the rank function $\operatorname{rk}_{G}$ of Example 2.5.

Let $\mathcal{Q}_{i}$ be the set of elements $q\in\mathcal{Q}$ with $v(q)\leq
p^{-i}$. In particular, $\mathcal{Q}_{0}$ is an $\mathbb{F}_{p}$-algebra
containing $\varphi(\mathbb{F}_{p}[\\![G]\\!])$ with ideals $\mathcal{Q}_{i}$
for each $i\geq 0$.

######  Claim 7.3.

There exists a division ring extension $\mathcal{Q}\leq\mathcal{Q}^{\prime}$
and an extension $w^{\prime}$ of the valuation $w$ to $\mathcal{Q}^{\prime}$
such that $\mathcal{Q}^{\prime}$ contains a central element $z$ with
$w^{\prime}(z)=p^{-1}$. In particular,
$\mathcal{Q}^{\prime}_{i}=z^{i}\mathcal{Q}^{\prime}_{0}$ for every
$i\in\mathbb{Z}$.

###### Proof.

Consider the polynomial ring $\mathcal{Q}[z]$ and define
$w^{\prime}\colon\mathcal{Q}[z]\to\\{p^{k}\mid k\in\mathbb{Z}\\}\cup\\{0\\}$
to be

$w^{\prime}\left(\sum_{i=0}^{n}q_{i}z^{i}\right)=\max\\{w(q_{i})p^{-i}\\}\,.$

We check that $w^{\prime}$ is an additive non-archimedian valuation on
$\mathcal{Q}[z]$, for it is immediate that it is discrete.

Let $r=\sum_{i=1}^{n}q_{i}z^{i}$ and
$r^{\prime}=\sum_{i=1}^{n}q_{i}^{\prime}z^{i}$ be two arbitrary elements in
$\mathcal{Q}[z]$. We have $w^{\prime}(r)=0$ if and only if all of the $q_{i}$
have valuation $0$, that is, if and only if $q_{i}=0$ for all $i$. Hence,
$w^{\prime}(r)=0$ implies $r=0$. Suppose the minimal $w^{\prime}$ valuation on
$r$ and $r^{\prime}$ is attained at the terms $q_{m}z^{m}$ and
$q_{n}^{\prime}z^{n}$ with $m$ and $n$ minimal for this properties. Then, for
all $0\leq i\,,j\leq n$, we have:

$w(q_{m})p^{-m}\geq w(q_{i})p^{-i}\,,$ $w(q^{\prime}_{n})p^{-n}\geq
w(q^{\prime}_{j})p^{-j}\,.$

Multiplying both inequalities implies:

$w(q_{m}q^{\prime}_{n})p^{-m-n}\geq w(q_{i}q^{\prime}_{j})p^{-i-j}\quad\forall
i\,,j\,.$

In particular, for every $1\leq k\leq 2n$, we have:

(12) $w(q_{m}q^{\prime}_{n})p^{-n-m}\geq\max\\{w(q_{i}q^{\prime}_{j})\mid
i+j=k\\}p^{-k}\geq w\left(\sum_{i+j=k}q_{i}q^{\prime}_{j}\right)p^{-k}\,.$

Suppose now, by contradiction, that there exists $i\neq m$ and $j\neq n$ with
$i+j=m+n$ such that $w(q_{i}q^{\prime}_{j})\geq w(q_{m}q^{\prime}_{n})$.
Without loss of generality, we may suppose that $i<m$, so that $n<j$. By the
minimality of $m$, we must have $w(q_{i})p^{-i}<w(q_{m})p^{-m}$, and by the
definition of $n$ we have $w(q^{\prime}_{j})p^{-j}\leq
w(q^{\prime}_{n})p^{-n}$. This implies:

$w(q_{i})/w(q_{m})<p^{i-m}=p^{n-j}\leq w(q^{\prime}_{n})/w(q^{\prime}_{j})\,,$

and thus $w(q_{i}q^{\prime}_{j})<w(q_{m}q^{\prime}_{n})$, a contradiction.
Therefore, if $i+j=m+n$, we have
$w(q_{i}q^{\prime}_{j})<w(q_{m}q^{\prime}_{n})$. Hence

(13)
$w(q_{m}q^{\prime}_{n})p^{-n-m}=w\left(\sum_{i+j=m+n}q_{i}q_{j}\right)p^{-m-n}\,.$

We have

$rr^{\prime}=\sum_{k=0}^{2n}\left(\sum_{i+j=k}q_{i}q^{\prime}_{j}\right)z^{k}\,.$

Since

$w\left(\sum_{i+j=k}q_{i}q^{\prime}_{j}\right)p^{-k}\leq
w(q_{m}q^{\prime}_{n})p^{-n-m}=w\left(\sum_{i+j=m+n}q_{i}q^{\prime}_{j}\right)p^{-n-m}\text{
by~{}(\ref{note7-eq1}) and~{}(\ref{note7-eq2})}\,,$

the minimal $w^{\prime}$ valuation on $rr^{\prime}$ must be attained in the
$z$-degree $m+n$, with value exactly $w^{\prime}(r)w^{\prime}(r^{\prime})$.

At last, we must show that
$w^{\prime}(r+r^{\prime})\leq\max\\{w^{\prime}(r)\,,w^{\prime}(r^{\prime})\\}$.
We may suppose, without loss of generality, that
$w(q_{m})p^{-m}=w^{\prime}(r)\leq
w^{\prime}(r^{\prime})=w(q^{\prime}_{n})p^{-n}$. Observe that in this case

$w(q_{i}+q^{\prime}_{i})p^{-i}\leq\max\\{w(q_{i})\,,w(q^{\prime}_{i})\\}p^{-i}\leq
w(q_{m})p^{-m}=w^{\prime}(r)\,.$

Since

$r+r^{\prime}=\sum_{i=1}^{n}(q_{i}+q^{\prime}_{i})z^{i}\,,$

we are done.

Now $\mathcal{Q}[z]$ is a Noetherian domain and thus we can construct the
division ring of rational functions $\mathcal{Q}^{\prime}=\mathcal{Q}(z)$ by
localizing at the non-zero elements. Hence, the valuation $w^{\prime}$ also
extends to $\mathcal{Q}^{\prime}$, which concludes the proof. ∎

By virtue of Claim 7.3, we can suppose that $\mathcal{Q}$ contains a central
element $z$ of valuation $p^{-1}$, for a division ring extension does not
change the induced rank function on $\mathbb{F}_{p}[\\![G]\\!]$. Consider now
$\mathcal{Q}_{0}$, the ring of elements with non-negative valuation. We note
that
$\varphi(\mathbb{F}_{p}[\\![G]\\!])\leq\mathcal{\mathcal{missing}}{Q}_{0}$ and
the non-zero ideals of $\mathcal{Q}_{0}$ are all of the form
$z^{i}\mathcal{Q}_{0}$ for $i\geq 0$. This implies that
$\mathcal{Q}_{0}/z\mathcal{Q}_{0}$ is the only simple
$\mathcal{Q}_{0}$-module, and the length of the $\mathcal{Q}_{0}$-modules
$\mathcal{Q}_{0}/z^{i}\mathcal{Q}_{0}$ are precisely $i$.

Let $A\in\operatorname{Mat}_{n\times n}(\mathbb{F}_{p}[\\![G]\\!])$ be a
matrix such that $\varphi(A)$ is invertible in $\mathcal{Q}$, that is,
$\varphi^{\\#}\operatorname{rk}_{\mathcal{Q}}(A)=n$. We define the left
$\mathbb{F}_{p}[\\![G]\\!]$-module
$M=\mathbb{F}_{p}[\\![G]\\!]^{n}/\mathbb{F}_{p}[\\![G]\\!]^{n}A$ and the
$\mathcal{Q}_{0}$-length of $M$ over an open subgroup $U$ to be

$l_{U}(M)=\operatorname{len}_{\mathcal{Q}_{0}}(\mathcal{Q}_{0}\otimes_{\mathbb{F}_{p}[\\![U]\\!]}M)\,,$

the length of the induced $\mathcal{Q}_{0}$-module. The proof henceforth will
follow similar arguments to the ones in [JZ20, Sec. 5].

Let $\dim_{G/G_{i}}$ and $\varphi^{\\#}\dim_{\mathcal{Q}}$ be the Sylvester
module rank functions associated to $\operatorname{rk}_{G/G_{i}}$ and
$\varphi^{\\#}\operatorname{rk}_{\mathcal{Q}}$. Then, since it suffices to
show that
$\operatorname{rk}_{G}(A)=\lim_{i\to\infty}\operatorname{rk}_{G/G_{i}}(A)=n$,
it only remains to show that

$\lim_{i\to\infty}\dim_{G/G_{i}}(M)=0\,.$

######  Claim 7.4.

We have that $l_{G}(M)$ is finite.

###### Proof.

Since $A$ is invertible over $\varphi$, we have

$\varphi^{\\#}\dim_{\mathcal{Q}}M=\dim_{\mathcal{Q}}\mathcal{Q}\otimes_{\mathbb{F}_{p}[\\![G]\\!]}M=0\,.$

Observing that

$\mathcal{Q}\otimes_{\mathbb{F}_{p}[\\![G]\\!]}M\simeq\mathcal{Q}\otimes_{\mathcal{Q}_{0}}\left(\mathcal{Q}_{0}\otimes_{\mathbb{F}_{p}[\\![G]\\!]}M\right)\,,$

this implies that the $\mathcal{Q}_{0}$-module
$\mathcal{Q}_{0}\otimes_{\mathbb{F}_{p}[\\![G]\\!]}M$ is torsion. Given that
$M$ is finitely generated and every proper quotient of $\mathcal{Q}_{0}$ is of
the form $\mathcal{Q}_{0}/z^{i}\mathcal{Q}_{0}$ and thus has finite length,
that module must have finite length over $\mathcal{Q}_{0}$. ∎

######  Claim 7.5.

Let $U\leq V$ be open subgroups of $G$ with $|V\colon U|=p$. Then,
$l_{U}(M)\leq pl_{V}(M)$. In particular, $l_{U}(M)\leq|G\colon U|l_{G}(M)$.

###### Proof.

Let $L=\mathcal{Q}_{0}\otimes_{\mathbb{F}_{p}[\\![U]\\!]}M$ and take any
element $g\in V\smallsetminus U$. Define the function
$\tau\colon\mathcal{Q}_{0}\times M\to L$ by

$\tau(q,m)=qg\otimes g^{-1}m-q\otimes m\,.$

Since $U$ must be normal in $V$, we have that $\tau(qu,m)=\tau(q,um)$ for
every $u\in U$, that is, $\tau$ induces a homomorphism of left
$\mathcal{Q}_{0}$-modules $\psi\colon L\to L$ such that

$\psi(q\otimes m)=qg\otimes g^{-1}m-q\otimes m\,.$

It is immediate to verify that $\psi^{i}(q\otimes m)=qg^{i}\otimes
g^{-i}m-q\otimes m$ and therefore $\psi^{p}=0$. Moreover,
$L/\psi(L)\simeq\mathcal{Q}_{0}\otimes_{\mathbb{F}_{p}[\\![V]\\!]}M$, and by
applying $\psi$ we find that this surjects onto $\psi^{i}(L)/\psi^{i+1}(L)$
for every $i$. Hence:

$l_{U}(M)=\operatorname{len}_{\mathcal{Q}_{0}}L\leq
p\cdot\operatorname{len}_{\mathcal{Q}_{0}}\mathcal{Q}_{0}\otimes_{\mathbb{F}_{p}[\\![V]\\!]}M=pl_{V}(M)\,.\qed$

Recall that $\overline{\mathcal{Q}}=\mathcal{Q}_{0}/\mathcal{Q}_{1}$ is a
division ring.

######  Claim 7.6.

We have that

$\dim_{\overline{\mathcal{Q}}}\overline{\mathcal{Q}}\otimes_{\mathbb{F}_{p}[\\![G_{i}]\\!]}M\leq\frac{l_{G_{i}}(M)}{i}\,.$

###### Proof.

When we view $M$ as a left $\mathbb{F}_{p}[\\![G_{i}]\\!]$-module, it has a
presentation of the form

$\mathbb{F}_{p}[\\![G_{i}]\\!]^{n|G\colon
G_{i}|}/\mathbb{F}_{p}[\\![G_{i}]\\!]^{n|G\colon G_{i}|}B$

for some matrix $B\in\operatorname{Mat}_{n|G\colon G_{i}|\times n|G\colon
G_{i}|}(\mathbb{F}_{p}[\\![G_{i}]\\!])$. By projecting the matrix onto
$\mathbb{F}_{p}$ and applying elementary matrix operations, we can find one
such matrix $B$ satisfying

$B=\begin{pmatrix}\operatorname{Id}_{a}+B_{1}&B_{2}\\\
B_{3}&B_{4}\end{pmatrix}$

where the $B_{i}$ are matrices with entries in the augmentation ideal
$I_{G_{i}}$ of $\mathbb{F}_{p}[\\![G_{i}]\\!]$ and $n|G\colon
G_{i}|-a=\dim_{\mathbb{F}_{p}}M/I_{G_{i}}$. Since the $B_{i}$ vanish when
acting on $\overline{\mathcal{Q}}$, one gets that

$\dim_{\overline{\mathcal{Q}}}\overline{\mathcal{Q}}\otimes_{\mathbb{F}_{p}[\\![G_{i}]\\!]}M=\dim_{\mathbb{F}_{p}}\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[\\![G_{i}]\\!]}M=\dim_{\mathbb{F}_{p}}M/I_{G_{i}}M=n|G\colon
G_{i}|-a\,.$

By definition, we have that for any $x\in I_{G_{i}}$, $v(\varphi(x))\geq i$,
so that $\varphi(I_{G_{i}})\leq z^{i}\mathcal{Q}_{0}$. Hence, there are
matrices $B_{2}^{\prime}$ and $B_{4}^{\prime}$ over $\mathcal{Q}_{0}$ such
that we can express $\varphi(B)$ as

$\varphi(B)=\underbrace{\begin{pmatrix}\operatorname{Id}_{a}+\varphi(B_{1})&B_{2}^{\prime}\\\
\varphi(B_{3})&B_{4}^{\prime}\end{pmatrix}}_{C_{1}}\underbrace{\begin{pmatrix}\operatorname{Id}_{a}&0\\\
0&z^{i}\operatorname{Id}_{n|G\colon G_{i}|-a}\end{pmatrix}}_{C_{2}}\,.$

This implies that

$(\mathcal{Q}_{0}/z^{i}\mathcal{Q}_{0})^{n|G\colon
G_{i}|-a}=\mathcal{Q}_{0}^{n|G\colon G_{i}|}/\mathcal{Q}_{0}^{n|G\colon
G_{i}|}C_{2}$

is a quotient of
$\mathcal{Q}_{0}\otimes_{\mathbb{F}_{p}[\\![G_{i}]\\!]}M\simeq\mathcal{Q}_{0}^{n|G\colon
G_{i}|}/\mathcal{Q}_{0}^{n|G\colon G_{i}|}\varphi(B)$. Therefore, we obtain
that

$\displaystyle l_{G_{i}}(M)$
$\displaystyle=\operatorname{len}_{\mathcal{Q}_{0}}\mathcal{Q}_{0}\otimes_{\mathbb{F}_{p}[\\![G_{i}]\\!]}M$
$\displaystyle\geq\operatorname{len}_{\mathcal{Q}_{0}}(\mathcal{Q}_{0}/z^{i}\mathcal{Q}_{0})^{n|G\colon
G_{i}|-a}$ $\displaystyle=i(n|G\colon
G_{i}|-a)=i\dim_{\overline{\mathcal{Q}}}\overline{\mathcal{Q}}\otimes_{\mathbb{F}_{p}[\\![G_{i}]\\!]}M\,.\qed$

Now we are ready to prove the theorem. Note that, by Lemma 2.4, to show that
$\varphi^{\\#}\operatorname{rk}_{\mathcal{Q}}\leq\operatorname{rk}_{G}$ it
suffices to show that they coincide over square matrices invertible under
$\varphi$. Indeed, any matrix $A\in\operatorname{Mat}_{n\times
m}(\mathbb{F}_{p}[\\![G]\\!])$ contains a maximal submatrix
$B\in\operatorname{Mat}_{k\times k}(\mathbb{F}_{p}[\\![G]\\!])$ such that

$\varphi^{\\#}\operatorname{rk}_{\mathcal{Q}}(B)=\varphi^{\\#}\operatorname{rk}_{\mathcal{Q}}A=k\,,$

and if $\operatorname{rk}_{G}(B)=k$ the lemma shows us that
$k\leq\operatorname{rk}_{G}(A)$.

Thus, let $M=\mathbb{F}_{p}[\\![G]\\!]^{n}/\mathbb{F}_{p}[\\![G]\\!]^{n}A$
where $A\in\operatorname{Mat}_{n\times n}(\mathbb{F}_{p}[\\![G]\\!])$ is
invertible under $\varphi$. We want to show that $\dim_{G}M=0$. We have that

$\displaystyle\frac{\dim_{\mathbb{F}_{p}}\mathbb{F}_{p}\otimes_{\mathbb{F}_{p}[\\![G_{i}]\\!]}M}{|G\colon
G_{i}|}$
$\displaystyle=\frac{\dim_{\overline{\mathcal{Q}}}\overline{\mathcal{Q}}\otimes_{\mathbb{F}_{p}[\\![G_{i}]\\!]}M}{|G\colon
G_{i}|}$ $\displaystyle\leq\frac{l_{G_{i}}(M)}{i|G\colon G_{i}|}\text{, by
Claim~{}\ref{lem:len-lemma},}$ $\displaystyle=\frac{l_{G}(M)}{i}\text{, by
Claim~{}\ref{lem:len-scaling}}$

Since $l_{G}(M)$ is finite by Claim 7.4, we have
$\lim_{i\to\infty}\frac{l_{G}(M)}{i}=0$, which concludes the proof. ∎

Now we are ready to finish the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

By Lemma 4.6 and Corollary 3.8, we can assume that $G$ is a mild flag free-
by-$\mathbb{Z}_{p}$ pro-$p$ group. Let $\mathcal{Q}$ and
$\mathbb{D}=\mathcal{D}[\\![t]\\!][\\![s;\sigma,\delta]\\!]$ be as in Section
5. Observe that the valuation $w$ on $\mathbb{D}$ naturally extends to
$\mathcal{Q}$ and the embedding $\mathbb{F}_{p}[\\![G]\\!]\to\mathcal{Q}$ is
continuous by Lemma 5.1. By Theorem 7.1,
$\varphi^{\\#}\operatorname{rk}_{\mathcal{Q}}\leq\operatorname{rk}_{G}\,$. In
the proof of Theorem 6.13 we showed that
$\varphi^{\\#}\operatorname{rk}_{\mathcal{Q}}=\operatorname{irk}_{\mathbb{F}_{p}[\\![G]\\!]}$.
Since
$\operatorname{irk}_{\mathbb{F}_{p}[\\![G]\\!]}\geq\operatorname{rk}_{G}$, we
have that
$\operatorname{irk}_{\mathbb{F}_{p}[\\![G]\\!]}=\operatorname{rk}_{G}$. ∎

##  8\. Abstract subgroups of free-by-$\mathbb{Z}_{p}$ pro-$p$ groups

In this section we discuss some consequences that Theorem 1.1 implies in the
study of the Lück approximation for finitely generated (abstract) subgroups of
a free-by-$\mathbb{Z}_{p}$ pro-$p$ groups.

Let us first recall the statement of the Lück approximation. Let $\Gamma$ be a
group and let $K$ be a field. For every matrix
$A\in\operatorname{Mat}_{n\times m}(K[\Gamma])$ and every normal subgroup $N$
of $\Gamma$ of finite index let us define

$\begin{array}[]{cccc}\phi_{\Gamma/N}^{A}:&K[\Gamma/N]^{n}&\to&K[\Gamma/N]^{m}\\\
&(x_{1},\ldots,x_{n})&\mapsto&(x_{1},\ldots,x_{n})A\end{array}.$

This is a $K$-linear map between two finite-dimensional $K$-vector spaces.
Thus, we can define a Sylvester rank function of $K[\Gamma]$ by means of

(14)
$\operatorname{rk}_{\Gamma/N}(A)=\frac{\dim_{K}\operatorname{Im}\phi_{\Gamma/N}^{A}}{|\Gamma:N|}=n-\frac{\dim_{K}\ker\phi_{\Gamma/N}^{A}}{|\Gamma:N|}.$

######  Conjecture 2 (The Lück approximation conjecture).

Let $\Gamma$ be a group, $K$ be a field and $\Gamma>N_{1}>N_{2}>\ldots$ a
descending chain of normal subgroups of $\Gamma$ of finite index with trivial
intersection. Let $A$ be a matrix over $K[\Gamma]$. Then the following holds.

  1. (1)

The sequence $\\{\operatorname{rk}_{\Gamma/N_{i}}(A)\\}_{i\geq 1}$ converges.

  2. (2)

The limit $\displaystyle\lim_{i\to\infty}\operatorname{rk}_{\Gamma/N_{i}}(A)$
does not depend on the chain $\Gamma>N_{1}>N_{2}>\ldots$.

  3. (3)

If moreover $\Gamma$ is locally indicable, then there exists a universal
embedding $K[\Gamma]\to\mathcal{Q}$ and
$\displaystyle\lim_{i\to\infty}\operatorname{rk}_{\Gamma/N_{i}}(A)=\operatorname{rk}_{K[\Gamma]}(A).$

If $K$ is of characteristic 0, the parts (1) and (2) of Conjecture 2 are known
to be true [JZ19a], and for part (3) we know that there exists an embedding
$\varphi:K[\Gamma]\to\mathcal{Q}$ into a division ring $\mathcal{Q}$ such that
$\displaystyle\lim_{i\to\infty}\operatorname{rk}_{\Gamma/N_{i}}(A)=\operatorname{rk}_{\mathcal{Q}}(\varphi(A))$
[JZLA20], but we still do not know whether $\varphi$ is universal in general
[JZ21]. If $K$ is of positive characteristic, the parts (1), (2) and (3) are
only known when $\Gamma$ is amenable ([JZ19b, JZ21]).

Let now $G$ be a free-by-$\mathbb{Z}_{p}$ pro-$p$ group and $\Gamma$ a
finitely generated (abstract) subgroup of $G$. It is clear that $\Gamma$ is
locally indicable and from [JZ21, Theorem 3.7] we know that there exist a
universal division $\mathbb{F}_{p}[\Gamma]$-ring of fractions
$\mathcal{D}_{\mathbb{F}_{p}[\Gamma]}$. The following theorem provides a
particular case of Conjecture 2 for $\Gamma$.

######  Theorem 8.1.

Let $G$ be a free-by-$\mathbb{Z}_{p}$ pro-$p$ group and $\Gamma$ a finitely
generated (abstract) subgroup of $G$. Let $G>U_{1}>U_{2}>\ldots$ be a chain of
normal open subgroups of $G$ with trivial intersections. Let $H_{i}=\Gamma\cap
U_{i}$. Then for every matrix $A$ over $\mathbb{F}_{p}[\Gamma]$,
$\displaystyle\lim_{i\to\infty}\operatorname{rk}_{\Gamma/H_{i}}(A)=\operatorname{rk}_{\mathbb{F}_{p}[\Gamma]}(A).$

###### Proof.

Since the closure of $\Gamma$ in $G$ is a finitely generated free-
by-$\mathbb{Z}_{p}$ pro-$p$ group, we can assume that $\Gamma$ is dense in
$G$, so that $G$ is topologically finitely generated. Hence, this implies that
the inclusion map $\Gamma\to G$ induces isomorphisms $\Gamma/H_{i}\simeq
G/U_{i}$ for every $i\geq 1$.

By Theorem 1.1, there exists $\mathcal{D}_{\mathbb{F}_{p}[\\![G]\\!]}$. Denote
by $\varphi:\mathbb{F}_{p}[\Gamma]\to\mathcal{D}_{\mathbb{F}_{p}[\\![G]\\!]}$
the corresponding embedding. Theorem 1.1 implies also that for every matrix
$A$ over $\mathbb{F}_{p}[\Gamma]$,

$\displaystyle\lim_{i\to\infty}\operatorname{rk}_{\Gamma/H_{i}}(A)=\lim_{i\to\infty}\operatorname{rk}_{G/U_{i}}(A)=\operatorname{rk}_{\mathcal{D}_{\mathbb{F}_{p}[\\![G]\\!]}}(\varphi(A)).$

Thus, we have to show that $\varphi$ is universal. Let $N$ be a normal free
pro-$p$ subgroup of $G$ such that $G/N\cong\mathbb{Z}_{p}$. Put
$H=N\cap\Gamma$. Denote by $\mathcal{D}_{\Gamma}$ (resp. $\mathcal{D}_{H}$)
the division closure of $\mathbb{F}_{p}[\Gamma]$ (resp. $\mathbb{F}_{p}[H]$)
in $\mathcal{D}_{\mathbb{F}_{p}[\\![G]\\!]}$ and by $R$ the subring generated
by $\mathcal{D}_{H}$ and $\mathbb{F}_{p}[\Gamma]$.

######  Claim 8.2.

$\mathcal{D}_{H}$ is the universal division ring of fraction of
$\mathbb{F}_{p}[H]$

###### Proof.

This is proved in [JZ24, Proposition 3.5]. ∎

######  Claim 8.3.

The induced map $\mathcal{D}_{H}*(\Gamma/H)\to R$ is an isomorphism.

###### Proof.

Let $\gamma_{1},\ldots,\gamma_{n}$ lie in different clases modulo $H$. There
exists an open normal subgroup $U$ of $G$ containing $N$ such that
$\gamma_{1},\ldots,\gamma_{n}$ lie in different clases modulo $U$. By
Proposition 3.4(d), $\displaystyle\sum_{i=1}^{n}\mathcal{D}_{H}\gamma_{i}$ is
direct. This implies the claim. ∎

The two claims toghether imply that the ring $R$ is isomorphic to a crossed
product $\mathcal{D}_{\mathbb{F}_{p}[H]}*(\Gamma/H)$. Since $\Gamma/H$ is
finitely generated, $R$ a Noetherian ring. Thus, $\mathcal{D}_{\Gamma}$ is
isomorphic to the Ore division ring of fractions of $R$. Hence, we obtain that
$\mathcal{D}_{\Gamma}$ will be the universal division ring of fraction of
$\mathbb{F}_{p}[\Gamma]$. ∎

##  Appendix A Localization of profinite rings

Let $R$ be an associative ring and $\Sigma$ a collection of square matrices
over $R$. The universal localization of $R$ with respect to $\Sigma$ is a ring
$R_{\Sigma}$ that comes with a ring homomorphism $\lambda\colon R\to
R_{\Sigma}$ such that $\lambda(A)$ is invertible for every $A\in\Sigma$, and
$R_{\Sigma}$ is universal for that property.

We denote by $\widetilde{\Sigma}$ the set of all square matrices over $R$ that
become invertible under $\lambda$. In general, $\widetilde{\Sigma}$ is greater
than $\Sigma$. However $R_{\widetilde{\Sigma}}$ is $R$-isomorphic to
$R_{\Sigma}$. We say that $\Sigma$ is complete if $\Sigma=\widetilde{\Sigma}$.

It is clear that a complete collection of square matrices is multiplicative,
that is it contains $\operatorname{GL}_{n}(R)$ for every $n$, is closed under
products when defined and if $A\,,B\in\Sigma$ then $\begin{pmatrix}A&C\\\
0&B\end{pmatrix}\in\Sigma$ for any matrix $C$ of the appropriate size. It also
has the following property regarding diagonal summands on $\Sigma$:

(DS) $\text{If }\begin{pmatrix}A&0\\\ 0&B\end{pmatrix}\in\Sigma\text{, then
}A\,,B\in\Sigma\,.$

J. Beachy has shown in [Bea93] that $R_{\Sigma}$ and $\lambda$ can be
constructed as follows. From now on we assume that $\Sigma$ is complete. Let
$R^{n}$ denote the rows of $n$ elements in $R$, ${}^{n}R$ the columns of $n$
elements in $R$ and $\Sigma_{n}$ denote the subset of $n$-by-$n$ matrices in
$\Sigma$. Consider the set $T_{\Sigma}$ given by the disjoint union of all the
products $R^{n}\times\Sigma_{n}\times{{}^{n}R}$, that is, the triples
$(a,C,x)$ where $a$ is a row matrix over $R$ of length $n$, $C\in\Sigma_{n}$
and $x$ is a column matrix over $R$ of height $n$. We define two binary
operations of sum and product in $T_{\Sigma}$ as follows:

$(a,C,x)+(b,D,y)=\left(\begin{pmatrix}a&b\end{pmatrix}\,,\begin{pmatrix}C&0\\\
0&D\end{pmatrix}\,,\begin{pmatrix}x\\\ y\end{pmatrix}\right)\,,$
$(a,C,x)\cdot(b,D,y)=\left(\begin{pmatrix}a&0\end{pmatrix}\,,\begin{pmatrix}C&-xb\\\
0&D\end{pmatrix}\,,\begin{pmatrix}0\\\ y\end{pmatrix}\right)\,,$

where the matrices make sense because $\Sigma$ is multiplicative, $x$ is an
$n\times 1$ matrix and $b$ is an $1\times m$ matrix.

We define an equivalence relation $\mathcal{R}_{1}\subset T_{\Sigma}\times
T_{\Sigma}$ on $T_{\Sigma}$ by means of $(a,C,x)\sim_{\mathcal{R}_{1}}(b,D,y)$
if there exists invertible matrices $U,V$ over $R$ such that $b=aU$, $y=Vx$
and $D=VCU$. Observe that in particular $C$ and $D$ must have the same size if
$(a,C,x)\sim_{\mathcal{R}_{1}}(b,D,y)$. We denote the quotient
$T_{\Sigma}/\sim_{\mathcal{R}_{1}}$ by $\Sigma^{-1}R$ and the equivalence
class of a triple $(a,C,x)$ in $\Sigma^{-1}R$ by $(a\colon C\colon x)$.

The operations of sum and product are well defined on the equivalence classes
and descend to a sum and a product on the quotient $T_{\Sigma}$. With the sum,
it becomes a commutative semigroup. To obtain an abelian additive group, we
further introduce a new equivalence relation.

Let $\Sigma^{-1}_{0}R$ be the subsemigroup of $\Sigma^{-1}R$ generated by all
the elements of the form $(a\colon C\colon 0)$ and $(0\colon C\colon y)$. It
is shown in [Bea93] that all the elements of $\Sigma^{-1}_{0}R$ are either of
that form or of the form
$\left(\begin{pmatrix}a&0\end{pmatrix}\colon\begin{pmatrix}A&0\\\
0&D\end{pmatrix}\colon\begin{pmatrix}0\\\ x\end{pmatrix}\right)$ for blocks of
the appropriate sizes. This subsemigroup allows us to define the congruence
relation $\mathcal{R}_{2}\subseteq\Sigma^{-1}R\times\Sigma^{-1}R$ on
$\Sigma^{-1}R$, where two elements
$\overline{p}\,,\overline{q}\in\Sigma^{-1}R$ are equivalent if there exists
$\overline{z}_{1}\,,\overline{z}_{2}\in\Sigma_{0}^{-1}R$ such that
$\overline{p}+\overline{z}_{1}=\overline{q}+\overline{z}_{2}$.

The quotient space $\Sigma^{-1}R/\sim_{\mathcal{R}_{2}}$ inherits the sum and
product, under which it becomes an associative ring with trivial element being
the class $[(1:1:0)]$ and the identity element being the class $[(1:1:1)]$.
The additive inverse of $[(a\colon C\colon x)]$ is $[(a\colon C\colon-x)]$.
One of the main results of [Bea93] is that this quotient space is precisely
the universal localization $R_{\Sigma}$, where the map $\lambda\colon R\to
R_{\Sigma}$ is given by $x\mapsto[(1\colon 1\colon x)]$.

Suppose now that $R$ is a topological ring. In that case, all the spaces
${}^{n}R$ and $R^{n}$ have a natural product topology, and $\Sigma_{n}$ also
becomes a topological space as it is contained in
$\operatorname{Mat}_{n}(R)\simeq R^{n^{2}}$. Hence, $T_{\Sigma}$ becomes a
topological space endowed with the direct union topology on which the
operations of sum and product defined are continuous. Under the quotient
topology, those operations are still continuous on $\Sigma^{-1}R$ and
$R_{\Sigma}$, so that $R_{\Sigma}$ becomes a topological ring. We call this
topology structure on $R_{\Sigma}$ the induced topological structure from $R$.
It is clear that $\lambda\colon R\to R_{\Sigma}$ is a continuous ring
homomorphism. The main result of the appendix is the following theorem.

######  Theorem A.1.

Let $R$ be a profinite ring and $\Sigma$ a collection of square matrices over
$R$. Then $R_{\Sigma}$ admits a Hausdorff ring topology such that
$\lambda\colon R\to R_{\Sigma}$ is continuous. In particular, every finitely
generated $R$-submodule of $R_{\Sigma}$ is profinite.

###### Proof.

Without loss of generality we assume that $\Sigma$ is complete. Since $R$ is
profinite, then $T_{\Sigma}$ is a Hausdorff space under the induced topology.

######  Claim A.2.

We have that $\Sigma^{-1}R$ is Hausdorff.

###### Proof.

We must show that the equivalence relation $\mathcal{R}_{1}$ is a closed
subspace of $T_{\Sigma}\times T_{\Sigma}$. Since $T_{\Sigma}$ is given the
union topology and each equivalence class under $\mathcal{R}_{1}$ is entirely
contained in one of the subsets $T_{n}=R^{n}\times\Sigma_{n}\times{{}^{n}R}$,
it suffices to prove that $X_{n}=(\mathcal{R}_{1}\cap(T_{n}\times T_{n}))$ is
closed in $T_{n}\times T_{n}$.

Consider the continuous maps $\psi\colon T_{n}\times
T_{n}\times\operatorname{GL}_{n}(R)^{2}\to
R^{n}\times\operatorname{Mat}_{n}(R)\times{{}^{n}R}$ and $\pi\colon
T_{n}\times T_{n}\times\operatorname{GL}_{n}(R)^{2}\to T_{n}\times T_{n}$
given by

$\psi(a,C,x,b,D,y,U,V)=(aU-b\,,VCU-D\,,Vx-y)\,,$
$\pi(a,C,x,b,D,y,U,V)=(a,C,x,b,D,y)\,.$

It is clear that $X_{n}=\pi(\psi^{-1}(0,0,0))$. Since
$\operatorname{GL}_{n}(R)^{2}$ is compact, $\pi$ is a closed map, so we are
done. ∎

The kernel of $\Sigma^{-1}R\to R_{\Sigma}$ is the subsemigroup

$\Sigma^{-1}_{0}R-\Sigma^{-1}_{0}R=\\{\overline{p}\in\Sigma^{-1}R\,\mid\,\exists\overline{z}\in\Sigma^{-1}_{0}R\text{
such that }\overline{p}+\overline{z}\in\Sigma^{-1}_{0}R\\}\,,$

which contains $\Sigma^{-1}_{0}R$ but it might be strictly larger.

######  Claim A.3.

We have that $\Sigma^{-1}_{0}R$ is closed in $\Sigma^{-1}R$.

###### Proof.

It again suffices to consider
$\Sigma^{-1}R_{n}=(R^{n}\times\Sigma_{n}\times{{}^{n}R})/\sim_{\mathcal{R}_{1}}$
and show that $Y_{n}=\Sigma^{-1}_{0}R\cap\Sigma^{-1}R_{n}$ is closed in
$\Sigma^{-1}R_{n}$, as we have a homeomorphism between $\Sigma^{-1}R$ and the
disjoint union $\bigcup_{n\geq 1}\Sigma^{-1}R_{n}$.

Decompose $Y_{n}=Z_{I}\cup Z_{II}\cup\bigcup_{i=1}^{n-1}Z_{III,i}$ where

$Z_{I}=\\{(a\colon C\colon 0)\in Y_{n}\\}\,,$ $Z_{II}=\\{(0\colon C\colon
x)\in Y_{n}\\}\,,$
$Z_{III,i}=\left\\{\left(\begin{pmatrix}a&0\end{pmatrix}\colon\begin{pmatrix}C&0\\\
0&D\end{pmatrix}\colon\begin{pmatrix}0\\\ x\end{pmatrix}\right)\in
Y_{n}\,\mid\,\begin{array}[]{l}a\in R^{i}\,,x\in{{}^{n-i}R},\\\
C\in\Sigma_{i}\,,D\in\Sigma_{n-i}\end{array}\right\\}\,.$

We only need to show that each one of the $Z_{I}$, $Z_{II}$ and $Z_{III,i}$ is
closed in $\Sigma^{-1}R_{n}$.

The inverse image of $Z_{I}$ in $R^{n}\times\Sigma_{n}\times{{}^{n}R}$ is the
set of triples of the form $(a,C,0)$, which immediately shows that it is
closed. An analogous description holds for the inverse image of $Z_{II}$,
hence it is also closed. The only case requiring a finer description is that
of $Z_{III,i}$.

The inverse image of $Z_{III,i}$ in $R^{n}\times\Sigma_{n}\times{{}^{n}R}$ is

$\left\\{\left(\begin{pmatrix}a&0\end{pmatrix}U\,,V\begin{pmatrix}C&0\\\
0&D\end{pmatrix}U\,,V\begin{pmatrix}0\\\
x\end{pmatrix}\right)\,\mid\,U\,,V\in\operatorname{GL}_{n}(R)\right\\}\,.$

Define

$Z_{i}=\left\\{\left(\begin{pmatrix}a&0\end{pmatrix}\,,\begin{pmatrix}C&0\\\
0&D\end{pmatrix}\,,\begin{pmatrix}0\\\ x\end{pmatrix}\right)\right\\}\subseteq
R^{n}\times\Sigma_{n}\times{{}^{n}R}\,.$

Then, the inverse image in question is the image of
$Z_{i}\times(\operatorname{GL}_{n}(R))^{2}$ under the continuous right group
action map
$\mu\colon(R^{n}\times\Sigma_{n}\times{{}^{n}R})\times(\operatorname{GL}_{n}(R))^{2}\to
R^{n}\times\Sigma_{n}\times{{}^{n}R}$ given by

$\mu(b,A,y,U,V)=(bU,V^{-1}AU,V^{-1}y)\,.$

Since $\Sigma$ satisfies property (DS), the set $Z_{i}$ is closed, and because
$(\operatorname{GL}_{n}(R))^{2}$ is compact the map $\mu$ is closed. Hence,
$\mu(Z_{i}\times(\operatorname{GL}_{n}(R))^{2})$ and $Z_{III,i}$ are both
closed. ∎

######  Claim A.4.

We have that $\\{[(1\colon 1\colon 0)]\\}$ is closed in $R_{\Sigma}$.

###### Proof.

We must show that $\Sigma_{0}^{-1}R-\Sigma_{0}^{-1}R$ is a closed subset of
$\Sigma^{-1}R$. Consider the map
$\varphi\colon\Sigma^{-1}R\times\Sigma^{-1}_{0}R\to\Sigma^{-1}R$ given by
$\varphi(\overline{p},\overline{z})=\overline{p}+\overline{z}$. The inverse
image $\varphi^{-1}(\Sigma^{-1}_{0}R)$ is closed in the product by Claim A.3.
Since this inverse image is precisely
$(\Sigma^{-1}_{0}R-\Sigma^{-1}_{0}R)\times\Sigma^{-1}_{0}R$, it proves that
$\Sigma^{-1}_{0}R-\Sigma^{-1}_{0}R$ is closed in $\Sigma^{-1}R$. ∎

By Claim A.4, the ring topology constructed on $R_{{\Sigma}}$ is such that
$\\{0\\}$ is closed. Since $(R_{{\Sigma}},+)$ is a topological abelian group,
it must also be Hausdorff.

For the last statement of the theorem, take one such $M\leq R_{\Sigma}$ and
observe that the kernel $I$ of any surjection $R^{n}\to M$ must be closed in
$R^{n}$. Therefore, $M\simeq R^{n}/I$ is profinite. ∎

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# Thermodynamics and weak cosmic censorship conjecture in (2+1)-dimensional
regular black hole with nonlinear electrodynamics sources

Yi-Wen Han  Ming-Jian Lan  Xiao-Xiong Zeng<EMAIL_ADDRESS>Department
of physics, college of computer science and electronic information
engineering, Chongqing Technology and Business University, Chongqing 400067,
China

###### Abstract

We study the dynamical behavior of spinor particles, and get the energy-
momentum relation of charged particles by solving the Dirac equation. Based on
the energy-momentum relation, we investigate the laws of thermodynamics and
the weak cosmic censorship conjecture for the (2+1)-dimensional regular black
hole with nonlinear electrodynamics sources in the normal phase space and the
extended phase space. Our results show that the first law of thermodynamics as
well as the weak cosmic censorship conjecture are valid in both the phase
spaces. However, the second law of thermodynamics is violated in the extended
phase space, implying that the first law and weak cosmic censorship conjecture
do not depend on the phase space while the second law depends. In addition, in
the extended phase space, we find the configurations of the extremal and near-
extremal black holes will not be changed for the final states and initial
states are the same.

###### keywords:

Thermodynamics; Weak cosmic censorship conjecture; Extended phase space;
Regular black hole

###### MSC:

[2010] 00-01, 99-00

††journal: Journal of LaTeX Templates

##  1 Introduction

Based on the ideas of Henneaux et al. [1], Caldarelli et al. [2] took the lead
in regarding the cosmological constant as a state parameter of the
thermodynamic systems, and shown that the first law of thermodynamics for the
Kerr-Newman-AdS space time was $dM=TdS+\Omega dJ+\Phi dQ+\Theta d\Lambda$, in
which, $\Lambda$ is the generalized force conjugating to the state parameter.
Since then, more and more investigations have shown that a variable
cosmological parameter can enrich the thermodynamic state functions [3, 4, 5,
6, 7, 8, 9] so that the phase spaces of the thermodynamic system are extended,
which is different from previous treatments where the cosmological parameter
is a constant [10, 11, 12, 13, 14, 15]. Recently, some authors claimed that
the cosmological constant can be used as the pressure of the thermodynamic
system [16, 17], and its thermodynamic conjugate can be defined naturally as
volume. In this framework, the mass is not the internal energy but the
enthalpy. On this premise, the thermodynamics and phase transitions of a class
of black holes with cosmological constants have been studied extensively in
the extended phase space [18, 19, 20, 21, 22, 23, 24, 25, 26]. However, most
of them focused on only the first law of thermodynamics. There was little work
to check whether the second law of thermodynamics as well as weak cosmic
censorship conjecture are valid in the extended phase space.

Recently, Gwak [27, 28] suggested that the laws of thermodynamics can be
checked by throwing charged particles into black holes in the framework of
extended phase space. Their results showed that the second law of
thermodynamics was violated as the contributions of pressure and volume are
considered. They also investigated the weak cosmic censorship conjecture and
found that it was valid. However, different from the case in the normal phase
space, the configurations of the black hole are found to be not changed for
both the extremal and near-extremal black holes. Subsequently, this work was
extended to Born-Infeld-anti-de Sitter black hole, in which the contribution
of the vacuum polarization energy was considered besides the pressure [29]. In
addition, [27] has also been extended to the case that the absorbed particles
are fermions, for which the Dirac equation was used to obtain the energy-
momentum relation [30, 31, 32]. In this paper, we intend to extend the idea in
[27] to the (2+1)-dimensional regular black hole with nonlinear
electrodynamics sources. Unlike previous studies, this space-time is coupled
by Einstein’s gravity and nonlinear electrodynamics. We want to explore
whether nonlinear electrodynamics affect the thermodynamics and the weak
cosmic censorship conjecture, which have not been reported so far. We will pay
attention to the case that the absorbed particles are fermions so that the
Dirac equation will be used to investigate the dynamical behavior of the
particles.

The present work is organized as follows. In section 2, the relation between
the energy and momentum of absorbed fermions is obtained by solving the Dirac
equation. In section 3, we will study thermodynamic laws and weak cosmic
censorship conjecture in the normal phase space. Section 4 will extend the
investigations on thermodynamic laws and weak cosmic censorship conjecture to
the extended phase space. Finally, the conclusion is presented in section 5.
Throughout the paper, the units $\pi=\hbar=c=1$ are used.

##  2 Energy and momentum of an absorbed fermion

The action of the $(2+1)$-Einstein gravity coupled with nonlinear
electrodynamics is given by [33, 34, 35]

$S=\int d^{3}x\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi}+L(F)\right],$ (1)

where $g$ is the determinant of the metric tensor, $R$ is the Ricci scalar,
$\Lambda=-1/l^{2}$ is the cosmological constant,
$F_{\mu\nu}=A_{\nu,\mu}-A_{\mu,\nu}$, and $L(F)$ is the Lagrangian of the
nonlinear electrodynamics with $F=F^{\mu\nu}F_{\mu\nu}$ [36]. From Eq.(1), we
can obtain the solutions of the (2+1)-dimensional regular black hole with
nonlinear electrodynamics sources, that is

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\phi^{2},$ (2)

where

$f(r)=r^{2}/l^{2}-m-q^{2}\log(q^{2}/a^{2}l^{2}+r^{2}/l^{2})$ (3)

in which $m$ and $q$ are the parameters that relate to the mass and charge of
the black hole. The strength of electric field is

$E(r)=\frac{a^{4}qr^{3}}{16\pi\left(q^{2}+a^{2}r^{2}\right)^{2}}.$ (4)

After integration with the relation $A_{t}(r)=\int_{r}^{\infty}E(x)dx$ [34],
we can get the non-vanishing component of the vector potential

$A_{t}(r)=-\frac{a^{2}qr^{2}-q^{3}\log\left[q^{2}+a^{2}r^{2}\right]}{32\pi}.$
(5)

By using the formula given in [37], we can get the electric charge of the
black hole, $Q=8q$. Comparing with the static, (2+1)-dimensional black hole,
we also can easily obtain the mass of the regular black holes, $M=m/8$.
Replacing $m$ and $q$ in Eq.(3) with $M$ and $Q$ respectively, the metric
function can be rewritten as

$F(r)=-8M+\frac{r^{2}}{l^{2}}-\frac{1}{64}Q^{2}\log\left(\frac{Q^{2}}{64a^{2}l^{2}}+\frac{r^{2}}{l^{2}}\right).$
(6)

Now, we investigate the dynamical of a charged fermion as it is swallowed by
the (2+1)-dimensional regular black hole with nonlinear electrodynamics
sources. The Dirac equations for electromagnetic field in curved spacetime can
be expressed as

$i\gamma^{\mu}\left(\partial_{\mu}+\Omega_{\mu}-\frac{i}{\hbar}eA_{\mu}\right)\Psi-\frac{\mu_{0}}{\hbar}\Psi=0,$
(7)

where $u_{0}$ and $e$ are respectively the rest mass and charge of a fermion,
$\Omega_{\mu}=\frac{i}{2}\Gamma_{\mu}^{\rho\tau}\sum_{\rho\tau}$,
$\sum_{\rho\tau}=\frac{i}{4}\left[{\gamma^{\rho},\gamma^{\tau}}\right]$,
$\gamma^{\mu}$ matrices satisfy
$\left\\{{\gamma^{\mu},\gamma^{\nu}}\right\\}=2g^{\mu\nu}I$, in which $I$ is
the unit matrix. In order to get the solution of the Dirac equation, we have
to choose matrices $\gamma^{\mu}$. Here, we set

$\gamma^{\mu}=\left(-iF^{-\frac{1}{2}}\sigma^{2},F^{\frac{1}{2}}\sigma^{1},\frac{1}{r}\sigma^{3}\right),$
(8)

where $\sigma^{\mu}$ is the Pauli matrix

$\sigma^{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\sigma^{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\sigma^{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (9)

A fermion have spin up state and a spin down state with spin $1/2$. For the
sake of simplicity, we only investigate the spin up case for the case of spin
down is similar. We use the ansatz for the two-component spinor $\Psi$ as

$\Psi=\left({{\begin{array}[]{*{20}c}{C\left({t,r,\phi}\right)\hfill}\\\
{D\left({t,r,\phi}\right)\hfill}\\\
\end{array}}}\right)\exp\left({\frac{i}{\hbar}G\left({t,r,\phi}\right)}\right).$
(10)

Substituting Eq.(10) into Eq.(7), we get the following two simplified
equations

$C\left(\mu_{0}+\frac{1}{r}\partial_{\phi}G\right)+D\left[\sqrt{F}\partial_{r}G-\left(\frac{1}{\sqrt{F}}\partial_{t}G-\frac{1}{\sqrt{F}}eA_{t}\right)\right]=0,$
(11)
$C\left[\sqrt{F}\partial_{r}G+\left(\frac{1}{\sqrt{F}}\partial_{t}G-\frac{1}{\sqrt{F}}eA_{t}\right)\right]+D\left(\mu_{0}-\frac{1}{r}\partial_{\phi}G\right)=0.$
(12)

The above two equations have a nontrivial solution for $C$ and $D$ if and only
if the determinant of the matrix coefficients is zero. So we get

$\frac{1}{r^{2}}\left(\partial_{\phi}G\right)^{2}-\mu_{0}^{2}+\left(\sqrt{F}\partial_{r}G\right)^{2}-\left(\frac{1}{\sqrt{F}}\partial_{t}G-\frac{1}{\sqrt{F}}eA_{t}\right)^{2}=0.$
(13)

There are two Killing vectors in the (2+1)-dimensional spacetime, so
$G\left(t,r,\phi\right)$ in Eq.(13) can be separated into

$G=-\omega t+L\phi+R\left(r\right)+\lambda,$ (14)

where $\omega$ and $L$ are Dirac particle’s energy and angular momentum
respectively, and $\lambda$ is a complex constant. Substituting Eq.(14) into
Eq.(13), we obtain

$\partial_{r}R\left(r\right)=\pm\frac{1}{F}\sqrt{\left(\omega+eA_{t}\right)^{2}+F\left(\mu_{0}^{2}-\frac{L^{2}}{r^{2}}\right)}.$
(15)

In order to study thermodynamics, we focus on only the radial momentum of
particles near the horizon, i.e. $p^{r}\equiv
g^{rr}p_{r}=\partial_{r}R\left(r\right)$. From Eq. (15), we get

$\omega=|p^{r}|-eA_{t}(r_{h}).$ (16)

As done in [27, 38, 39, 40], we choose the positive sign in front of
$|p^{r}|$.

##  3 Thermodynamics and weak cosmic censorship conjecture of
(2+1)-dimensional black hole in the normal phase space

The electrostatic potential between the black hole horizon and the infinity is
defined by $\Phi=A_{t}(\infty)-A_{t}(r_{h})$. After integration, we can obtain

$\Phi_{h}=-\frac{Q\left(l^{2}Q^{2}+\left(l^{2}Q^{2}+64a^{2}r_{h}^{2}\right)\log\left(\frac{Q^{2}}{64a^{2}}+\frac{r_{h}^{2}}{l^{2}}\right)\right)}{256\left(l^{2}Q^{2}+64a^{2}r_{h}^{2}\right)},$
(17)

where $r_{h}$ is the event horizon of the (2+1)-dimensional regular black hole
with nonlinear electrodynamics sources, which is determined by $F(r_{h})=0$.
According to the definition of surface gravity, the Hawking temperature can be
written as

$T_{h}=\frac{r_{h}}{128l^{2}\pi}\left(64-\frac{64a^{2}Q^{2}r_{h}^{2}}{l^{2}Q^{2}+64a^{2}r_{h}^{2}}\right).$
(18)

By using the Bekenstein-Hawking entropy area relation, the black hole entropy
can be written as

$S_{h}=\frac{1}{2}\pi r_{h}.$ (19)

From Eq.(6), we can get the mass of the (2+1)-dimensional regular black hole
with nonlinear electrodynamics sources, which can be expressed as

$M=\frac{r_{h}^{2}}{8l^{2}}-\frac{Q^{2}}{512}{\log}\left[\frac{1}{l^{2}}\left(\frac{Q^{2}}{64a^{2}}+r_{h}^{2}\right)\right].$
(20)

As a charged fermion is swallowed by the regular black holes, the variations
of the internal energy and charge of the black hole system should satisfy

$\omega=dM,\quad e=dQ,$ (21)

where the conservations of energy and charge have been considered. Now,
Eq.(16) can be rewritten as

$dM=\Phi_{h}dQ+p^{r}.$ (22)

Similarly, the event horizon of a black hole changes as it absorbs particles,
leading to change in $F(r)$. In the new horizon, there is also a relation,
$F(r_{h}+dr_{h})=0$. In other words, the change of the horizon should satisfy

$dF_{h}=\frac{\partial F_{h}}{\partial M}dM+\frac{\partial F_{h}}{\partial
Q}dQ+\frac{\partial F_{h}}{\partial r_{h}}dr_{h}=0.~{}~{}~{}~{}$ (23)

Inserting Eq.(22) into Eq.(23), $dM$ will be deleted. Interestingly, $dQ$ will
de deleted too. We can get $dr_{h}$ directly

$dr_{h}=\frac{4p^{r}\left(l^{4}Q^{2}+64a^{2}l^{2}r_{h}^{2}\right)}{r_{h}\left(\left(1-a^{2}\right)l^{2}Q^{2}+64a^{2}r_{h}^{2}\right)}.$
(24)

With Eqs.(19) and (24), the variation of entropy can be expressed as

$dS_{h}=\frac{2\pi
p^{r}\left(l^{4}Q^{2}+64a^{2}l^{2}r_{h}^{2}\right)}{r_{h}\left(\left(1-a^{2}\right)l^{2}Q^{2}+64a^{2}r_{h}^{2}\right)}.$
(25)

From Eqs.(18) and (25), we get

$T_{h}dS_{h}=p^{r}.$ (26)

So, the internal energy in Eq.(22) can be rewritten as

$dM=TdS+\Phi dQ,$ (27)

which is the first law of the (2+1)-dimensional regular black hole with
nonlinear electrodynamics sources in the normal phase space.

So far, we have derived the first law of thermodynamics. Next, we are going to
discuss the second law of the black hole thermodynamics. For the extreme black
holes, the temperature is zero, which implies.

$r_{h}=\frac{lQ}{\sqrt{-64a^{2}+a^{2}Q^{2}}}.$ (28)

The second law for the extremal black holes is meaningless strictly for the
temperature of the system is zero. So we are interested only the non-extremal
black hole, where the temperature is larger then zero. According to the
Eq.(18), the event horizon should satisfy $r>r_{h}$. In this case, $dS_{h}$ is
positive in Eq.(25), which shows that the second law of thermodynamics is also
valid.

Furthermore, we can investigate the weak cosmic censorship conjecture, which
states that the singularity should be hidden by the event horizon of the black
hole for an observer located at infinity. To ensure the validity of the weak
cosmic censorship conjecture, the existence of an event horizon is needed. To
do this, we try to test whether an event horizon exists when a fermion is
absorbed by the black hole. That is, whether there are solutions for $F(r)$.

For the (2+1)-dimensional regular black hole with nonlinear electrodynamics
sources, there is a minimum value for $F(r)$ with the radial coordinate
$r_{min}$. For the case $F(r_{min})>0$, there is not a horizon while for the
case $F(r_{min})\leq 0$, there are horizons always. At $r_{min}$, the
following relations should be satisfied, yielding

$\displaystyle F|_{r=r_{min}}\equiv F_{min}=\alpha\leq 0,$
$\displaystyle\partial_{r}F|_{r=r_{min}}\equiv F^{\prime}_{min}=0,$
$\displaystyle(\partial_{r})^{2}F|_{r=r_{min}}\equiv
F^{\prime\prime}_{min}>0.$ (29)

For the extreme black hole, $\alpha=0$, and for the near extreme black hole,
$\alpha$ is a small quantity. The inner horizon and outer horizon are
distributed at either side of $r_{min}$. As a fermion is swollowed into the
black hole, the change in mass and charge of the black hole can be written as
$(M+dM,Q+dQ)$. The locations of minimum value and the event horizon are
written as $r_{min}+dr_{min}$, $r_{h}+dr_{h}$ correspondingly. Here, we have a
transformation of $F(r)$, which is labeled as $dF_{min}$. By using condition
$F^{\prime}_{min}=0$, at $r_{min}+dr_{min}$, $F(r)$ can be expressed as

$\displaystyle
F|_{r=r_{min}+dr_{min}}=F_{min}+dF_{min}=\alpha+\left(\frac{\partial
F_{min}}{\partial M}dM+\frac{\partial F_{min}}{\partial Q}dQ\right),$ (30)

For the extremal black hole, the horizon is located at $r_{min}$, the Eq.(22)
is valid at $r_{min}$. Substituting Eq.(22) into Eq.(30), $dQ$ will be deleted
at the same time. Therefore, the Eq.(30) can be reduced to

$F_{min}+dF_{min}=-8p^{r}.$ (31)

Obviously, $F(r_{min}+dr_{min})$ is smaller than $F(r_{min})$ when a charged
fermion is engulfed by the black hole. This will cause extremal black holes to
become into non-extremal black holes. In other words, the weak cosmic
censorship conjecture is valid in the normal phase space.

##  4 Thermodynamics and weak cosmic censorship conjecture in the extended
phase space

Now, we turn to investigate thermodynamics and weak cosmic censorship
conjecture with a variable cosmological parameter, namely $\Lambda$ is treated
as the thermodynamic pressure, and its conjugate variable is the thermodynamic
volume, which can be represented as respectively

$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi l^{2}},$ (32)
$\displaystyle V_{h}$ $\displaystyle=$ $\displaystyle\frac{\pi
r_{h}^{2}\left(\left(1-a^{2}\right)l^{2}Q^{2}+64a^{2}r_{h}^{2}\right)}{l^{2}Q^{2}+64a^{2}r_{h}^{2}}.$
(33)

From Eqs. (18), (19), (32) and (33), we obtain the Smarr formula

$T_{h}S_{h}=2V_{h}P.$ (34)

Considering that $M$ has the physical significance of enthalpy in the extended
phase space[16, 17], it has a relation to internal energy as

$M=U+PV_{h}.$ (35)

As a charged fermion is swollowed into the black hole, the energy and charge
are supposed to be conserved. So, the change in energy and charge of the black
hole system should be equal to the energy and charge of the particle. Namely

$\omega=dU=d(M-PV_{h}),\quad e=dQ.$ (36)

The energy in Eq.(22) changes correspondingly into

$dU=\Phi_{h}dQ+|p^{r}|.$ (37)

Because of the backreaction, the absorbed fermions will change the event
horizon of the black hole. But the event horizon is always determined by
$F(r)$. In other words, in the new horizon, there is also a relation
$F(r_{h}+dr_{h})=F(r_{h})+dF_{h}=0$, namely

$dF_{h}=\frac{\partial F_{h}}{\partial M}dM+\frac{\partial F_{h}}{\partial
Q}dQ+\frac{\partial F_{h}}{\partial l}dl+\frac{\partial F_{h}}{\partial
r_{h}}dr_{h}=0.$ (38)

With Eq.(35), the Eq.(37) can be rewritten as as

$dM-d(PV_{h})=\Phi_{h}dQ+|p^{r}|.$ (39)

From Eq.(39), we can get $dl$, and substituting $dl$ into Eq.(38), we can
delete it directly. Interestingly, $dQ$, and $dM$ are also eliminated
simultaneously. Based on this, there is only a relation between $|p^{r}|$ and
$dr_{h}$, that is

$dr_{h}=-\frac{p^{r}\left(l^{2}Q^{2}+64a^{2}r_{h}^{2}\right){}^{2}}{16a^{4}Q^{2}r_{h}^{3}}.$
(40)

On this basis, the variations of entropy and volume of the black hole can be
expressed as

$dS_{h}=-\frac{\pi
p^{r}\left(l^{2}Q^{2}+64a^{2}r_{h}^{2}\right){}^{2}}{32a^{4}Q^{2}r_{h}^{3}},$
(41)

$dV_{h}=\frac{\pi
p^{r}\left(\left(-1+a^{2}\right)l^{4}Q^{4}-128a^{2}l^{2}Q^{2}r_{h}^{2}-4096a^{4}r_{h}^{4}\right)}{8a^{4}Q^{2}r_{h}^{2}}.$
(42)

According to above formulations, we find

$T_{h}dS_{h}-PdV_{h}=|p^{r}|.$ (43)

The internal energy in Eq.(37) thus would change into

$dU=\Phi_{h}dQ+T_{h}dS_{h}-PdV_{h}.$ (44)

With Eq.(35), we can obtain the relation between the enthalpy and internal
energy in the extended phase space, that is

$dM=dU+PdV_{h}+V_{h}dP.$ (45)

Substituting Eq.(44) into Eq.(45), we get

$dM=T_{h}dS_{h}+\Phi_{h}dQ+V_{h}dP,$ (46)

which is consistent with that in Eq.(27). That is, in the extended phase
space, the first law of thermodynamics still holds when charged fermions are
absorbed by the black hole.

We are willing to discuss the second law of thermodynamics in extended phase
space. From Eq.(41), we find $dS_{h}<0$ . Obviously, in the extended phase
space, the change of entropy of the (2+1)-dimensional regular black hole with
nonlinear electrodynamics sources is negative. In other words, the entropy of
the black hole unavoidably decreases, which is contrary to the second law of
thermodynamics. We suspect that the reason might be the contribution of the
$PV_{h}$ term.

Next, we also can discuss the weak cosmic censorship conjecture in the
extended phase space. Considering the backreaction, the mass $M$, charge $Q$,
and AdS radius $l$ of the black hole will change into $(M+dM,Q+dQ,l+dl)$ as a
charged fermion is swallowed by the black hole. The locations of the minimum
value and the event horizon will change into $r_{min}+dr_{min}$,
$r_{h}+dr_{h}$ correspondingly. At $r_{min}+dr_{min}$, the change of $F(r)$ is
subject to

$F|_{r=r_{min}+dr_{min}}=\alpha+(\frac{\partial F_{min}}{\partial
M}dM+\frac{\partial F_{min}}{\partial Q}dQ+\frac{\partial F_{min}}{\partial
l}dl).$ (47)

By using $F^{\prime}_{min}=0$ in Eq.(47), there is also a relation at the new
lowest point, which can be expressed as

$\partial_{r}F|_{r=r_{min}+dr_{min}}=F^{\prime}_{min}+dF^{\prime}_{min}=0,$
(48)

implying

$dF^{\prime}_{min}=\frac{\partial F^{\prime}_{min}}{\partial
Q}dQ+\frac{\partial F^{\prime}_{min}}{\partial l}dl+\frac{\partial
F^{\prime}_{min}}{\partial r_{min}}dr_{min}=0,$ (49)

where we have used the condition $F^{\prime}(r_{min})=0$ in Eq.(3). Solving
Eq.(49), we get

$\displaystyle dl$ $\displaystyle=$
$\displaystyle\frac{\left(\left(1-a^{2}\right)l^{5}Q^{4}+64a^{2}\left(2+a^{2}\right)l^{3}Q^{2}r_{min}^{2}+4096a^{4}lr_{min}^{4}\right)}{2r_{min}\left(\left(1-a^{2}\right)l^{4}Q^{4}+128a^{2}l^{2}Q^{2}r_{min}^{2}+4096a^{4}r_{min}^{4}\right)}{dr}$
(50) $\displaystyle=$
$\displaystyle\frac{-128a^{4}l^{3}Qr_{min}^{3}}{2r_{min}\left(\left(1-a^{2}\right)l^{4}Q^{4}+128a^{2}l^{2}Q^{2}r_{min}^{2}+4096a^{4}r_{min}^{4}\right)}{dQ}.$

For the extremal black hole, the horizon is located at $r_{min}$. That is,
Eq.(37) is applicable at $r_{min}$. Substituting Eq.(37) into Eq.(47), we
obtain

$F_{min}+dF_{min}=-8p^{r}+\frac{2drr_{h}}{l^{2}}\left(-1+\frac{a^{2}l^{4}Q^{4}}{\left(l^{2}Q^{2}+64a^{2}r_{h}^{2}\right){}^{2}}\right).$
(51)

Substituting Eq.(43) into Eq.(51), we find

$F_{min}+dF_{min}=\frac{2r_{\min}\left(\left(-1+a^{2}\right)l^{2}Q^{2}-64a^{2}r_{\min}^{2}\right){dr}}{l^{4}Q^{2}+64a^{2}l^{2}r_{\min}^{2}}.$
(52)

In addition, according to the condition $F^{\prime}_{min}=0$, we can get $Q$,
that is

$Q=\frac{8ar}{\sqrt{-l^{2}+a^{2}l^{2}}}.$ (53)

Substituting Eq.(53) into Eq.(52), we get lastly

$F_{min}+dF_{min}=0.$ (54)

Eq.(54) shows that when a charged fermion is absorbed by the black hole, there
is not shift for $F_{min}$. In other words, the configurations of the black
holes are not changed, the extremal black holes are still extremal black holes
as fermions are absorbed.

For the near-extremal black hole, Eq.(37) is not applicable at $r_{min}$ for
it is valid only at $r_{h}$. But we can expand it near $r_{min}$ with
$r_{h}=r_{min}+\epsilon$. Note that $p^{r}$ should also be expanded for it is
a function of the horizon $r_{h}$. To the first order, we get

$\displaystyle
dM=\frac{A+B+E}{256l^{3}\left(l^{2}Q^{2}+64a^{2}r_{\min}^{2}\right)}+\frac{(X+Y+Z)\epsilon}{4l^{3}\left(l^{2}Q^{2}+64a^{2}r_{\min}^{2}\right){}^{2}}+O(\epsilon)^{2},$
(55)

in which

$\displaystyle
A=l^{5}\left(Q^{3}{dQ}+Q^{3}\log\left[\frac{Q^{2}}{64a^{2}}+\frac{r_{\min}^{2}}{l^{2}}\right]{dQ}\right),$
$\displaystyle
B=-4096a^{2}lr_{\min}^{3}{dr}+4096a^{2}r_{\min}^{4}{dl}+l^{2}\left(64Q^{2}r_{\min}^{2}{dl}-64a^{2}Q^{2}r_{\min}^{2}{dl}\right),$
$\displaystyle
E=l^{3}(-64Q^{2}r_{\min}{dr}+64a^{2}Q^{2}r_{\min}{dr}+64a^{2}Q\log\left[\frac{Q^{2}}{64a^{2}}+\frac{r_{\min}^{2}}{l^{2}}\right]r_{\min}^{2}{dQ}),$
$\displaystyle
X=-128a^{4}l^{3}Qr_{\min}^{3}{dQ}+4096a^{4}lr_{\min}^{4}{dr}-8192a^{4}r_{\min}^{5}{dl},$
$\displaystyle
Y=Q^{4}\left(l^{5}{dr}-a^{2}l^{5}{dr}-2l^{4}r_{\min}{dl}+2a^{2}l^{4}r_{\min}{dl}\right),$
$\displaystyle
Z=Q^{2}\left(128a^{2}l^{3}r_{\min}^{2}{dr}+64a^{4}l^{3}r_{\min}^{2}{dr}-256a^{2}l^{2}r_{\min}^{3}{dl}\right).$

Substituting Eq.(55) into Eq.(49), we can get $F_{min}+dF_{min}$. Substituting
$dl$ in Eq.(50) in to this equation, we find

$\displaystyle
F_{min}+dF_{min}=\alpha+\frac{4r_{\min}^{2}\left(\left(1-a^{2}\right)l^{2}Q^{2}+64a^{2}r_{\min}^{2}\right)K}{H}+O(\epsilon)^{2},$
(56)

where we have defined

$\displaystyle K$ $\displaystyle=$
$\displaystyle-\left(1-a^{2}\right)l^{4}Q^{4}{dl}+64a^{2}l^{2}Q\left(a^{2}l{dQ}+2{dl}Q\right)r_{\min}^{2}+4096a^{4}r_{\min}^{4}{dl},$
$\displaystyle H$ $\displaystyle=$
$\displaystyle\left(-1+a^{2}\right)l^{9}Q^{6}-192a^{2}l^{7}Q^{4}r_{\min}^{2}$
$\displaystyle-$ $\displaystyle
4096a^{4}\left(3+a^{2}\right)l^{5}Q^{2}r_{\min}^{4}-262144a^{6}l^{3}r_{\min}^{6}.$

Substituting Eq.(53) into Eq.(56), we find lastly

$\displaystyle F_{min}+dF_{min}=\alpha+O(\epsilon)^{2},$ (57)

which is the same as that for the extremal black holes in Eq.(54) for
$O(\epsilon)^{2}$ is the high order terms of $\epsilon$, that is, the near-
extremal black holes are also stable. So we can conclude that the weak cosmic
censorship conjecture holds for both the extremal and near-extremal black
holes in the extended phase space as fermions are adsorbed.

##  5 Conclusion

In this work, we first studied the dynamics of charged fermions being
swallowed by the (2+1)-dimensional regular black holes with nonlinear
electrodynamics sources. We obtained the energy-momentum relation of the
spinor particles by solving the Dirac equation. With this relation, we got the
first law, and the second law of thermodynamics in the normal phase space
further. We found that the laws of thermodynamics for the regular black holes
held. We also discussed the weak cosmic censorship conjecture and found it was
valid in the normal phase space for there exist horizons always stoping the
singularities to be naked. Especially, the extremal black holes will evolve
into the non-extremal black holes.

We also investigated the laws of thermodynamics and weak cosmic censorship
conjecture in the extended phase space, where the cosmological parameter is
regarded as an extensive variable of the thermodynamic system. We derived the
first law of thermodynamics by making use of the conservations of energy and
charge. The second law of thermodynamics was checked too by studying the
variation of the entropy of the (2+1) dimensional black hole with nonlinear
electrodynamic sources. The result showed that the entropy decreased
unavoidably. The second law thus is violated in the extended phase space,
which is different from the case in the normal phase space. We investigated
the weak cosmic censorship conjecture for the extremal and near-extremal black
holes in extended phase space and found that it was valid too. But different
from that in the normal phase space, the configurations of the regular black
holes will not be changed as charged fermions are absorbed for the final
states are still extremal and near-extremal black holes.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(Grant No. 11875095), and Basic Research Project of Science and Technology
Committee of Chongqing (Grant No. cstc2018jcyjA2480).

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