# SS-CADA: A Semi-supervised cross-anatomy domain adaptation for coronary
artery segmentation

###### Abstract

The segmentation of coronary arteries by convolutional neural network is
promising yet requires a large amount of labor-intensive manual annotations.
Transferring knowledge from retinal vessels in widely-available public labeled
fundus images (FIs) has a potential to reduce the annotation requirement for
coronary artery segmentation in X-ray angiograms (XAs) due to their common
tubular structures. However, it is challenged by the _cross-anatomy domain
shift_ due to the intrinsically different vesselness characteristics in
different anatomical regions under even different imaging protocols. To solve
this problem, we propose a Semi-Supervised Cross-Anatomy Domain Adaptation
(SS-CADA) which requires only limited annotations for coronary arteries in
XAs. With the supervision from a small number of labeled XAs and publicly
available labeled FIs, we propose a vesselness-specific batch normalization
(VSBN) to individually normalize feature maps for them considering their
different cross-anatomic vesselness characteristics. In addition, to further
facilitate the annotation efficiency, we employ a self-ensembling mean-teacher
(SE-MT) to exploit abundant unlabeled XAs by imposing a prediction consistency
constraint. Extensive experiments show that our SS-CADA is able to solve the
challenging cross-anatomy domain shift, achieving accurate segmentation for
coronary arteries given only a small number of labeled XAs.

Index Terms—  Coronary artery segmentation, cross-anatomy domain shift, domain
adaptation

##  1 Introduction

The segmentation of coronary arteries in X-ray angiograms (XAs) is
indispensable for both coronary artery disease diagnosis and percutaneous
coronary intervention navigation. Convolutional neural networks have achieved
remarkable progress in this task [1, 2], yet requiring a large amount of
manual annotations for training images, which is highly time-consuming and
labor-intensive due to the complex coronary-tree structures with varying lumen
sizes and inhomogeneous contrast agent inflow. Therefore, it is desired to
reduce the manual annotations for coronary arteries in XAs while maintaining
the segmentation performance.

Fig. 1: Illustration of the challenging cross-anatomy domain shift between
fundus image (FI) and X-ray angiogram (XA).

Considering that the tubular feature of coronary artery is similar to that of
vascular structures in other part of the body, e.g., retinal vessels in fundus
images (FIs), and there are several publicly available annotated FI datasets,
e.g., DRIVE [3], STARE [4] and RITE [5], it is promising to transfer the
knowledge from annotated FIs to facilitate the coronary artery segmentation
task in XAs. These publicly available annotated FI data is regarded as a well-
established source domain, which enables a potentially cost-free knowledge
transfer to the target domain (i.e., XAs). However, data distributions of FIs
and XAs mismatch significantly due to their intrinsically different vesselness
characteristics in different anatomical regions (i.e., the retinal fundus and
the heart) under different imaging protocols (i.e., optical and radiological
imaging), which is called the _cross-anatomy domain shift_ as shown in Fig. 1.
On the contrary, existing unsupervised domain adaptation (UDA) methods [6, 7,
8, 9] are devoted to solving only cross-modality domain shift, i.e., the same
anatomical region under different imaging modalities. They align the feature
space (or even the image appearance [9]) across modalities explicitly by
minimizing the maximum mean discrepancy [6] and implicitly by adversarial
learning [7, 8], without using target domain labels. However, compared with
such common cross-modality domain shift, the cross-anatomy discrepancy in our
task is much more challenging and under-studied. Yu _et al._ [10] presented
the first attempt to mitigate the cross-anatomy discrepancy by forcing shape
consistency in UDA yet limited by the lack of reliable synthetic labels for
semantic guidance. Therefore, existing UDA methods tend to be misguided by the
intractable cross-anatomy domain shift between FIs and XAs, leading to
performance degradation for coronary artery segmentation.

To solve this problem, we propose a Semi-Supervised Cross-Anatomy Domain
Adaptation (SS-CADA) for coronary artery segmentation. Only a few labeled XAs
are required, while the public labeled FIs and abundant unlabeled XAs can be
obtained without annotation cost. Considering the cross-anatomical vesselness
difference of coronary arteries and retinal vessels, we propose a vesselness-
specific batch normalization (VSBN) to individually normalize the feature maps
of them by using the supervision from a small number of labeled XAs and
publicly avaliable FIs. In addition, to further facilitate the annotation
efficiency, we adopt a self-ensembling mean-teacher (SE-MT) architecture to
leverage abundant unlabeled XAs by encouraging the prediction consistency with
random perturbations. The proposed SS-CADA is compact and flexible without
using adversarial learning [11] that suffers from the intractable Nash
equilibrium solution [12], the additional network designs and the complex
training process. Extensive experiments demonstrate that our SS-CADA achieves
accurate segmentation for coronary arteries given only very limited manual
annotations for XAs.

##  2 Methodology

Let $\mathcal{S}$ and $\mathcal{T}$ denote the source domain of FIs with
retinal vessels and the target domain of XAs with coronary arteries,
respectively. We are given publicly available annotated FIs
$S_{L}=\\{(x_{i}^{S_{L}},y_{i}^{S_{L}})\\}^{n^{S_{L}}}_{i=1}\\}$, a small set
of annotated XAs
$T_{L}=\\{(x_{i}^{T_{L}},y_{i}^{T_{L}})\\}^{n^{T_{L}}}_{i=1}\\}$ and abundant
unlabeled XAs $T_{U}=\\{(x_{i}^{T_{U}})\\}^{n^{T_{U}}}_{i=1}$. The proposed
semi-supervised cross-anatomy domain adaptation (SS-CADA) is depicted in Fig.
2, which consists of two parts: 1) a segmentation network with vesselness-
specific batch normalization (VSBN) that learns from $S_{L}$ and $T_{L}$ to
provide a supervision guidance to bridge the cross-anatomy discrepancy; and 2)
a self-ensembling mean-teacher (SE-MT) that imposes an unsupervised
consistency constraint on $T_{U}$ for more annotation efficiency.

Fig. 2: Flow chart of the proposed SS-CADA that consists of two parts: 1) a
segmentation network with vesselness-specific batch normalization (VSBN); and
2) a self-ensembling mean-teacher (SE-MT) architecture.

###  2.1 Vesselness-specific batch normalization (VSBN)

Without loss of generality, we adopt a classical U-Net [13] as the network
backbone for segmentation. Batch normalization (BN) [14] is used in the
network to reduce internal convariate shift, leading to improved model
generalization and faster convergence. In our cross-anatomy problem setting,
it is interesting to observe that when two separated networks with BN layers
are used to learn individually from $S_{L}$ and $T_{L}$, the BN statistic
(batch-wise mean and variance) are highly separable, as shown in Fig. 3. It
suggests that the separable distribution of BN statistic is able to account
for the cross-anatomy discrepancy between domain $\mathcal{S}$ and
$\mathcal{T}$ to some extent. As a result, a joint training manner that
directly mixes $S_{L}$ and $T_{L}$ would have limited performance, since the
shared kernels are misguided by statistical variations between domain
$\mathcal{S}$ and $\mathcal{T}$ and thus fail to grasp the generic
representations.

Fig. 3: t-SNE [15] visualization of BN statistic distributions in shallow,
intermediate and deep layer from FIs and XAs.

To solve this problem, we propose a vesselness-specific batch normalization
(VSBN) layer in the network, which adopts individual BN parameters for domain
$\mathcal{S}$ and $\mathcal{T}$ considering their cross-anatomical vesselness
difference. Meanwhile, convolutional kernels are shared for domain
$\mathcal{S}$ and $\mathcal{T}$ to learn a general representation, such as the
similar tubular feature. Formally, let
$f^{d}=[f^{d}_{1},...,f^{d}_{c},...,f^{d}_{C}]$ denote one feature map given
an input from domain $d\in{\\{\mathcal{S},\mathcal{T}\\}}$, where $f^{d}_{c}$
is the feature map in channel $c$. The proposed VSBN normalizes each channel
respectively and then applies affine transformation with trainable parameters
that are specific to a certain domain $d$, i.e., rescale parameters
$\gamma^{d}=[\gamma^{d}_{1},...,\gamma^{d}_{c},...,\gamma^{d}_{C}]$ and bias
parameters $\beta^{d}=[\beta^{d}_{1},...,\beta^{d}_{c},...,\beta^{d}_{C}]$:

$\hat{f}_{c}^{d}=\gamma_{c}^{d}\cdot\overline{f}_{c}^{d}+\beta^{d}_{c},\quad\text{
where
}\quad\overline{f}_{c}^{d}=\frac{f_{c}^{d}-\mu_{c}^{d}}{\sqrt{(\sigma_{c}^{d})^{2}+\varepsilon}},$
(1)

where $\hat{f}_{c}^{d}$ is the channel $c$ of VSBN output. $\mu_{c}^{d}$ and
$\sigma_{c}^{d}$ are the mean and standard deviation of $f_{c}^{d}$, and
$\varepsilon$ is an infinitesimal.

The proposed VSBN enables a compact dual-model architecture to deal with FIs
and XAs respectively using different parameter sets, i.e.,
$\Theta^{\mathcal{S}}=[\theta,\gamma^{\mathcal{S}},\beta^{\mathcal{S}}]$ for
FIs and
$\Theta^{\mathcal{T}}=[\theta,\gamma^{\mathcal{T}},\beta^{\mathcal{T}}]$ for
XAs, which share the convolution parameters $\theta$ while use different BN
parameters specific to FIs and XAs, respectively. Given image-annotation pairs
$(x_{i}^{S_{L}},y_{i}^{S_{L}})\in S_{L}$ from domain $\mathcal{S}$ and
$(x_{i}^{T_{L}},y_{i}^{T_{L}})\in T_{L}$ from domain $\mathcal{T}$, we define
a loss function to jointly optimize these parameter sets:

$L_{s}=\sum_{i=1}^{n^{S_{L}}}L_{seg}(p_{i}^{S_{L}},y_{i}^{S_{L}})+\sum_{i=1}^{n^{T_{L}}}L_{seg}(p_{i}^{T_{L}},y_{i}^{T_{L}}),$
(2)

where $p_{i}^{S_{L}}=\psi(x_{i}^{S_{L}};\Theta^{\mathcal{S}})$ and
$p_{i}^{T_{L}}=\psi(x_{i}^{T_{L}};\Theta^{\mathcal{T}})$ are predictions of
$x_{i}^{S_{L}}$ and $x_{i}^{T_{L}}$ using the corresponding parameter sets
$\Theta^{\mathcal{S}}$ and $\Theta^{\mathcal{T}}$. Moreover, $L_{seg}$ denotes
a hybrid segmentation loss that consists of the cross-entropy loss and dice
loss.

###  2.2 Self-ensembling mean teacher (SE-MT)

The proposed VSBN relieves the cross-anatomy domain shift by the supervision
guidance of $S_{L}$ and $T_{L}$. However, the small number of annotated XAs in
$S_{L}$ may limit the performance of the model. To deal with this problem, we
employ a self-ensembling mean teacher (SE-MT) architecture to exploit abundant
unlabeled XAs in $T_{U}$ without annotation cost. Specifically, we define the
sub-model with parameter set $\Theta^{\mathcal{T}}$ specific to XAs as the
student model. Then, we calculate an exponential moving average (EMA) for it
and generate a teacher model, which produces better predictions due to the
temporal self-ensembling [16]. Finally, we define an unsupervised consistency
loss between the predictions of student model and teacher model given the same
input $x_{i}^{T_{U}}\in T_{U}$ from domain $\mathcal{T}$ yet under different
perturbations $\delta$ and $\delta^{\prime}$:

$L_{c}=\sum_{i=1}^{n^{T_{U}}}L_{mse}\left(\psi(x_{i}^{T_{U}};\Theta^{\mathcal{T}},\delta),\psi(x_{i}^{T_{U}};(\Theta^{\mathcal{T}})^{\prime},\delta^{\prime})\right)$
(3)

where $(\Theta^{\mathcal{T}})^{\prime}$ is the EMA result of
$\Theta^{\mathcal{T}}$, and it is used for the teacher model.
$\psi(x_{i}^{T_{U}};\Theta^{\mathcal{T}},\delta)$ and
$\psi(x_{i}^{T_{U}};(\Theta^{\mathcal{T}})^{\prime},\delta^{\prime})$ are the
student and teacher model prediction, respectively, which are used to compute
the mean squared error loss $L_{mse}$.

Overall, integrating the supervised loss $L_{s}$ in Eq. 2 and the unsupervised
consistency loss $L_{c}$ in Eq. 3, the complete training objective of the
proposed SS-CADA is formulated as:

$L=L_{s}+\lambda\cdot L_{c}$ (4)

where $\lambda$ acts as an ramp-up trade-off parameter. Once the training
process is completed, an inference process obtains coronary artery
segmentation result by performing forward propagation on the student model
with parameter set $\Theta^{\mathcal{T}}$.

Fig. 4: Two examples of segmentation results. The true positives, false
negatives and false positives are colored in green, red and orange,
respectively. The zoomed views are appended below each case to highlight the
segmentation details.

##  3 experiments and results

###  3.1 Dataset and implementation details

We used the FI dataset DRIVE [3] with annotated retinal vessels as $S_{L}$ in
source domain. It contains 40 FIs that were acquired from a Canon CR5
nonmydriatic 3CCD camera at 45o field of view. We used their green channel and
pre-processed them by contrast limited adaptive histogram equalization and
gamma correction. In addition, we collected 191 XAs of 30 patients with
coronary arteries as the target domain, using a Philips UNIQ FD10 C-arm
system. Among them, for training, an expert radiologist randomly annotated 20
XAs of 3 patients that is regarded as $T_{L}$, and we took 92 XAs of 14
patients without annotations as $T_{U}$. The annotation process was performed
on a PyQT GUI, where 5$\times$ resolution was used for the clear visualization
of even thin branches. In addition, we used the remaining 43 XAs of 7 patients
for validation and 36 XAs of 6 patients for testing. All FIs and XAs were
resized to 512$\times$512 before feeding them in the network.

In SE-MT, the EMA decay rate was empirically set to 0.99 [16], and a time-
dependent Gaussian warming up function
$\lambda(t)=0.1*e^{\left(-5\left(1-t/t_{max}\right)^{2}\right)}$ was used to
dynamically change hyperparameter $\lambda$, where $t$ denotes the current
training epoch and $t_{max}=50$ is the last epoch. Eq. 4 was optimized with
stochastic gradient descent with momentum 0.9, batch size 6, max iteration
30000 and initial learning rate 0.001 that was decayed exponentially with
power 0.95.

###  3.2 Results and analysis

To demonstrate the effectiveness of our SS-CADA for solving cross-anatomy
domain shift, we compare it with several other methods which can be divided
into four categories: 1) using only $T_{L}$: A standard U-Net [13] is learned
only from the small set of labeled XAs, which is denoted as L-SUP; 2) using
$S_{L}$ and $T_{U}$: this is a standard UDA problem, and we use SC-UDA [10]
that incorporates shape consistency between two domains for this purpose; 3)
using $S_{L}$ and $T_{L}$: this is a multi-domain learning problem, and we
adopt three methods for this purpose such as a joint training paradigm (JOINT)
without considering the domain discrepancy, a X-shape multi-modality learning
network (MML) [17] and an U-Net with the proposed VSBN; and 4) using $T_{L}$
and $T_{U}$: this is a standard semi-supervised learning problem, which is
addressed by the self-ensembling mean-teacher and denoted as SE-MT.

As shown in Table 1, the low Dice of L-SUP indicates that using only the small
set of labeled XAs cannot obtain a sufficient model generalization capability.
SC-UDA leads to even lower Dice when additional abundant unlabeled XAs are
used, due to the misguidance of the challenging cross-anatomy domain shift.
JOINT, MML and VSBN achieve better results than SC-UDA, demonstrating that a
small set of labeled XAs can provide effective supervision guidance for
bridging the cross-anatomy domain shift. However, they are still largely
inferior to our SS-CADA due to the ignorance of unlabeled XAs. On the
contrary, SE-MT uses only the knowledge specific to coronary arteries in XAs
yet without transferable knowledge from retinal vessels in FIs, leading to
2.06$\%$ decrease in Dice compared with our SS-CADA. The comparison between
VSBN, SE-MT and SS-CADA is also regarded as the ablation study, which
validates the feasibility of each key component of our method. The highest
recall, precision and Dice of our SS-CADA emphasize that the cross-anatomy
domain shift can be effectively solved in an annotation-efficient manner,
i.e., manual annotations are required for only 18% of the XAs.

Table 1: Quantitative comparison with other methods, where the ablation study
is also performed between VSBN, SE-MT and SS-CADA. The best results are
highlighted in bold font.

Training Set | Method | Recall (%) | Precision (%) | Dice (%)
---|---|---|---|---
$T_{L}$ | L-SUP |  75.12$\pm$4.83 |  65.87$\pm$8.47 |  69.81$\pm$5.35
$S_{L}$,$T_{U}$ | SC-UDA |  80.87$\pm$7.69 |  57.88$\pm$8.73 |  66.92$\pm$6.83
| JOINT |  81.14$\pm$6.75 |  62.66$\pm$9.35 |  70.37$\pm$7.37
$S_{L}$,$T_{L}$ | MML |  83.10$\pm$4.11 |  65.11$\pm$10.92 |  72.49$\pm$7.69
| VSBN |  80.66$\pm$4.58 |  69.12$\pm$9.53 |  74.12$\pm$6.62
$T_{L}$,$T_{U}$ | SE-MT |  82.55$\pm$3.89 |  72.14$\pm$8.40 |  76.70$\pm$5.43
$S_{L}$,$T_{L}$,$T_{U}$ | SS-CADA | 83.27$\pm$3.95 | 75.12$\pm$6.59 | 78.84$\pm$4.60

We visualize the segmentation results in Fig. 4. Due to the page limitation,
among all methods that use $S_{L}$ and $T_{L}$, only VSBN is shown due to its
highest Dice. SC-UDA exhibits obvious false negatives for terminal vessels
with thin scale and attenuated contrast, as well as false positives scattered
in semi-transparent background. Less false positives can be achieved in L-SUP,
VSBN and SE-MT, while false negatives for bifurcation points and thin vessels
seem to be intractable, breaking the normal connectivity in coronary-tree
structures. Given the small set of labeled XAs, only the proposed SS-CADA
enables accurate extraction of coronary arteries with the best connectivity
and the least false positives.

##  4 Conclusion

This paper propose a semi-supervised cross-anatomy domain adaptation for
coronary artery segmentation with knowledge transfer from retinal vessels in
fundus image. With the proposed vesselness-specific batch normalization and
self-ensembling mean-teacher, it overcomes the challenging cross-anatomy
domain shift and facilitate the annotation efficiency. Experimental results
show that our method can effectively achieve accurate coronary artery
segmentation with only very limited manual annotations required, bringing
advantages for alleviating the annotation burden in clinical practice. In the
further, it is of interest to leverage more different types of vessels for
multi-source domain adaptation to further improve the segmentation accuracy
and robustness.

##  5 COMPLIANCE WITH ETHICAL STANDARDS

This study was performed in line with the principles of the Declaration of
Helsinki. Approval was granted by the Ethics Committee of School of Biomedical
Engineering in Shanghai Jiao Tong University (No. 2020004).

##  6 Acknowledge

This research is partially supported by Beijing Natural Science Foundation
Haidian Original Innovation Collaborative Fund (No. L192006), the National Key
research and development program (No.2016YFC0106200), the National Natural
Science Foundation of China (No. 81771921 and No.61901084), and the funding
from Institute of Medical Robotics of Shanghai Jiao Tong University as well as
the 863 national research fund (No.2015AA043203). The authors declare that
they have no conflict of interest.

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111The research was funded by Sobolev Institute of Mathematics, project
FWNF-22-0003.

# On the Andrews-Curtis groups: non-finite presentability

Vitaly Roman’kov

###### Abstract.

The Andrews-Curtis conjecture remains one of the outstanding open problems in
combinatorial group theory. It claims that every normally generating $r$-tuple
of a free group $F_{r}$ of rank $r\geq 2$ can be reduced to a basis by means
of Nielsen transformations and arbitrary conjugations. These transformations
generate the so-called Andrews-Curtis group AC($F_{r}$). The groups
AC($F_{r}$) ($r=2,3,\ldots$) are actively investigated and allows various
generalizations, for which there are a number of results. At the same time,
almost nothing is known about the structure and properties of the original
groups AC($F_{r}$). In this paper we define a class $\\{A_{r,s}:r,s\geq 1\\}$
of generalized Andrews-Curtis groups in which any group $A_{r,r}$ is
isomorphic to the Andrews-Curtis group AC($F_{r}$). We prove that every group
$A_{2,s}$ ($s\geq 1$) is non-finitely presented. Hence the Andrews-Curtis
group AC($F_{2}$) $\simeq A_{2,2}$ is non-finitely presented. Thus, we give a
partial answer to the well-known question about the finite presentability of
the groups AC($F_{r}$), explicitly stated by J. Swan and A. Lisitsa in the
Kourovka notebook [28] (Question 18.89).

##  1\. Introduction

###  1.1. Andrews-Curtis groups

Further $F_{r}$ means a free group of rank $r$. In this paper we prove that
the Andrews-Curtis group AC($F_{2}$) is non-finitely presented. In fact, we
first show that the widely considered general Andrews-Curtis groups
GAC($F_{r}$) are isomorphic to the corresponding groups AC($F_{r}$). Second,
we prove that any Andrews-Curtis group AC($F_{r}$) is isomorphic to a member
$A_{r,r}$ of an infinite series of finitely generated groups
$\\{A_{r,s}:r,s\geq 1\\}$. At last, we prove that each of groups $A_{2,s}$ is
non-finitely presented. It follows that AC($F_{2}$) $\simeq A_{2,2}$ is non-
finitely presented. We also discuss some interesting properties of Andrews-
Curtis groups AC$(F_{r})$, $r\geq 2$, and related open problems.

The Andrews-Curtis groups AC($F_{r}$) for $r\geq 2$ were introduced in their
connection with the famous Andrews-Curtis Conjecture (ACC) named after James
J. Andrews and Morton L. Curtis who proposed it in 1965 [1]. The ACC claims
that every balanced presentation of the trivial group can be reduced to the
standard one by a finite sequence of “elementary AC-transformations”, which
are Nielsen transformations augmented by conjugations. This problem is of
interest in topology as well as in group theory. A topological interpretation
of this conjecture was given in the original paper by Andrews and Curtis [1].

See also [17]. For any $r\geq 2$, the AC-transformations generate the group
AC($F_{r}$), which acts on the set NGen($F_{r}$) of all normally generating
$r$-tuples $u=(u_{1},\ldots,u_{r})\in F_{r}^{r}$. AC-transformations also act
on the set $F_{r}^{r}$ of all $r$-tuples, where they generate the general
Andrews-Curtis group GAC($F_{r}$).

These definitions generalize naturally from the group $F_{r}$ to any group
$G$.

To explain, fix a natural number $r\geq 2$. The following are _elementary
Andrews-Curtis transformations_ (or _elementary AC-moves_) on the set $G^{r}$
of $r$-tuples of $G$:

  * •

$R_{ij}^{\pm 1}$:
$(u_{1},\ldots,u_{i},\ldots,u_{r})\mapsto(u_{1},\ldots,u_{i}u_{j}^{\pm
1},\ldots,u_{r}),\,i\neq j;$

  * •

$L_{ij}^{\pm 1}$:
$(u_{1},\ldots,u_{i},\ldots,u_{r})\mapsto(u_{1},\ldots,u_{j}^{\pm
1}u_{i},\ldots,u_{r}),\ i\neq j;$

  * •

$I_{i}$ :
$(u_{1},\ldots,u_{i},\ldots,u_{r})\mapsto(u_{1},\ldots,u_{i}^{-1},\ldots,u_{r});$

  * •

$C_{i,w}$:
$(u_{1},\ldots,u_{i},\ldots,u_{r})\mapsto(u_{1},\ldots,u_{i}^{w},\ldots,u_{r}),\
w\in G.$

(For $g,f\in G$ $g^{f}$ means $fgf^{-1}$.)

Transformations $R_{ij}^{\pm 1},L_{ij}^{\pm 1},I_{i}$ are called _elementary
Nielsen transformations_. Composition of a finite sequence of elementary
Andrews-Curtis moves is called an AC-transformation. Each inverse of an AC-
transformation is AC-transformation.

Every such AC-move gives a bijection on the set $G^{r}$ of all $r$-tuples of
elements of $G$, which is an element of the symmetric group $Sym(G^{r})$. The
subgroup generated by all the AC-moves in $Sym(G^{r})$ is called the _general
AC-group of $G$ of dimension $r$_ and is denoted by $GAC_{r}(G)$. The set
$NGen_{r}(G)$ of all $r$-tuples in $G^{r}$ that generate $G$ as a normal
subgroup, is invariant under all AC-moves, so every AC-move gives a bijection
of the set $NGen_{r}(G)$. The subgroup generated by all the AC-moves in
$Sym$(NGenr($G$)) is called the _AC-group of $G$ of dimension $r$_ and is
denoted by ACr($G$). The group ACr($F_{r}$) is the true Andrews-Curtis group
of $F_{r}$ directly related to ACC in $F_{r}$. However, the set NGenr($F_{r}$)
may have a much more complicated structure than the set $F_{r}^{r}$, in this
regard it would be easier to study the group GACr($F_{r}$). Observe, that the
restriction of an element $\alpha\in$ GACr($F_{r}$), viewed as a bijection on
$F_{r}^{r}$, gives a bijection $\bar{\alpha}$ on NGenr($F_{r}$), which is an
element of ACr($F_{r}$). It is easy to see that the map
$\alpha\to\bar{\alpha}$ gives a homomorphism $\phi:$ GACr($F_{r}$) $\to$
ACr($F_{r}$), which is onto. Of course, we are interested only in the case
when the set $NGen_{r}(G)$ is not empty.

Note that the set Genr($G$) of generating $r$-tuples of $G$ is closed under
Nielsen moves, while the set NGenr($G$) of normal-generating $r$-tuples of $G$
is closed under AC-moves.

Two tuples $u,v\in G^{r}$ are called _AC-equivalent_ (symbolically
$u\sim_{AC}v$) if there is an AC-transformation which moves $u$ to $v$.

If $G$ is finitely generated, then it suffices to have a finite number of
elementary AC-moves, where the element $w$ in $C_{i,w}$ or its inverse
$w^{-1}$ belongs to a fixed finite set generating $G$.

###  1.2. Andrews-Curtis Conjecture

Let $F_{r}$ be a free group with basis $f=(f_{1},\ldots,f_{r})$. Then a tuple
of elements $u=(u_{1},\ldots,u_{r})$ generates $F_{r}$ as a normal subgroup if
and only if $u\sim_{AC}f$.

To state ACC in the original form recall that a presentation is called
balanced if the number of generators is equal to the number of relators.
Furthermore, a balanced presentation $\langle f\mid u\rangle$, where
$f=(f_{1},\ldots,f_{r}),u=(u_{1},\ldots,u_{r})$ is called _AC-trivializable_
if $u\sim_{AC}f$. The original ACC states that every balanced presentation of
the trivial group is trivializable.

In 1985 Akbulut and Kirby [2] came up with a series of “potential
counterexamples” to ACC:

$AK(n)=\langle x,y|~{}x^{n}=y^{n+1},~{}xyx=yxy\rangle,~{}n\geq 2.$

Presentations $AK(n)$ are balanced presentations of the trivial group. Akbulut
and Kirby conjectured that they are not trivializable, i.e., that the pair of
its relators $(x^{n}y^{-n-1},xyxy^{-1}x^{-1}y^{-1})$ is not AC-equivalent to
the pair of generators $(x,y)$. It turned out later that the presentation
$AK(2)$ is AC-trivializable (see [18]), so $AK(2)$ is not a counterexample to
ACC. The question whether or not the presentations $AK(n)$ with $n>2$ are
trivializable is still open despite an ongoing effort by the research
community (see [19]). Note that currently $AK(3)$ is the shortest (in the
total length of relators) potential counterexample to ACC. Indeed, in [11]
Havas and Ramsay proved that if $\langle x,y\mid u=1,v=1\rangle$ is a
presentation of the trivial group with $|u|+|v|\leq 13$ then either
$(u,v)\sim_{AC}(x,y)$ or $(u,v)\sim_{AC}(x^{3}y^{-4},xyxy^{-1}x^{-1}y^{-1})$.

Other infinite series of potential counterexamples are given in [6], [20] and
in [19]. Finally, we mention a positive solution of a similar problem for free
solvable groups by A. Myasnikov [21]. See papers [6] and [19] for more details
and some particular results.

Currently there are several approaches to resolve ACC. One is based on the
following observation: if $r$-tuples $u$ and $v$ in a free group $F_{r}$ are
AC-equivalent then for any group homomorphism $\phi:F_{r}\to G$ one has
$\phi(u)\sim_{AC}\phi(v)$ in $G$. Therefore, to show, for example, that
$AK(3)$ is a counterexample to ACC it suffices to find a group $G$ and a pair
of elements $a,b\in G$ such that
$(a,b)\not\sim_{AC}(a^{3}b^{-4},abab^{-1}a^{-1}b^{-1})$ in $G$. This brought
up an interesting research on Andrews-Curtis Conjecture in arbitrary groups
$G$. Observe, that ACC can be easily reformulated in a direct way for an
arbitrary group $G$, by saying that any two $r$-tuples of elements from $G$
that generate $G$ as a normal subgroup are AC-equivalent in $G$. However, this
straightforward version of ACC immediately fails for some groups, in
particular, for some finitely generated abelian groups with torsion (see [4]),
and as a consequence, for some groups $G$ with torsion in their
abelianization. Fortunately, one can slightly adjust the formulation of ACC to
accommodate torsion in the abelianization of $G$. It turns out that the
resulting version of ACC, termed the _generalized Andrews-Curtis Conjecture_ ,
holds in nilpotent and solvable groups [21], [10], finite groups [5], and some
other groups. Up to now, this approach provides more and more groups
satisfying the generalized ACC, and no counterexamples to the original one.

Another approach calls to study the group structure of AC-transformations and
apply this knowledge to ACC. Almost nothing is known in this direction and our
goal here is to present some results on algebraic structure of the group of
AC-transformations of NGenr($F_{r}$).

The group ACr($F_{r}$) for $r\geq 2$ is the classical AC-group that was
introduced in relation with ACC. In Section 2 we prove the following result on
AC-groups.

Theorem A. For any $r\geq 2$ the homomorphism $\phi:$ GACr($F_{r}$) $\to$
AC($F_{r}$) is an isomorphism.

In Section 3 we define a class $\\{A_{r,s}:r,s\geq 1\\}$ of generalized
Andrews-Curtis groups in which any group $A_{r,r}$ is isomorphic to the
Andrews-Curtis group AC($F_{r}$). Section 4 is devoted to preliminary results.
In Section 5 the following main results are proved.

Theorem B. The groups $A_{2,s},s\geq 1,$ are not finitely presentable.

Since $AC(F_{2})$ is antiisomorphic to $A_{2,2}$ we get the following result
that solves the Problem 18.89 posed by J. Swan and A.P. Lisitsa in the
Kourovka Notebook [28].

Corollary C. The group $AC(F_{2})$ is not finitely presentable.

Section 6 contains open problems proposed in collaboration with A.G.
Myasnikov.

##  2\. AC-groups and automorphisms of free groups

There is a deep and interesting relationship between the AC-groups $AC(F_{r})$
and automorphisms of free groups.

Let $G$ be a group, and Aut($G$) the automorphism group of $G.$ Denote by
IAut($G$) the subgroup of Aut($G$) consisting of those automorphisms of $G$
which induce the identity map on the commutator quotient group
(abelianization) $G/G^{\prime}.$

As above $F_{r}$ is a free group of rank $r$ with basis $f=(f_{1},...,f_{r})$.
J. Nielsen, in 1924 (compare [22], [15]), using hyperbolic geometry gave
finite sets of generators and relations for the automorphism group
Aut($F_{r}$). Specifically Aut($F_{r}$) is generated by automorphisms of the
following three forms (all indexes $i,j,l$ range over the set $\\{1,...,r\\}$
subject only the condition that $i\not=j,$ every automorphism changes exactly
one free generator):

  * •

$\rho_{i,j}:f_{i}\mapsto f_{i}f_{j},f_{l}\mapsto f_{l}$ for $l\not=i,$

  * •

$\lambda_{i,j}:f_{i}\mapsto f_{j}f_{i},f_{l}\mapsto f_{l}$ for $l\not=i,$

  * •

$\iota_{i}\,\,\,\,\,:f_{i}\mapsto f_{i}^{-1},f_{l}\mapsto f_{l}$ for
$l\not=i.$

It was generally agreed that both Nielsen’s methods and his results were
complicated (see [15], p. 164), and the subject did not progress for long
time. In 1974 J. McCool [16] has found a new finite presentation for
Aut($F_{r}$). McCool’s generators were the Whitehead’s automorphisms and his
relations were of two types: the relations among the integral monomial
$r\times r$ matrices and relations he called R1 - R6 according to two types of
the Whitehead’s automorphisms [29]. See details in [13].

It was shown by W. Magnus [14], using a work of J. Nielsen [23], that
IAut($F_{r}$) has a finite generating set of the following two forms (all
indexes $i,j,l$ range over the set $\\{1,...,r\\}$ subject only the condition
that $i\not=j,k;j\not=k$ every automorphism changes exactly one free
generator):

  * •

$\xi_{i,j}:f_{i}\mapsto f_{i}^{f_{j}},f_{l}\mapsto f_{l}$ for $l\not=i,$

  * •

$\rho_{i,j,k}:f_{i}\mapsto f_{i}[f_{j},f_{k}],f_{l}\mapsto f_{l}$ for
$l\not=i$

Here $[g,h]$ denotes commutator $ghg^{-1}h^{-1}$ of elements $g,h.$ The
statement includes the result of J. Nielsen [22] that IAut($F_{2}$) =
Inn($F_{2}$), the subgroup of inner automorphisms of $F_{2}.$

Since the group IAut($F_{2}$) is isomorphic to Inn($F_{2}$) and so to $F_{2},$
it is finitely presented. W. Magnus [14] raised the question on the finite
presentability of IAut$(F_{r})$ for $r\geq 3.$ The question attracted the
attention of researchers, among them O. Chein [7], S. Bachmuth [3], J. Smillie
and K. Vogtman [26].

The first result in solving this problem has been obtained by S. Krstić and J.
McCool.

######  Theorem 2.1.

([12].) Let $G=F_{3}/N$, where $N$ is a characteristic subgroup of $F_{3}$
such that $N$ is contained in the second derived subgroup
$F_{3}^{\prime\prime}$ of $F_{3}.$ Let
$\bar{\xi}_{1,2},\bar{\xi}_{1,3},\bar{\rho}_{1,2,3}$ be the elements of
Aut($G$) induced by $\xi_{1,2},\xi_{1,3},\rho_{1,2,3}$ respectively. Then any
subgroup of IAut($G$) which contains the set
$\\{\bar{\xi}_{1,2},\bar{\xi}_{1,3},\bar{\rho}_{1,2,3}\\}$ is not finitely
presentable.

This statement covers the cases of the free group $F_{3},$ of the free
metabelian group $M_{3}=F_{3}({\mathcal{A}}^{2}),$ and more generally every
relatively free group $F_{3}({\mathcal{C}})$ of rank $3$ where ${\mathcal{C}}$
is a variety of groups containing the variety ${\mathcal{A}}^{2}$ of all
metabelian groups.

Let $F_{r,s}$ be a free group of finite rank $r+s$ with a set of free
generators $f\cup y$, where
$f=\\{f_{1},\ldots,f_{r}\\},y=\\{y_{1},\ldots,y_{s}\\}$. We assume that $r\geq
1$ and $s\geq 0$. The group $\tilde{A}_{r,s}$ is defined as a subgroup of the
automorphism group Aut($F_{r,s}$) generated by the automorphism group
Aut($F_{r}$), more precisely, the subgroup of all automorphisms fixing the
generators $y$ in $F_{r,s}$, and mapping the subgroup generated by the
elements of $f$ (isomorphic to the group $F_{r}$) onto itself, together with
automorphisms $\xi_{i,k}$ for $i=1,...,r$ and $k=1,...,s,$ that are defined on
the basic elements as follows:

(2.1) $\xi_{i,k}:f_{i}\mapsto f_{i}^{y_{k}},f_{l}\mapsto f_{l}\ \textrm{and}\
y_{t}\mapsto y_{t},\ \textrm{for}\ i,l=1,...,r;\ l\not=i,\ \textrm{and}\
k,t=1,...,s.$

Thus the group $\tilde{A}_{r,s}$ is isomorphic to a subgroup of the group
Aut($F_{r,s}$). Then $\tilde{A}_{r,s}$ acts on its orbit $O_{r,s}$ generated
by all tuples of the form $(u_{1},\ldots,u_{r},y_{1},\ldots,y_{s}),$, where
$u=(u_{1},\ldots,u_{r})\in F_{r}^{r}$. Denote by $O_{r}$ the set of
corresponding truncated tuples $u$. Then $\tilde{A}(r,s)$ also acts on the set
$O_{r}$. Of course, $\tilde{A}_{r,s}$ also acts on the set
$F_{r,s}^{r}\supseteqq O_{r}.$

Let us define the AC-transformation group $A_{r,s}$, which naturally
corresponds to the group $\tilde{A}_{r,s}$. To explain we need a more detailed
notation of AC-moves in $F_{r,s}$. For any tuple of the form
$u=(u_{1},\ldots,u_{r},y_{1},\ldots,y_{s})\in F_{r,s}^{r+s}$ we put (all moves
change exactly one component of the tuple):

  * •

AC1($i,j$) replaces $u_{i}$ by $u_{i}u_{j},$ where $1\leq i,j\leq r,i\not=j.$

  * •

AC2($i,j$) replaces $u_{i}$ by $u_{j}u_{i},$ where $1\leq i,j\leq r,i\not=j.$

  * •

AC3($i$) replaces $u_{i}$ by $u_{i}^{-1}$, where $1\leq i\leq r.$

  * •

AC4($i,k$) replaces $u_{i}$ by $u_{i}^{y_{k}},$ where $1\leq i\leq r\,1\leq
k\leq s.$

By definition, the group $A_{r,s}$ is generated by these transformations. Then
this group also acts on the orbit $O_{r,s}$, on $O_{r}$ and on $F_{r,s}^{r}$.
It is well known that $A_{r,s}$ is antiisomorphic to $\tilde{A}_{r,s}$ via the
following map:

  * •

AC${}_{1}(i,j)\mapsto\rho_{i,j},$

  * •

AC${}_{2}(i,j)\mapsto\lambda_{i,j},$

  * •

AC${}_{3}(i)\mapsto\iota_{i},$

  * •

AC${}_{4}(i,k)\mapsto\xi_{i,k}.$

The proof one can find in the monograph in the monograph [15] (p. 130) or in
the monograph [13] (section “Automorphisms of free groups”).

##  3\. Proof of Theorem A

Let $s=r$ and $A_{r}$ denote the restriction of $A_{r,r}$ to $O_{r}.$ Theorem
A is obtained as a consequence of the following assertions, which have
independent significance.

######  Lemma 3.1.

For any $\alpha\in A_{r}$, if $\alpha(x)=x$, where $x=(x_{1},\ldots,x_{r})$ is
a basis of $F_{r}$, then $\alpha$ acts identically to $F_{r,s}^{r}$, in
particular to $O_{r}$. Moreover, if we change in each transformation of the
form AC4($i,k$) the conjugator $y_{k}$ by $x_{k}$ in $\alpha$ and denote the
resulting transformation by $\beta$, then $\beta$ is identical on $F_{r}^{r}.$
Hence $\nu:\alpha\mapsto\beta$ is a homomorphism $A_{r}\rightarrow$
GACr($F_{r}$).

###### Proof.

Let $v=(v_{1},\ldots,v_{r})\in F_{r,s}^{r}$. We define a homomorphism
$\varphi:F_{r}^{r}\rightarrow F_{r,s}^{r},\,x_{i}\mapsto
v_{i}\,(i=1,\ldots,r).$ Then

$\varphi(\alpha(x))=\alpha(\varphi(v))=\alpha(v)=\varphi(x)=v.$

Moreover, since $\alpha(x)=x$ all $y_{k}$ in $\alpha$ cancel among themselves.
Then the elements $x_{k}$ corresponding to $y_{k}$ in $\beta$ cancel among
themselves too. Define homomorphism $\mu:F_{r,r}\rightarrow F_{r}$ such that
$\varphi(x_{i})=v_{i},\varphi(y_{k})=x_{k}$ for $i,k=1,\ldots,r.$ Then
$\mu(\alpha(x))=\alpha(\mu(x))=\beta(v)=v.$ Then $\beta$ is identical as
element of GACr($F_{r}$) and $\nu$ is a homomorphism. ∎

######  Lemma 3.2.

If $\alpha\in A_{r}$ is not identical on $O_{r}$ then the transformation
$\beta\in A_{r,r}$ derived from $\alpha$ as in Lemma 3.1 is not identical on
$F_{r}^{r}$. Moreover, $\beta$ is not identical on NGenr($F_{r}$). It follows
that $A_{r,r}$ is isomorphic to GACr($F_{r}$) and AC($F_{r}$). Therefore,
GACr($F_{r}$) $\simeq$ AC($F_{r}$).

###### Proof.

Since $\alpha$ is not identical on $O_{r}$, then by lemma 3.1 $\alpha$ is not
identical on $F_{r}^{r}$. Then $\alpha(f)\not=f.$ Suppose that all occurences
of $y_{1},\ldots,y_{r}$ in formal expression of $\alpha$ cancel among
themselves. Then all corresponding occurences of $f_{1},\ldots,f_{r}$ in
$\beta=\nu(\alpha)$ also cancel among themselves. Then $\beta(f)\not=f$ and
$\beta$ is non-trivial.

Now suppose that not all occurences of $y_{1},\ldots,y_{r}$ in formal
expression of $\alpha$ cancel among themselves and $\beta(f)=f$. Suppose that
$\alpha$ includes $m$ occurences of AC4-transformations or their inverses.
Denote $u=(u_{1},\ldots,u_{r})$, where
$u_{1}=f_{2}^{2m}f_{1}f_{2}^{2m},u_{2}=f_{3}^{2m}f_{2}f_{3}^{2m},\ldots,u_{r-1}=f_{r}^{2m}f_{r-1}f_{r}^{2m},u_{r}=f_{1}^{2m}f_{r}f_{1}^{2m}$.
Then elements $u$ generate a free subgroup of rank $r.$ Obviously, the
reduction process for components of elements coming from AC4-transformations
of $\alpha(u)$ does not cancel more than $m$ letters from a side of any entry
of $u_{i}$ for every $i.$ It follows that $\alpha(u)\not=u.$ Note that $u\in$
NGen($F_{r}$). Thus $\beta$ is non-trivial as an element of AC($F_{r}$).
Therefore $A_{r}\simeq$ GAC($F_{r})\simeq$ AC($F_{r}$). ∎

Therefore Theorem A is proved.

##  4\. Auxiliary results

###  4.1. Essentially infinite sets of relations

In this Section we are following to [12]. Let $F(X)$ be the free group with
basis $X,$ and let $\mu:F(X)\rightarrow G$ be a group epimorphism. If $S$ is a
subset of $X,$ then elements of ker$(\mu)\cap$ gp($S$) are called relations of
$G$ on the generators $S$. If $Q$ is a set of such relations, we say that $Q$
is essentially infinite if there is no finite subset $W$ of ker($\mu$) such
that $Q$ is contained in the normal closure ncl($W$) of $W$ in $F(X).$

It was shown in [12] that this notion is independent of the choice of the
generating set $X$ of $X,$ containing $S.$ Also, it was noted that if $Q$ is
an essentially infinite set of relations of $G$ on the finite set $S,$ then no
subgroup $H$ of $G$ containing the image $\mu(S)$ can be finitely presentable.
Indeed, if $H$ is finitely presentable, then there is a finite preimage in
$F(X)$, containing $S$, of a generating set of $H$. Since $H$ is finitely
presentable on this set of generating elements, it would follow that $Q$ is
contained in the norml closure of a finite set of relations of $G$, which is a
contradiction.

Furthemore, it was proved in [12] that if we have two free groups $F(X_{1})$
and $F(X_{2})$, two subsets $S_{1}\subseteq X_{1}$ and $S_{2}\subseteq X_{2}$
and

(4.1) $\begin{array}[]{ccc}\psi^{\prime}:F(S_{1})&\rightarrow&F(S_{2})\\\
\downarrow&&\downarrow\\\ \psi:G_{1}&\rightarrow&G_{2}\\\ \end{array}$

be a commutative diagram of groups, $Q\subseteq F(S_{1})$ be a set of
relations of $G_{1},$ $Q_{2}=\psi^{\prime}(Q_{1})$ be an essential infinite
set of relations of $G_{2},$ then $Q_{1}$ is an essential infinite set of
relations of $G_{1}.$

###  4.2. Fox derivatives

For a given positive integer $r,$ we consider the free group $F_{r}$ with
basis $\\{f_{1},...,f_{r}\\}$ and the integral group ring $\mathbb{Z}F_{r}.$
We use the partial derivatives introduced by Fox [9]. An exellent introduction
to the theory of the Fox derivatives and possible applications of them can be
found in [8] (see also [25] and [27]). In our notation, these are defined as
follows.

For $j=1,...,r,$ the (left) Fox derivative associated with $f_{j}$ is the
linear map $D_{j}:\mathbb{Z}F_{r}\rightarrow\mathbb{Z}F_{r}$ satisfying the
conditions

(4.2) $D_{j}(f_{j})=1,D_{j}(f_{i})=0\ \textrm{for}\ i\not=j$

and

(4.3) $D_{j}(uv)=D_{j}(u)+uD_{j}(v)\ \textrm{for all}\ u,v\in F_{r}.$

Obviously, an element $u\in F_{r}$ is trivial if and only if $D_{i}(u)=0$ for
all $i=1,...,r.$ Also note that for an arbitrary element $g$ of $F_{n}$ and
every $j=1,...,n,$ $D_{j}(g^{-1})=-g^{-1}D_{j}(g).$

The trivialization homomorphism
$\varepsilon:\mathbb{Z}F_{r}\rightarrow\mathbb{Z}$ is defined on the
generators of $F_{n}$ by $\varepsilon(f_{i})=1$ for all $i=1,...,r$ and
extended linearly to the group ring $\mathbb{Z}F_{n}.$

The Fox derivatives appear in another setting as well. Let $\Delta F_{r}$
denote the fundamental ideal of the group ring $\mathbb{Z}F_{r}.$ It is a free
left $\mathbb{Z}F_{r}-$module with a free basis consisting of
$\\{f_{1}-1,...,f_{r}-1\\}.$ This it leads us to the following formula which
is called the main identity for the Fox derivatives:

(4.4) $\sum_{i=1}^{r}D_{i}(\alpha)(f_{i}-1)=\alpha-\varepsilon(\alpha),$

where $\alpha\in\mathbb{Z}F_{r}.$ Conversely, if for any element $f\in F_{r}$
and $\alpha_{i}\in\mathbb{Z}F_{r}$ we have equality

(4.5) $\sum_{i=1}^{r}\alpha_{i}(f_{i}-1)=f-1,$

then $D_{i}(f)=\alpha_{i}$ for $i=1,...,r.$

More generally, we call a linear map
$D:\mathbb{Z}F_{r}\rightarrow\mathbb{Z}F_{r}$ the free Fox derivative if $D$
satisfies the property

(4.6) $D(uv)=D(u)+uD(v)$

for all $u,v\in F_{r}.$ Every such derivative has the form

(4.7) $D=\alpha_{1}D_{1}+...+\alpha_{n}D_{r},$

where $\alpha_{i}=D(f_{i})$ for $i=1,...,r.$ By definition $(\alpha
D)(u)=D(u)\alpha$ for any $\alpha\in\mathbb{Z}F_{r},u\in F_{r}.$ Conversely,
we can define a derivative $D=\sum_{i=1}^{r}\alpha_{i}D_{i}$ for arbitrary
tuple of elements $\alpha_{i}\in\mathbb{Z}F_{r}.$

The notion of the free Fox derivatives, defined above for free groups, can be
generalized to groups of the type $F_{r}/R^{\prime},$ where $R$ is any normal
subgroup of $F_{r}.$ First we give some notation.

Let $F_{r}$ be a free group with basis $\\{f_{1},...,f_{r}\\}.$ Let $R$ be a
normal subgroup of $F_{r}$ and that $R^{\prime}$ is its derived subgroup (it
is also normal subgroup of $F_{r}).$ Let $\bar{G}=F_{r}/R^{\prime},$
$G=F_{r}/R$ and let $\mathbb{Z}G$ be the group ring of $G.$ By
$\mu:F_{r}\rightarrow G$ we denote the standard epimorphism, as well as its
linear extension to $\mu:\mathbb{Z}F_{r}\rightarrow\mathbb{Z}G.$ By
$\bar{\mu}:\mathbb{Z}\bar{G}\rightarrow\mathbb{Z}G$ we mean the epimorphism
induced by $\mu$. Also we denote by $\mu^{\prime}:F_{r}\rightarrow\bar{G}$ the
other standard epimorphism. Obviously $\mu$ is a superposition of
$\mu^{\prime}$ and $\bar{\mu}.$

An abelian normal subgroup $\bar{R}=R/R^{\prime}$ of the group $\bar{G}$ is
considered as a module over the group ring $\mathbb{Z}G$ where action of
$\mathbb{Z}G$ on $\bar{R}$ is induced by conjugation in $\bar{G}.$ This module
is called the relation module of $\bar{G}.$

Every free Fox partial derivative $D_{j}$ induces a linear map
$d_{j}:\mathbb{Z}F_{r}\rightarrow\mathbb{Z}G$ via $\mu.$ These linear maps
$d_{j}$ also are called the free Fox derivatives (or the free partial
derivatives). Every $d_{j}$ is well defined via $\bar{\mu}$ on
$\mathbb{Z}\bar{G}.$

An arbitrary element $u$ in $\bar{G}$ is trivial if and only if $d_{i}(u)=0$
for all $i=1,...,r.$ Thus two elements $u,v$ are equal in $\bar{G}$ if and
only if $d_{i}(u)=d_{i}(v)$ for all $i=1,...,r.$

The main identity for the Fox derivatives (4.4) gives a similar one in the
considered case too:

(4.8)
$\sum_{i=1}^{r}d_{i}(\alpha)(\mu(f_{i})-1)=\bar{\mu}(\alpha)-\varepsilon(\bar{\mu}(\alpha)),$

where $\alpha\in\mathbb{Z}\bar{G}$ and
$\varepsilon:\mathbb{Z}G\rightarrow\mathbb{Z}$ is the trivialization
homomorphism on $\mathbb{Z}G.$

Let $u\in\bar{R}$ and $\alpha\in\mathbb{Z}G.$ Then

(4.9) $d_{i}(u^{\alpha})=\alpha d_{i}(u)\ \textrm{for all}\ i=1,...,r.$

###  4.3. Matrix representations

#### Magnus representations for automorphism groups.

Let $F_{r}$ be a free group with basis $\\{f_{1},...,f_{r}\\}.$

######  Definition 4.1.

The Magnus representation for Aut$(F_{r})$ is the map

(4.10) $\alpha:\mathrm{Aut}(F_{r})\rightarrow GL_{r}(\mathbb{Z}F_{r})$

assigning to $\varphi\in\mathrm{Aut}(F_{r})$ the Jacobi matrix

(4.11) $J(\varphi)=(D_{j}(\varphi(f_{i}))$

where $D_{j}$ are the free Fox derivatives for $j=1,...,r,$ and
$D_{j}(\varphi(f_{i}))$ is the $ij-$th entry of the matrix.

The Magnus representation $\alpha$ for Aut$(F_{r})$ is injective since
$\varphi(f_{i})$ is recovered from $J(\varphi)$ by applying the main identity
for free Fox derivatives to the $i-$th row for each
$\varphi\in\mathrm{Aut}(F_{r}).$

The Magnus representation is not homomorphism as is seen from the following
assertion.

######  Proposition 4.2.

For $\varphi,\psi\in\mathrm{Aut}(F_{r}),$ the equality

(4.12) $J(\varphi\psi)=\psi(J(\varphi))\cdot J(\psi)$

holds, where $\psi(J(\varphi))$ means that $\psi$ is applied to every entry in
$J(\varphi).$

In particular, it follows that the image of $\alpha$ is contained in the group
$GL_{r}(\mathbb{Z}F_{r})$ of invertible matrices.

To obtain a genuine representation, a homomorphism, we need to change $F_{r}$
to a group $\bar{G}$ of type $F_{r}/R^{\prime},$ where $R$ is a normal
subgroup in $F_{r}.$ Let $G=F_{r}/R$ and $\bar{G}=F_{r}/R^{\prime}.$ Then
$\bar{R}=R/R^{\prime}$ is normal abelian subgroup of $\bar{G},$ which is
considered as a module over the group ring $\mathbb{Z}G.$ Let
$\\{x_{1},...,x_{r}\\}$ be the generators of $\bar{G}$ corresponding to the
basic elements $\\{f_{1},...,f_{r}\\}$ of $F_{r}.$

######  Definition 4.3.

The Magnus representation for Aut$(\bar{G})$ is the map

(4.13) $\alpha_{A}:\mathrm{Aut}(\bar{G})\rightarrow GL_{r}(\mathbb{Z}G)$

assigning to $\varphi\in\mathrm{Aut}(\bar{G})$ the Jacobi matrix
$J(\varphi)=(d_{j}(\varphi(x_{i}))$ of $\varphi$ over $\mathbb{Z}G.$ Here
$d_{j}$ are the induced free Fox derivatives on $\bar{G}$ with values in
$\mathbb{Z}G$ with respect to generators $x_{i}$ for $i,j=1,...,r,$ and
$d_{j}(\varphi(x_{i}))$ is the $ij-$th entry in the matrix.

For $R$ a normal subgroup of a group $H,$ let IRAut$(H)$ denote the group of
all automorphisms of $H$ for which $R$ is invariant, and which induce the
identical map on the quotient $H/R.$

Then the map $\alpha_{AR}:\mathrm{IRAut}(\bar{G})\rightarrow
GL_{r}(\mathbb{Z}G)$ induced by the map $\alpha_{A}$ is homomorphism.

Let $M_{r}$ be the free metabelian group of rank $r$ with basis
$\\{x_{1},...,x_{r}\\},$ and $A_{r}=M_{r}/M_{r}^{\prime}$ be the abealization
of $M_{r}$ with the corresponding basis $\\{a_{1},...,a_{r}\\}.$ The group
ring $\mathbb{Z}A_{r}$ can be considered as a Laurent polynomial ring
$\Lambda_{r}=\mathbb{Z}[a_{1}^{\pm 1},...,a_{r}^{\pm 1}].$ Let $\psi$ be an
automorphism of $M_{r}.$ We define the Jacobi matrix $J(\psi)$ corresponding
to $\psi$ as above. Then the map

(4.14) $\beta:\psi\rightarrow J(\psi)$

gives an injective homomorphism (embedding) of $\mathrm{IAut}(M_{r})$ into the
group of matrices of size $r$ over the Laurent polynomial ring $\Lambda_{r}.$
This homomorphism is called Bachmuth’s embedding.

Every automorphism $\varphi\in$ Aut$(F_{r})$ induces an automorphism
$\bar{\varphi}\in$ Aut$(M_{r}).$ Let $r=3.$ We compute the matrices
$J(\bar{\xi}_{1,2}),J(\bar{\xi}_{1,3})$ and $J(\bar{\rho}_{1,2,3})$ where
$\bar{\xi}_{1,3},\bar{\xi}_{2,3},\bar{\rho}_{1,2,3}$ are induced by
$\xi_{1,3},\xi_{2,3},\rho_{1,2,3}\in$ IAut$(F_{3}),$ respectively.

(4.15) $J(\bar{\xi}_{1,2})=\left(\begin{array}[]{ccc}a_{2}&1-a_{1}&0\\\
0&1&0\\\ 0&0&1\\\ \end{array}\right),$

(4.16) $J(\bar{\xi}_{1,3})=\left(\begin{array}[]{ccc}a_{3}&0&1-a_{1}\\\
0&1&0\\\ 0&0&1\\\ \end{array}\right),$

(4.17)
$J(\bar{\rho}_{1,2,3})=\left(\begin{array}[]{ccc}1&a_{1}(1-a_{3})&a_{1}(a_{2}-1)\\\
0&1&0\\\ 0&0&1\\\ \end{array}\right).$

Let $\eta:\Lambda_{3}\rightarrow\mathbb{Z}[t,t^{-1}]$ be a homomorphism of the
Laurent rings defined by the map

(4.18) $\eta:a_{i}\mapsto 1\ \mathrm{for}\ i=1,2;a_{3}\mapsto t.$

Now we can define by ( 4.15 \- 4.18) a homomorphism $\bar{\eta}:$
IAut$(F_{3})\rightarrow GL_{2}(\mathbb{Z}[t,t^{-1}])$ such that

(4.19)
$\bar{\eta}(\xi_{1,2})=1,\bar{\eta}(\xi_{1,3})=\left(\begin{array}[]{cc}t&0\\\
0&1\\\
\end{array}\right),\bar{\eta}(\rho_{1,2,3})=\left(\begin{array}[]{cc}1&1-t\\\
0&1\\\ \end{array}\right).$

The following lemma, which has been proved in [12], provides essentially
infinite sets of relations of $PGL_{2}(\mathbb{Z}[t,t^{-1}]).$ We write
$d,e_{0},u$ for the matrices

(4.20) $d=\left(\begin{array}[]{cc}t&0\\\ 0&1\\\
\end{array}\right),e_{0}=\left(\begin{array}[]{cc}1&1\\\ 0&1\\\
\end{array}\right),u=\left(\begin{array}[]{cc}1&t-1\\\ 0&1\\\
\end{array}\right).$

######  Lemma 4.4.

[12]. Let $K$ be a field and let $Q$ be any infinite subset of

(4.21) $Q_{1}=\\{[e_{0},d^{k}e_{0}d^{-k}]|k\geq 1\\}$

or

(4.22) $Q_{2}=\\{d^{k}(ud)^{-k}d^{k}(u^{-1}d)^{-k}|k\geq 1\\}.$

Then $Q$ is an essentially infinite set of relations in
$PGL_{2}(K[t,t^{-1}]).$

Lemma 4.4 has been used in[12] in establishing Theorem 2.1. We will use it in
establishing our main result on non-finite presentability of the generalized
Andrews-Curtis groups.

##  5\. Proof of Theorem B

The homomorphism $\bar{\eta}$ defined by (4.19) can be naturally extended via
the Magnus representation and setting $\xi_{i,j}\mapsto 1$ for
$i=1,2;j=2,...,m$ to a homomorphism

(5.1) $\nu:A_{2,m}\rightarrow PGL_{2}(\mathbb{Z}[t,t^{-1}]).$

Let $\lambda_{1,2}$ be an automorphism in $A_{2,m}$ which maps $x_{1}\mapsto
x_{2}x_{1}$ and fixes other generators. Also let $\rho_{1,2}$ be an
automorphism in $A_{2,m}$ which maps $x_{1}\mapsto x_{1}x_{2}$ and fixes other
generators. Then $\lambda_{1,2}$ commutes with all automorphisms of the form
$\xi_{2,3}^{k}\rho_{1,2}\xi_{2,3}^{-k}$ for every integer $k.$

By direct computation we get that

(5.2)
$\nu(\lambda_{1,2})=e_{0},\nu(\xi_{2,3}^{k}\rho_{1,2}\xi_{2,3}^{-k})=d^{k}e_{0}d^{-k}\
\textrm{for every integer }\ k.$

Since the set $Q_{1}$ in (4.21) is essentially infinite the set of relations
$\\{[\lambda_{1,2},\xi_{2,3}^{k}\rho_{1,2}\xi_{2,3}^{-k}]|k\in\mathbb{N}\\}$
which maps onto $Q_{1}$ by $\nu$ is essentially infinite in $A_{2,m}.$ Hence
the group $A_{2,m}$ is non-finitely presented.

Therefore Theorem B is proved.

Remark For any $m,$ $A_{1,m}$ is finite presentable, and for any $n,$
$A_{n,0}$ is finite presentable.

Indeed, for any $m,$ $A_{1,m}$ is isomorphic to a direct product of groups
$C_{2}\times F_{m},$ where $C_{2}\simeq\ \textrm{Aut}(F_{1})$ is the cyclic
group of order $2$ and $F_{m}=$ gp$(\xi_{1,1},...,\xi_{1,m})$ is the free
group with basis $\xi_{1,1},...,\xi_{1,m}.$ Hence every group $A_{1,m}$ is a
finitely presentable group.

##  6\. Open problems

In this section we state several open problems on Andrews-Curtis groups.

######  Problem 6.1.

Are any of the groups $AC(F_{r})$, $r>2$, finitely presentable?

######  Problem 6.2.

For which normally $r$-generated groups $G$ the canonical epimorphism
$\phi:GAC_{r}(G)\to AC(G)$ is an isomorphism? In particular:

  * •

Is $\phi:GAC_{r}(G)\to AC(G)$ an isomorphism for a torsion-free non-elementary
hyperbolic group $G$?

  * •

For which $r$-generated partially commutative groups $G=G(\Gamma)$ the
canonical epimorphism $\phi:GAC_{r}(G)\to AC(G)$ is an isomorphism?

######  Problem 6.3.

Find ”good” (quasi-geodesic) normal forms of elements in $AC(F_{r})$

Solution to this problem will enhance efficacy of computations with ACC.

######  Problem 6.4.

For which $r$ does the group $AC(F_{r})$ have Kazhdan property (T)?

Aknowledgments.

The author is grateful to A. Myasnikov for invaluable help in preparing the
paper.

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Author’s information

Vitaly Roman’kov:

Sobolev Institute of Mathematics (Omsk Branch), Pevtsova str. 13, 644099,
Omsk, Russia.

E-mail<EMAIL_ADDRESS>


# Dissociative recombination of NS+ in collisions with slow electrons

R. Hassaine1<EMAIL_ADDRESS>F. Gauchet1 F. Iacob2 J. Zs
Mezei1,3<EMAIL_ADDRESS>E. Roueff4 J. Tennyson1,3,5 I. F. Schneider1,6
<EMAIL_ADDRESS>1LOMC-UMR6294, CNRS, Université Le Havre
Normandie, 76600 Le Havre, France  2Physics Faculty, West University of
Timisoara, 300223,Timisoara, Romania  3HUN-REN Institute for Nuclear Research
(ATOMKI), H-4001 Debrecen, Hungary  4LERMA CNRS-UMR8112, Observatoire de
Paris, Université PSL, F-92190, Meudon, France  5Department of Physics and
Astronomy, University College London, WC1E 6BT London, UK  6LAC-UMR9188, CNRS
Université Paris-Saclay, F-91405 Orsay, France

###### Abstract

Cross sections and rate coefficients for the Dissociative Recombination (DR)
of the NS+ ion induced by collisions with low-energy electrons are reported
for temperatures between 10 and 1000 K, relevant to a large range of
interstellar cloud temperatures. Uncertainties are discussed for these rates.
Comparisons are made with DR rates for the isovalent NO+ molecular ion which
are found to be much faster. The present findings lead to a moderate
dissociative reaction rate coefficient, smaller by a factor of 2 than the
current estimates reported in the different kinetic databases for a
temperature of 10 K. We consider that our rate coefficients obtained through
multichannel quantum defect theory for NS+ are likely to be better than those
displayed in the different kinetic databases.

###### pacs:

33.80. -b, 42.50. Hz

††preprint: APS/123-QED

##  I Introduction

Nitrogen sulfide (NS) was first detected in the early years of radio astronomy
toward ‘SgrB2 interstellar cloud Gottlieb1975  and its presence has later been
found in giant molecular clouds McGonagle1997 , dense cores HilyBlant2022  as
well as in comets Irvine2000 . Recently, new observations have led to the
discovery of the nitrogen sulfide cation NS+ thanks to laboratory spectroscopy
Cernicharo2018 ; the ion is found to be ubiquitous, from cold molecular cores,
prestellar cores, and to shocks Cernicharo2018  and photon dominated regions
Riviere2019 . Dissociative recombination (DR):

$\mbox{NS}^{+}(v_{i}^{+})+e^{-}(\varepsilon)\longrightarrow\mbox{N}+\mbox{S}$
(1)

\- where $v_{i}^{+}$ is the initial vibrational number of the target and
$\varepsilon$ is the energy of the incident electron - is likely the major
removal mechanism for the ion. It competes with autoinization, resulting in
vibrational transitions:

$\mbox{NS}^{+}(v_{i}^{+})+e^{-}(\varepsilon)\longrightarrow\mbox{NS}^{+}(v_{f}^{+})+e^{-}({\varepsilon}^{\prime})$
(2)

i.e. vibrational excitation - when $v_{f}^{+}$ is larger than $v_{i}^{+}$ and
$\varepsilon$ exceeds the threshold of reaching this excited level - or
vibrational de-excitation - when $v_{f}^{+}$ is smaller than $v_{i}^{+}$ \-
$\varepsilon^{\prime}$ being the energy of the scattered electron.

Here computations are presented which were performed using our step-wise
Multichannel Quantum Defect Theory (MQDT) Giusti1980 ; mezei2019  and
references therein. We focus on calculating dissociative recombination (eq.
(1)) cross sections and rate coefficients for collision energies/temperatures
up to 1 eV/1000 K. The first attempt to provide uncertainties for our MQDT
method is presented. Due to the lack of other theoretical or experimental
results we will compare our results with those obtained for NO+, another
closed shell molecular cation relevant for the upper atmosphere of Earth, that
has similar electron structure and chemical properties to NS+.

The organisation of the paper is as follows: In section II, a succinct
presentation of our theoretical approach is given. Section LABEL:sec:results
contains the major results - cross sections and rate coefficients - and
section III summarizes the concluding remarks.

##  II Theoretical Method

The effectiveness of our theoretical approach based on the step-wise MQDT
Giusti1980 ; mezei2019  for modeling electron/diatomic cation collisions has
been demonstrated through numerous previous studies involving various species.
Examples of these include H${}_{2}^{+}$ and its isotopologues, where we
account for rotational and isotope effects across low to medium-high collision
energies motapon2008 ; waffeutamo2011 ; chakrabarti2013 ; motapon2014 ;
Epee2016 ; djuissi2020 . Additionally, we have explored core-excited effects
for CH+ mezei2019  and for SH+ Kashinski2017 ; boffelli2023 . As the main
elements of our approach have been extensively presented in previous works,
such as mezei2019 , we will focus here on only outlining its main steps.

Reactions (1) and (2) encompass ionization channels, characterizing the
interaction of an electron with the target cation, as well as dissociation
channels, relating to atom-atom scattering. The interaction of these channels
gives rise to quantum interference between the direct mechanism relying on
open channels—where capture occurs into a doubly excited dissociative state of
the neutral system— and the indirect one—where capture takes place through a
Rydberg bound state of the molecule pertaining to closed channels. These
Rydberg states are predissociated by the dissociative channels. In both
mechanisms, autoionization, which relies on the presence of open ionization
channels, competes with predissociation, resulting in the cation’s vibrational
excitation or de-excitation-reaction (2). Each individual ionization channel
is constructed by adding an electron to the NS+ ion within its fundamental
electronic state $X$ ${{}^{1}}\Sigma{{}^{+}}$ at a specific ($v_{i}^{+}$)
vibrational level. These ionization channels interact with all available
dissociation exit channels, via Rydberg-valence interaction, for all relevant
molecular symmetries - ${}^{2}\Sigma^{+}$, ${}^{2}\Pi$, and ${}^{2}\Delta$,
labeled by ”sym” within this article.

The MQDT approach for studying dissociative recombination relies on prior
knowledge of the potential energy curves (PECs) corresponding to the ion’s
ground state as well as that of the dissociative states of the neutral
molecule and of its mono-excited Rydberg states. These latter states, which we
can organize in Rydberg series, each of them characterized by geometry-
dependent quantum defects. Additionally, the electronic couplings between the
dissociation and ionization continua (referred as Rydberg-valence couplings)
are needed.

Figure 1: Energy diagram of the NS+/NS molecular system, relevant for
dissociative recombination, compiled from iacob2022 . Given are the potential
energy curves (PECs) and the atomic states of their corresponding asymptotic
limits are given as short horizontal lines at large internuclear separation.
The thick black solid curve represents the PEC of the ground electronic state
of the NS+ ion, while the solid colored curves represent NS∗∗ dissociative
states computed by the R-matrix method. The thin black horizontal lines denote
the lowest vibrational levels of the ion.

A summary of the relevant PECs, taken from Iacob et al. iacob2022 , can be
seen in Fig. 1. The black curve stands for the ground electronic state of the
cation, while the colored solid, dashed and dotted lines represent the
dissociative states of NS belonging to the ${}^{2}\Sigma^{+}$, ${}^{2}\Pi$ and
${}^{2}\Delta$ symmetries.

The current MQDT calculations neglect rotational effects. Within each of the
symmetries involved, focus is directed towards the few lowest dissociative
states accessible at low energy of the incident electron - below 1 eV -
relevant for the interstellar media. For a given total symmetry, here we
consider ${}^{2}\Sigma^{+}$, ${}^{2}\Pi$ and ${}^{2}\Delta$, the initial step
of our methodology involves constructing the interaction matrix $\cal V$,
which serves as the driving force during the collision process. Its elements
quantify the interconnections between the ionization and the dissociation
channels.

The next step consists of construction of the short-range reaction matrix
$\cal K$. This is achieved using a second-order perturbative solution of the
Lippmann-Schwinger equation. The $\cal K$-matrix is then diagonalized, its
eigenvalues being related to the long-range phase-shifts of the
eigenfunctions. Applying a Cayley transformation on these matrices
cayley1894collected  allows the generalized scattering matrix $X$ to be
constucted. Elimination of the closed channels Seaton1983  is then performed,
yielding the physical scattering matrix $\cal S$:

$\mbox{\boldmath$\cal
S$}=\mbox{\boldmath$X$}_{oo}-\mbox{\boldmath$X$}_{oc}\frac{1}{\mbox{\boldmath$X$}_{cc}-\exp({\rm-i2\pi}\mbox{\boldmath$\nu$})}\mbox{\boldmath$X$}_{co}\,,$
(3)

which relies on the block-matrices built not only for the open channels,
$X$oo, but also for the closed ones, $X$oc, $X$co and $X$cc. The diagonal
matrix $\nu$ appearing in the denominator of equation (3) incorporates the
effective quantum numbers corresponding to the vibrational thresholds of the
closed ionization channels at the given total energy of the system.

Figure 2: Relevant molecular data sets used in the nuclear dynamics
calculations. In the top row, the thick red solid curves ($D_{1}$) stand for
the PEC of NS∗∗ $1\,^{2}\Sigma^{+}$ (left panel) and $1\,^{2}\Pi$ (right
panel), while the thick blue solid curve ($D_{2})$ represents NS∗∗
$2\,^{2}\Pi$ state. The middle row displays the electronic couplings between
these dissociative states and the ionization continuum. The quantum defects
$\mu$ characterizing the Rydberg series of dominant partial waves $s$, $p$,
and $d$ are shown in the bottom row. Table 1: Asymptotic states of NS∗∗
relevant for low-energy impact collisions. The corresponding energies are
given with respect to NS∗∗ lowest limit: S(3P)+N(4S).

State | Energy (eV) | Symmetry
---|---|---
S(3P)+N(2Do) | 2.381 |  1${}^{2}\Sigma^{+}$, 1${}^{2}\Pi$
S(1D)+N(2Do) | 3.524 |  1${}^{2}\Pi$

Finally, the global cross section for the dissociative recombination is:

$\begin{split}\sigma_{diss\leftarrow
v_{i}^{+}}&=\frac{\pi}{4\varepsilon}\sum_{sym}\rho^{(sym)}\sum_{l,j}\mid
S^{(sym)}_{d_{j},v_{i}^{+}l}\mid^{2},\end{split}$ (4)

where $\rho$ represents the ratio of multiplicities between the neutral system
and the ion for a specific neutral symmetry. The thermal rate coefficients are
determined by convoluting the cross section with the Maxwell energy
distribution function of the free electrons:

$\alpha(T)=\frac{2}{kT}\sqrt{\frac{2}{\pi
mkT}}\int_{0}^{+\infty}\sigma(\varepsilon)\varepsilon\exp(-\varepsilon/kT)d\varepsilon,$
(5)

where $m$ is the electron’s mass, $T$ the temperature, and $k$ the Boltzmann
constant. Applying the step-wise MQDT method outlined in the previous section,
we have calculated the dissociative recombination (eq. (4)) cross sections of
NS+ for the lowest two vibrational levels of the ground electronic state of
the cation. Since NS+ molecular cation has been observed in the diffuse
interstellar medium (ISM) for temperatures between $10$ and $300$ K
Cernicharo2018 , corresponding to electron energies in the range between
$0.01$ meV and $1$ eV, here we focus on producing cross sections for the
lowest relevant open dissociative states of the neutral. For the ionization
channels, the partial waves we took into account for the incident electron are
$s$, $p$, and $d$ for the ${}^{2}\Sigma^{+}$ states and $p$ and $d$ for the
${}^{2}\Pi$ ones iacob2022 .

Figure 3: Dissociative recombination of NS+ in its ground state with an
electron into NS states - $D_{1}$ or/and $D_{2}$ of ${}^{2}\Pi$ symmetry -
restricted to the contribution of the $d$ partial wave: Total vs direct
mechanism.  Figure 4: Dissociative recombination cross section of NS+ for the
${}^{2}\Pi$ molecular symmetry for $p$, $d$ and mixed ”M”$=(1/2)\times
p+(1/2)\times d$ partial waves for the incident electron.

Starting from Fig. 1 containing the PECs of NS+ and doubly excited
dissociative states of NS, and considering the temperature ranges relevant for
the ISM we reduced the input molecular data to those curves presented in Fig.
2. We also limited our calculations to collision energies up to 1 eV shown
with dark green line in Figs. 1 and 2. This restricts the open dissociation
channels to those whose asymptotic limits are given in the first column of
table 1. In addition, the unfavourable crossing points - far from the Franck-
Condon region - between the PEC’s of the remaining open dissociative channels
1 ${}^{2}\Delta$, 2 ${}^{2}\Delta$, 3 ${}^{2}\Sigma$, 3 ${}^{2}\Pi$ and that
of the target ion - make them irrelevant for dissociative recombination at low
energy. Consequently, $1\,^{2}\Sigma^{+}$, $1\,^{2}\Pi$, $2\,^{2}\Pi$ are the
only relevant dissociative paths to be considered in this study. Figure 2
contains the molecular data characterizing these dissociative states. The
middle panels represent their electronic couplings with the ionization
continuum. The quantum defects generating the complete Rydberg series
corresponding to the partial waves $s$, $p$, $d$ for ${}^{2}\Sigma^{+}$
symmetry and $p$, $d$ for ${}^{2}\Pi$ symmetry are displayed in the bottom
panels.

Figure 5: Rate coefficients for dissociative recombination of NS+ cation from
its lowest two vibrational levels and uncertainties due to the use of
different partial waves for the incident electrons. Curves give the partial
wave contributions for the incident electron. ”M” denotes mixed partial waves
(($1/2)\times p+(1/2)\times d$ for ${}^{2}\Pi$ states and $(1/3)\times
s+(1/3)\times p+(1/3)\times d$ for ${}^{2}\Sigma^{+}$ states). The lower limit
of the gray band stands for the minimum rate coefficient as a function of
temperature while its upper limit stands for the maximum rate coefficient as a
function of temperature.

Figure 3 shows the DR cross sections obtained for the two dissociative states
of the ${}^{2}\Pi$ symmetry corresponding to the (most important) $d$ partial
wave of the incoming electron. We give here the direct \- smooth cross section
due to the direct capture of the electron into doubly excited dissociative
states - and total or direct and indirect cross sections - infinite number of
resonances, due to the temporary capture into highly excited Rydberg states.
The $D_{1}$ dissociative state (black curve) has a more favourable crossing
with the cation compared to $D_{2}$ (blue curve) leading thus to larger DR
cross sections, although the vibronic coupling of $D_{1}$ is a factor of two
smaller than the coupling of $D_{2}$.

Figure 4 shows the total DR cross sections associated to the ${}^{2}\Pi$
molecular states considering three scenarios for the partial waves of the
incoming electron. Dark green stands for $p$ wave, blue for the $d$ and red
for the $0.5\times p+0.5\times d$ mixture. The overall behaviour of the three
cross sections seems the same, at least starting from 100 meV collision
energies. For very low collision energies we obtained smaller cross sections
for the $p$ incident electrons.

Figure 6: Potential energy curves of the NS+/NS (left figure) and NO+/NO
molecular (right figure) systems relevant for low energy dissociative
recombination compiled from iacob2022  and Motapon2006  respectively. The
solid curves stand for NS while the dashed ones for NO. The black curves show
the ground electronic state of the target, while in red and blue we present
the most important dissociative molecular states with ${}^{2}\Pi$ symmetry.

The present nuclear dynamics study relies on the molecular data sets recently
calculated in the framework of the R-matrix theory iacob2022 . We found one
ion core (NS+ $X\,^{1}\Sigma^{+}$), three molecular symmetries (NS
${}^{2}\Sigma^{+}$, ${}^{2}\Pi$, and ${}^{2}\Delta$) and two or three partial
waves for the incident electron for the three symmetries relevant for low-
energy DR. Unfortunately, the electron structure and scattering calculations
iacob2022  provide global autoionization widths for the doubly excited
dissociative resonant states, meaning that the electronic couplings are not
resolved for the angular momentum of the incident electron. As a consequence,
neither our cross sections nor rate coefficients are resolved. In order to
overcome this we assume certain partial wave distributions for the vibronic
couplings. This is the case for example in Fig. 4, where we have assumed
either 100% $p$ or $d$ waves or $0.5\times p+0.5\times d$ mixture.

Following this idea, we made a detailed study considering ${}^{2}\Sigma^{+}$
and ${}^{2}\Pi$ symmetries of NS and $s$, $p$, $d$ or $p$ and $d$ partial
waves for the two symmetries. The calculated rate coefficients are presented
in Fig. 5 for different partial wave distribution scenarios. The recommended
rate coefficients for the DR of NS+ $X^{1}\Sigma^{+}$ ($v_{i}^{+}=0,1$) are
given in blue dashed-dotted lines labeled as M$(^{2}\Pi)+$M$(^{2}\Sigma^{+})$.
The grey area enclosed by the different scenarios can be regarded as
uncertainties of MQDT calculations caused by the unresolved electronic
couplings.

Due to the lack of any theoretical or experimental studies regarding the
electron induced reactive processes of NS+, we have instead compared our
present NS+ results with those previously obtained for NO+ schneider2000 ;
Motapon2006 , which due to its chemical similarity could be a proxy for NS+.
In Fig. 6 we have compared the molecular states most relevant for DR of the
NS+/NS and NO+/NO molecular systems. In black we represent the molecular state
of the target cations, in red and blue the first two dissociative states of
the neutral belonging to the ${}^{2}\Pi$ symmetry. The solid line stands for
NS, while the dashed line for NO. The two target states are different, the NS+
has a shallower but wider PEC, leading to less vibrational levels, thus fewer
ionization channels. While the second pair of dissociative states (2 and L
${}^{2}\Pi$) presented in blue show similarities: they have unfavourable
crossing points with the target states, similar shapes and asymptotic limits,
we can see more important differences for the first two states (1 and B
${}^{2}\Pi$) given in red. These have different shapes and asymptotic limits,
but most importantly the NO dissociative state has more favourable crossing
with the cation. This latter is mostly responsible for the larger rate
coefficients obtained for the DR of NO+.

The comparison between the NO+ DR rate coefficient and the NS+ one for
$v_{i}^{+}=0$ is performed in Fig. 7. We also compare here our MQDT-based rate
coefficients with those displayed in the KIDA database Wakelam2012 ;
prasad1980 . Our thermal rates are lower than the latter ones with a factor up
to 7 for NO+, and with a factor between 3 and 9 for NS+. It is stated that the
KIDA results are just estimates Wakelam2012 ; prasad1980 , without further
information. As a conclusion, we consider that our rate coefficients obtained
through MQDT for NS+ are likely to be more reliable than those given by KIDA.
As for NO+, we note that our MQDT cross sections and rate coefficients
schneider2000 ; Motapon2006  show excellent agreement with the experimental
values and should also be more reliable than those given by KIDA.

Figure 7: Dissociative recombination rate coefficients of NS+ (in black) and
NO+ schneider2000 ; Motapon2006  (in red) molecular targets for $v_{i}^{+}=0$
compared with the rates available in the KIDA database Wakelam2012 .

Finally, in order to simplify the use of our recombination rate coefficients
for kinetic modelling, we have fitted their temperature dependence by using a
simplified Arrhenius-type formula:

$\alpha(T)=a_{0}\left(\frac{T}{300}\right)^{a_{1}}$ (6)

where $T$ is in Kelvin and $\alpha$ in cm3s-1. The fitting coefficients from
the exponent became very small numbers, so they were omitted from the formula.
The fitting parameters for the DR of the lowest 2 vibrational levels of the
target, are summarized in table 2.

Table 2: Fitting parameters for the equation (6), corresponding to the rate coefficients for DR on NS+, with $v_{i}^{+}=0$ and $v_{i}^{+}=1$. Electron temperature range was divided in two regions and they vary between 10 and 1000 K. Model | Temperature | $v_{i}^{+}$ | $a_{0}\times 10^{7}$ | $a_{1}$ | RMS
---|---|---|---|---|---
| (K) |  |  (cm3s${}^{-1})$ |  |
$\mbox{M}(^{2}\Pi)+\mbox{M}(^{2}\Sigma^{+})$ |  10$\leq$ T $\leq$ 33 | 0 | 0.1426 | -0.963555 | 0.0058
|  33$<$ T $\leq$ 78 |  | 0.23193 | -0.741244 | 0.0127
|  78$<$ T $\leq$ 1000 |  | 0.348726 | -0.43795 | 0.0075
$\mbox{M}(^{2}\Pi)+\mbox{M}(^{2}\Sigma^{+})$ |  10$\leq$ T $\leq$ 31 | 1 | 1.00126 | -0.509637 | 0.0089
|  31 $<$ T $\leq$ 160 |  | 1.22272 | -0.420919 | 0.0025
|  160 $<$ T $\leq$ 1000 |  | 1.32716 | -0.280971 | 0.0057

According to the RMS values given in the table, the fitted parameters
reproduce our MQDT rate coefficients within 1.3% in the whole temperature
range $10<T<1000$ K.

##  III Conclusions

Using the step-wise multichannel quantum defect theory, we calculate cross
sections and thermal rate coefficients for the dissociative recombination of
electrons with NS${}^{+}(X\Sigma^{+})$ ions. The cross sections are computed
within the energy range of $0.01$ meV to $1$ eV, and the corresponding thermal
rate coefficients were evaluated for temperatures ranging from 10 to 1000 K,
focusing on the two lowest vibrational levels of the ions.

Our model considers all pertinent electronic states and symmetries of the
cation target, along with the relevant vibronic electronic couplings, by
taking into account the quantum interference among the direct and indirect
mechanisms. We estimate the uncertainty of our MQDT calculations caused by the
global electronic couplings unresolved in the angular momentum of the incident
electron. Comparing our results with and those obtained for the similar
molecular cation NO+ and with values given in the KIDA database, suggests that
the database values are too high. We suggest that rate coefficients obtained
through MQDT for NS+ and NO+ are likely to be more reliable than those
currently recommended in the KIDA database. The relatively moderate
dissociative recombination rate we evaluate explains - at least partially -
the ubiquitous presence of NS+ in interstellar space.

The numerical data, ready to be used in the kinetic modelling in
astrochemistry and plasma physics is available upon request to the authors.

## Acknowledgements

The authors acknowledge support provided by the Fédération de Recherche Fusion
par Confinement Magnétique (CNRS and CEA), La Région Normandie, FEDER, and
LabEx EMC3 through projects PTOLEMEE, Bioengine COMUE Normandie Université,
the Institute for Energy, Propulsion and Environment (FR-IEPE), as well as the
European Union through COST actions TUMIEE (CA17126), MW-Gaia (CA18104), and
MD-GAS (CA18212). Additional support comes from l’Agence Universitaire de la
Francophonie en Europe Centrale et Orientale (AUF ECO) through project
CE/MB/045/2021 CiCaM – ITER. The authors acknowledge the contribution of the
Agence Nationale de la Recherche (ANR) through the MONA project. This work has
received support from the Programme National ”Physique et Chimie du Milieu
Interstellaire” (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES.
J.Zs.M. is grateful for financial support from the National Research,
Development and Innovation Fund of Hungary, under the K18 and FK 19 funding
schemes with project numbers K 128621 and FK 132989. JT and JZsM extend their
appreciation to the Distinguished Guest Scientists Fellowship Programme-2022
of the Hungarian Academy of Sciences.

## Data availability

Upon a reasonable request, the data supporting this article will be provided
by the corresponding author.

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