1)}$ if $i\neq j_{r+1}$ and
$\widetilde{Y}_{j_{r+1}}=U_{j_{r+1}}$. As we chose the sets of representatives
$S_{j}$ in such a way that the identity is contained in them, for every
$\ell\in\\{1,\ldots,m\\}$, either $s_{\ell,j_{r+1}}$ is trivial or
$|\pi(s_{\ell,j_{r+1}})|=1$, so in particular $s_{\ell,j_{r+1}}$ is a good
representative. Thus, by (3.8) and (3.9), we deduce from Lemma 3.7 that
$\varphi^{*}_{J_{r}}(s_{\ell,i};\overline{Y}_{i})\subseteq N_{\sigma(r+1)}$,
where $\overline{Y}_{i}=Y_{i}^{(r+1)}$ if $i\neq j_{r+1}$ and
$\overline{Y}_{j_{r+1}}=s_{\ell,j_{r+1}}U_{j_{r+1}}$. Since this holds for
every $\ell\in\\{1,\ldots,m\\}$, and since $G_{j_{r+1}}=\bigcup_{s\in
S_{j_{r+1}}}sU_{j_{r+1}}$, we obtain
$\varphi^{*}_{J_{r}}(s_{\ell,i};Y_{i}^{(r+1)})\subseteq N_{\sigma(r+1)}$, as
we wanted.

Now using (3.5) and Lemma 3.13, we conclude exactly as in the case $r=n$ that
there exists a finite set $\sigma(r)$ of primes such that
$\varphi^{*}_{J_{r}}(s_{\ell,i};Y_{i}^{(r)})\subseteq N_{\sigma(r)}$ for every
$\ell\in\\{1,\ldots,m\\}$.

This completes the reverse induction on $r$. In particular, for $r=0$, it
follows that $\varphi^{*}_{J}(G_{i};U_{i})\subseteq N_{\sigma(0)}$, so this,
in turn, concludes the inductive step on $|J|$, and the claim is proved.

Finally, taking $J$ in such a way that $\mathcal{I}\cup J=\\{1,\ldots,t\\}$,
we obtain a finite set of primes $\sigma(J)$ and a finite set
$W\subseteq\varphi^{*}(G_{i})$ such that
$\varphi^{*}(G_{1},\ldots,G_{t})\subseteq N_{\sigma(J)}\langle W\rangle^{G}$,
as desired. ∎

Recall that if $G$ is a profinite group of Fitting height $k+1$, then there
exist pronilpotent subgroups $T_{0},\ldots,T_{k}$ of $G$ such that $T_{j}\leq
N_{G}(T_{i})$ for every $j\leq i$. Moreover, $G=T_{0}\cdots T_{k}$,
$T_{k}=\delta_{k}^{*}(G)$ and
$T_{i}=[T_{0},\ldots,T_{i}]\pmod{\delta_{i+1}^{*}(G)}$ for every
$i\in\\{0,\ldots,k\\}$ (see the beginning of Section 3.2).

We remark that $\varphi$ and $\varphi^{*}$ were defined with variables
$\\{x_{i}\mid i=1,\ldots,t\\}$ for a generic positive integer $t$. Since we
now want to apply the previous results to the subgroups $T_{0},\ldots,T_{k}$,
we will set $t=k+1$ and we will write $\varphi(T_{i-1})$ for
$[T_{0},\ldots,T_{k}]$ and $\varphi^{*}(T_{i-1})$ for
$\varphi((T_{0}\times\cdots\times T_{k})\cap\mathcal{C})$, where $\mathcal{C}$
is defined as in (3.1).

######  Lemma 3.15.

Let $G=T_{0}\cdots T_{k}$ be as above with $T_{k}=\delta_{k}^{*}(G)$ abelian,
and assume $|G_{\delta_{k}^{*}}|<2^{\aleph_{0}}$. Let
$g\in\varphi^{*}(T_{i-1})$. Then, there exists a finite normal subgroup
$N\trianglelefteq G$ such that $g\in N$.

###### Proof.

Write $g=[x_{0},\ldots,x_{k}]$, where $x_{j}\in T_{j}$ for all $j$ and
$(|x_{\ell}|,|x_{\ell+1}|)=1$ for all $\ell\in\\{0,\ldots,k-1\\}$. In the same
way as in the proof of Lemma 3.10, we see that $[x_{0},\ldots,x_{j}]$ is a
$\delta_{j}^{*}$-value for every $j\in\\{0,\ldots,k\\}$.

In particular $x:=[x_{0},\ldots,x_{k-1}]$ is a $\delta_{k-1}^{*}$-value, and
let $H$ be the minimal Hall subgroup of $\delta_{k}^{*}(G)$ containing
$x_{k}$, so that $(|x|,|H|)=1$. Since, again, $[x,h]$ is a
$\delta_{k}^{*}$-value for every $h\in H$, the set $K:=\\{[x,h]\mid h\in H\\}$
has less than $2^{\aleph_{0}}$ values, and, since $H$ is abelian and normal in
$G$, it follows that $K$ is actually a subgroup of $G$. In particular, $K$ is
finite, so every element of $K$ has finite order. Thus, we deduce from Lemma
2.3 that the set $S:=\bigcup\\{k^{G}\mid k\in K\\}$ is finite, and therefore
$N=\langle S\rangle$ is finite. ∎

We are now ready to prove the strong conciseness of $\delta_{k}^{*}$ in
prosoluble groups of Fitting height $k+1$.

######  Proposition 3.16.

Let $G$ be a prosoluble group of Fitting height $k+1$. Assume that
$|G_{\delta^{*}_{k}}|<2^{\aleph_{0}}$. Then $\delta_{k}^{*}(G)$ is finite.

###### Proof.

In view of Lemma 3.1, we may assume that $\delta_{k}^{*}(G)$ is abelian. Thus,
we can take $T_{0},\ldots,T_{k}\leq G$ as in Lemma 3.15, so that
$G=T_{0}\cdots T_{k}$ with $T_{k}=\delta_{k}^{*}(G)$ abelian.

We claim that for every family of subgroups $G_{i-1}\leq T_{i-1}$ with
$i\in\\{1,\ldots,k+1\\}$ such that $G_{j}\leq N_{G}(G_{i})$ for $j\leq i$, we
have $|\varphi^{*}(G_{i-1})|<\infty$. We argue by induction on
$|\mathcal{I}|$, where

$\mathcal{I}=\\{i\in\\{1,\ldots,k+1\\}\mid|\pi(G_{i-1})|=\infty\\}.$

If $|\mathcal{I}|=0$, then Lemma 3.14 gives the result since for every finite
set $W\subseteq\varphi^{*}(G_{i-1})$, the normal subgroup $\langle
W\rangle^{G}$ is finite by Lemma 3.15, and since, by definition,
$N_{\sigma}=1$ for every finite set of primes $\sigma$. Suppose thus
$|\mathcal{I}|\geq 1$. Then, Lemma 3.14 produces a finite set of primes
$\sigma$ and a finite set $W\subseteq\varphi^{*}(G_{i-1})$ such that
$\varphi^{*}(G_{i-1})\subseteq N_{\sigma}\langle W\rangle^{G}$. Observe that
by induction, for every $j\in\mathcal{I}$, we have
$|\varphi^{*}_{\\{j\\}}(H_{i-1};G_{i-1})|<\infty$, where $H_{i-1}$ is the Hall
$\sigma$-subgroup of $G_{i-1}$, and therefore $N_{\sigma}$ is finite by Lemma
3.15. Again by Lemma 3.15, $\langle W\rangle^{G}$ is also finite, and the
claim follows.

In particular, we have shown that $|\varphi^{*}(T_{i-1})|<\infty$. Let
$U_{0}=T_{0}$; for $j\geq 1$ we construct inductively a series of subgroups
$U_{j}\leq T_{j}$ in the following way. Let $H_{j-1}(p^{\prime})$ and
$Q_{j}(p)$ be, respectively, the Hall $p^{\prime}$-subgroup of $U_{j-1}$ and
the Sylow $p$-subgroup of $T_{j}$, and define

$U_{j}=\prod_{p\in\pi(G)}[H_{j-1}(p^{\prime}),Q_{j}(p)].$

Denote by $P_{j}(p)=[H_{j-1}(p^{\prime}),Q_{j}(p)]$ the Sylow $p$-subgroup of
$U_{j}$. We claim that $Q_{j}(p)\equiv P_{j}(p)\pmod{\delta_{j+1}^{*}(G)}$ for
every $p\in\pi(G)$ and every $j\in\\{0,\ldots,k\\}$, and in particular that
$T_{j}\equiv U_{j}\pmod{\delta_{j+1}^{*}(G)}$. The case $j=0$ follows
trivially, so let $j\geq 1$ and assume by induction that the congruences hold
for $j-1$. Let $K_{j}(p^{\prime})$ be the Hall $p^{\prime}$-subgroup of
$T_{j-1}T_{j}$. Since
$\gamma_{\infty}(T_{j-1}T_{j})=T_{j}\pmod{\delta_{j+1}^{*}(G)}$, Lemma 2.6
yields

$Q_{j}(p)\equiv[K_{j-1}(p^{\prime}),Q_{j}(p)]\equiv[H_{j-1}(p^{\prime}),Q_{j}(p)]\pmod{\delta_{j+1}^{*}(G)},$

where the last congruence holds because $T_{j-1}T_{j}\equiv
U_{j-1}T_{j}\pmod{\delta_{j+1}^{*}(G)}$ by induction, and hence
$K_{j-1}(p^{\prime})\equiv
H_{j-1}(p^{\prime})H^{*}_{j}(p^{\prime})\pmod{\delta_{j+1}^{*}(G)}$, where
$H^{*}_{j}(p^{\prime})$ is the Hall $p^{\prime}$-subgroup of $T_{j}$. The
claim follows by the definition of $P_{j}(p)$. In particular, this shows that
$U_{k}=T_{k}$.

Furthermore, as $(|Q_{j}(p)|,|H_{j-1}(p^{\prime})|)=1$, we have

$P_{j}(p)=[Q_{j}(p),H_{j-1}(p^{\prime})]=[Q_{j}(p),H_{j-1}(p^{\prime}),H_{j-1}(p^{\prime})]=[P_{j}(p),H_{j-1}(p^{\prime})]$

by [16, Lemma 4.29], and thus, for every $j\in\\{1,\ldots,k\\}$ and very
$p\in\pi(G)$ we obtain

$P_{j}(p)=\prod_{\begin{subarray}{c}q\in\pi(G)\\\ q\neq
p\end{subarray}}[P_{j-1}(q),P_{j}(p)].$

Therefore, for every $q_{k}\in\pi(G)$,

$P_{k}(q_{k})=\prod_{(q_{0},\ldots,q_{k-1})\in
S_{q_{k}}}[P_{0}(q_{0}),\ldots,P_{k}(q_{k})],$

where

$S_{q_{k}}=\\{(q_{0},\ldots,q_{k-1})\in\pi(G)^{k-1}\mid q_{j}\neq q_{j+1}\
\text{for every}\ j=0,\ldots,k-1\\}.$

By Lemma 3.6, this implies that
$P_{k}(q_{k})\leq\langle\varphi^{*}(T_{i-1})\rangle$ for every
$q_{k}\in\pi(G)$, and so

$\delta_{k}(G)=T_{k}=\prod_{p\in\pi(G)}P_{k}(p)=\langle\varphi^{*}(T_{i-1})\rangle.$

The proposition follows from Lemma 3.15. ∎

##  4\. The main theorems

We recall that a _minimal simple group_ is a finite non-abelian simple group
all of whose proper subgroups are soluble. These groups have been classified
by Thompson in [26].

######  Lemma 4.1.

In every minimal simple group there exist an involution $e$ and an element $h$
of odd order such that $h^{e}=h^{-1}$. Moreover, for every positive integer
$k$, the element

$g_{k}=[h,e,\overset{k-1}{\ldots},e]$

is both a non-trivial $\gamma_{k}^{*}$-value and a non-trivial
$\delta_{k-1}^{*}$-value.

###### Proof.

The first claim follows from Theorem 2.13 of [16] and the fact that non-
abelian simple groups are of even order. Notice that $g_{k}=h^{(-2)^{k-1}}$,
so $g_{k}\neq 1$ for every positive integer $k$. Clearly $g_{i}$ and $e$ are
coprime for every $i\in\\{1,\ldots,k-1\\}$, so $g_{k}$ is a
$\gamma_{k}^{*}$-value. Thus, it suffices to prove that the same is true for
$\delta^{*}_{k-1}$. By Proposition 25 of [4], every involution of a minimal
simple group is a $\delta^{*}_{\ell}$-value for every $\ell\in\mathbb{N}$.
Hence, we can use Lemma 2.5 with $g_{1}=\cdots=g_{k-1}=e$ and $H=\langle
h\rangle$ and conclude the proof. ∎

We are now ready to prove our main results. As in [10], we start showing that
the Fitting subgroup of any profinite group $G$ with
$|G_{\gamma_{k}^{*}}|<2^{\aleph_{0}}$ or $|G_{\delta_{k}^{*}}|<2^{\aleph_{0}}$
is infinite.

######  Proposition 4.2.

Let $G$ be an infinite profinite group and let $w^{*}=\delta_{k}^{*}$ or
$w^{*}=\gamma_{k}^{*}$. Suppose that $|G_{w^{*}}|<2^{\aleph_{0}}$. Then the
Fitting subgroup $F$ of $G$ is infinite.

###### Proof.

We first show that $F$ is non-trivial. Assume by contradiction that $F=1$. For
every $w^{*}$-value $x$ of $G$, the normal closure $\langle x^{G}\rangle$ is
finite. Indeed, by Lemma 2.3, $x^{G}$ is finite, so in particular
$|G:C_{G}(\langle x^{G}\rangle)|<\infty$. As a consequence, the index
$|\langle x^{G}\rangle:Z(\langle x^{G}\rangle)|$ is also finite, but
$Z(\langle x^{G}\rangle)$ is contained in $F=1$. This implies that $G$
possesses finite minimal normal subgroups, so let $N$ be the product of all of
them.

If $N$ is finite, then there exists a normal open subgroup
$K\trianglelefteq_{o}G$ such that $K\cap N=1$. Such a subgroup cannot contain
any $w^{*}$-value since otherwise, repeating the same argument as before, we
would obtain a minimal normal subgroup of $G$ contained in $K$, contradicting
that $K\cap N=1$. If $w^{*}(K)=1$, then by Theorem 2.1 $K$ is either
pronilpotent (if $w^{*}=\gamma_{k}^{*}$) or prosoluble of Fitting height $k$
(if $w^{*}=\delta_{k}^{*}$), and this contradicts the fact that $F\cap K=1$.
This proves that $N$ is an infinite subgroup of $G$.

None of the infinitely many minimal normal subgroups of $G$ is abelian because
$F=1$, so each of these minimal normal subgroup contains a section isomorphic
to a minimal simple group. For each minimal normal subgroup $N_{i}$, with
$i\in I$, choose a section isomorphic to a minimal simple group $S_{i}$. We
remark that by the previous discussion $I$ is an infinite set and so the
Cartesian product of $S_{i}$ is a section of $G$. By Lemma 4.1 in each of
these groups $S_{i}$ there exist an involution $e_{S_{i}}\in S_{i}$ and an
element $h_{S_{i}}\in S_{i}$ of odd order with
$h_{S_{i}}^{e_{S_{i}}}=h_{S_{i}}^{-1}$ such that
$g_{S_{i}}:=[h_{S_{i}},e_{S_{i}},\overset{k-1}{\ldots},e_{S_{i}}]$ is a non-
trivial $w^{*}$-value. Now, for each subset $J\subseteq I$, the element
$c_{J}=\prod_{j\in J}g_{S_{j}}$ can be written as

$c_{J}=\Big{[}\prod_{j\in J}h_{S_{j}},\prod_{j\in
J}e_{S_{j}},\ldots,\prod_{j\in J}e_{S_{j}}\Big{]}.$

Clearly $\prod_{j\in J}e_{S_{j}}$ is an involution normalizing the cyclic
subgroup generated by the element of odd order $\prod_{j\in J}h_{S_{j}}$, and
hence it is a $w^{*}$-value (if $w^{*}=\delta_{k}^{*}$ we also need to use
Lemma 2.5). However, there exist at least $2^{\aleph_{0}}$ distinct subsets
$J\subseteq I$ that give rise to different $c_{J}$, against the assumption
that $|G_{w^{*}}|<2^{\aleph_{0}}$, so $F\neq 1$.

If we assume by contradiction that the Fitting subgroup $F$ is finite, then
there would be a subgroup $K\trianglelefteq_{o}G$ with $K\cap F=1$, so that
$K$ has trivial Fitting subgroup. By the previous argument, this can happen
only if $w^{*}(K)=1$, so $K$ is either pronilpotent or prosoluble of Fitting
height $k$, contradicting that $K\cap F=1$ and proving the proposition. ∎

###### Proofs of Theorems 1.1 and 1.2.

In view of Theorem 2.1 it is sufficient to show that if
$|G_{w^{*}}|<2^{\aleph_{0}}$, then $G$ is finite-by-pronilpotent in the case
$w^{*}=\gamma_{k}^{*}$ or finite-by-(prosoluble of Fitting height $k$) if
$w^{*}=\delta_{k}^{*}$. We can assume $G$ to be infinite, otherwise the
theorem is trivially true.

We will denote the Fitting subgroup of $G$ by $F$, and for $i\geq 2$, let
$F_{i}$ be the $i$-th Fitting subgroup of $G$. By Proposition 4.2, $F$ is
infinite (and hence the same is true for all $F_{k}$). Let $n=2$ if
$w^{*}=\gamma_{k}^{*}$ and $n=k+1$ if $w^{*}=\delta_{k}^{*}$. By Propositions
3.4 and 3.16, $w^{*}(F_{n})$ is finite. Therefore there exits an open normal
subgroup $R\trianglelefteq_{o}F_{n}$ with $R\cap w^{*}(F_{n})=1$. Theorem 2.1
implies that the Fitting height of $R$ is at most $n-1$, and hence $R$ is
contained in $F_{n-1}$. However, since $F_{n}/F_{n-1}$ is the Fitting subgroup
of $G/F_{n-1}$, it follows that $G/F_{n-1}$ has finite Fitting subgroup, and
by Proposition 4.2 this can only happen if $G/F_{n-1}$ is finite.

Thus, we will prove the result by induction on $|G:F_{n-1}(G)|$, with the base
case $G=F_{n-1}(G)$ being trivial. Assume then that $|G:F_{n-1}(G)|>1$ and
suppose first that $G/F_{n-1}$ has a nontrivial proper normal subgroup $N$.
The inductive hypothesis yields $|w^{*}(N)|<\infty$, and working in
$G/w^{*}(N)$, we obtain by Theorem 2.1 that $N/w^{*}(N)$ is prosoluble of
Fitting height at most $n-1$. This implies that $N/w^{*}(N)$ is contained in
the $(n-1)$-th Fitting subgroup of $G/w^{*}(N)$ and by inductive hypothesis
$w^{*}(G/w^{*}(N))$ must be finite, so $w^{*}(G)$ is finite too.

We can hence assume that $G/F_{n-1}$ is a simple group. Notice that if
$G/F_{n-1}$ is abelian, then we can conclude simply by applying Proposition
3.4 or Proposition 3.16. Thus, the only case left is when $G/F_{n-1}$ is a
finite non-abelian simple group. By Theorem 2.1 we have
$w^{*}(G/F_{n-1})=G/F_{n-1}$, so there is a finite set $S$ consisting of
$w^{*}$-values such that $G=\langle S\rangle F_{n-1}$. By Lemma 2.3 the set
$T:=\bigcup\\{s^{G}\mid s\in S\\}$ is finite, so the index $|G:C_{G}(T)|$ is
also finite. This implies that the center of $\langle T\rangle$ has finite
index in $\langle T\rangle$, so by Schur’s theorem $\langle T\rangle^{\prime}$
is finite too. Note that $\langle T\rangle^{\prime}$ is normal in $G$.
Factoring out $\langle T\rangle^{\prime}$, we can assume $G/F_{n-1}$ to be
abelian, and we conclude the proof as before. ∎

## References

  * [1] Acciarri, Cristina and Shumyatsky, Pavel and Thillaisundaram, Anitha, Conciseness of coprime commutators in finite groups. Bull. Aust. Math. Soc. 89(2) (2014), 252–258.
  * [2] Acciarri, Cristina and Shumyatsky, Pavel, On words that are concise in residually finite groups. J. Pure Appl. Algebra 218 (2014), 130–134.
  * [3] Azevedo, Joao and Shumyatsky, Pavel, On finiteness of some verbal subgroups in profinite groups. J. Algebra 574 (2021), 573–583.
  * [4] Bastos, Raimundo and Monetta, Carmine, Coprime commutators in finite groups. Comm. Algebra 47(10) (2019), 4137–4147.
  * [5] de las Heras, Iker and Morigi, Marta, Lower central words in finite $p$-groups. Publ. Mat. 65(1) (2021), 243–269.
  * [6] Detomi, Eloisa and Morigi, Marta and Shumyatsky, Pavel, Words of Engel type are concise in residually finite groups. Bulletin of Mathematical Sciences 9 (2019) 1950012 (19 pages).
  * [7] Detomi, Eloisa and Morigi, Marta and Shumyatsky, Pavel, Words of Engel type are concise in residually finite groups. Part II. Groups Geom. Dyn. 14 (2020), 991-1005.
  * [8] Detomi, Eloisa, and Morigi, Marta and Shumyatsky, Pavel, On bounded conciseness of Engel-like words in residually finite groups. J. Algebra 521 (2019), 1–15.
  * [9] Detomi, Eloisa and Klopsch, Benjamin and Shumyatsky, Pavel, Strong conciseness in profinite groups. J. Lond. Math. Soc. 102(3) (2020), 977–993.
  * [10] Detomi, Eloisa and Morigi, Marta and Shumyatsky, Pavel, Strong conciseness of coprime and anti-coprime commutators. Ann. Mat. Pura Appl. 200(3) (2021) 945–952.
  * [11] Detomi, Eloisa and Morigi, Marta and Shumyatsky, Pavel, Bounding the order of a verbal subgroup in a residually finite group. Isr. J. Math. (2022). https://doi.org/10.1007/s11856-022-2378-3.
  * [12] Detomi, Eloisa and Morigi, Marta and Shumyatsky, Pavel, Commutators, centralizers, and strong conciseness in profinite groups. Accepted in Mathematische Nachrichten, arXiv:2212.09665 [math.GR]
  * [13] Detomi, Eloisa, A note on strong conciseness in virtually nilpotent profinite groups. _Arch. Math._ (2022). https://doi.org/10.1007/s00013-022-01808-z
  * [14] Fernández-Alcober, Gustavo A. and Shumyatsky, Pavel, On bounded conciseness of words in residually finite groups. J. Algebra 500 (2018), 19–29.
  * [15] Guralnick, Robert and Shumyatsky, Pavel, On rational and concise word. J. Algebra 429 (2015), 213–217.
  * [16] Isaacs, Martin, Finite group theory. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 92, (2008).
  * [17] Ivanov, Sergej, P. Hall’s conjecture on the finiteness of verbal subgroups. _Soviet Math. (Iz. VUZ)_ 33 (1989), 59–70.
  * [18] Jaikin-Zapirain, Andrei, On the verbal width of finitely generated pro-p groups. Rev. Mat. Iberoam. 168 (2008), 393–412.
  * [19] Khukhro, Evgeny and Shumyatsky, Pavel, Compact groups with countable Engel sinks. Bull. Math. Sci., 11 (2021), no. 3, 28 pp.
  * [20] Khukhro, Evgeny and Shumyatsky, Pavel, Strong conciseness of Engel words in profinite groups, Accepted in Mathematische Nachrichten, arXiv:2108.11789 [math.GR]
  * [21] Robinson, Derek J. S., _A course in the theory of groups_. Graduate Texts in Mathematics, Springer-Verlag, New York, 80 (1996).
  * [22] Ribes, Luis and Zalesskii, Pavel, _Profinite groups_. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin (2010).
  * [23] Ol’shanskii, Alexander Y., Geometry of Defining Relations in Groups. Mathematics and its applications 70 (Soviet Series), Kluwer Academic Publishers, Dordrecht, 1991.
  * [24] Segal, Dan, Words: notes on verbal width in groups. LMS Lecture Notes 361, Cambridge Univ. Press, Cambridge, 2009.
  * [25] Shumyatsky, Pavel, Commutators of elements of coprime orders in finite groups. _Forum Math._ , 27(1) (2015), 575–583.
  * [26] Thompson, John G., Nonsoluble finite groups all of whose local subgroups are soluble. Bull. Amer. Math. Soc., 74 (1968), 383–437.