# New type of rogue waves

N. V. Ustinov<EMAIL_ADDRESS>Joint Institute for Nuclear Research, 141980
Dubna, Russia

###### Abstract

New type of localized solutions for the two-dimensional multicomponent
Yajima–Oikawa system is presented. The dynamics of solutions of this type
occurs on the zero background and is similar to that of rogue waves.

###### pacs:

05.45.Yv, 42.65.Tg, 42.81.Dp

I. INTRODUCTION

Much attention of researchers has been paid in the recent decades to the study
of rogue waves OOS ; DKM ; AAT ; KPS ; ORBMA ; DDEG ; BCGWC ; ORB ; CBSGM ;
DGMCD ; SPP ; LSM . Various mechanisms of formation of these waves were
suggested. The occurrence of rogue waves is most often investigated on the
basis of the mechanisms of modulation instability and superposition of waves
OOS ; ORBMA ; BCGWC ; ORB ; SPP ; LSM . In both cases, an evolution of rogue
waves takes place against the background of a wave field, which is reflected
in the definitions of such waves DKM ; ORBMA ; ORB . In this report, the
localized waves developed in the absence of the background wave fields are
considered. At the same time, their dynamics corresponds to the dynamics of
rogue waves that ”appear from nowhere and disappear without a trace” AAT .

A search among solutions of the multi-dimensional nonlinear equations for ones
suitable for describing the behavior of rogue waves is of great interest. The
solutions having dynamics similar to the dynamics of rogue waves were obtained
as particular cases of lump (rational) solutions, semi-rational ones and their
generalizations (see, e.g., Refs. DM ; MS ; OY1 ; OY2 ; LQMH ). It is
important to find other types of solutions describing the dynamics of rogue
waves. The mechanisms generating such waves may be different.

The investigation of the two-dimensional multicomponent Yajima-Oikawa (YO)
system attracts significant attention in the recent years OMO ; RKLG ; KVSL ;
SK ; KVL ; KKT ; CCFM_1 ; CCFM_2 ; CCFMa ; RPHK ; SU4 . This system comprises
multiple (say $N$) short-wave components and a single long-wave one. It
generalizes the scalar ($N=1$) two-dimensional YO system ZMNP  and is often
called the 2D coupled long-wave–short-wave resonance interaction system.

The two-dimensional multicomponent YO system belongs to the class of equations
integrable by the inverse scattering transformation method M . Also, it arose
in different physical contexts. The two-component system and the
multicomponent one were derived by applying the reductive perturbation method
in Refs. OMO  and KVL , respectively, as the governing equations for the
interaction of dispersive waves in a weak Kerr-type nonlinear medium. In these
systems, the short waves propagate in anomalous dispersion regime while the
long wave propagates in the normal dispersion regime. A generation of the
terahertz radiation by optical pulses in a medium of asymmetric quantum
particles is described under the quasi-resonance conditions by the two-
dimensional two-component YO system SU4 .

Various types of solutions of the two-dimensional multicomponent YO system
were found. So, rational and semi-rational solutions mentioned above due to
their role under considering rogue waves were investigated in Refs. CCFM_2
and RPHK , respectively. The rational solutions include the fundamental
(simplest) and general (multi- and higher-order) lumps and line rogue waves
derived from the lumps under the certain parameter conditions CCFM_2 . It was
shown that the fundamental lumps and rogue waves have three different
patterns: bright, intermediate and dark states. The fundamental semi-rational
solutions considered in RPHK  can describe the fission of a dark soliton into
a lump and a dark soliton or the fusion of one lump and one dark soliton into
a dark soliton. The nonfundamental semi-rational solutions were shown to fall
into three subclasses: higher-order, multi- and mixed-type semi-rational
solutions.

The solutions discussed above of the two-dimensional multicomponent YO system
were found using the bilinear method. In this report, we exploit the Darboux
transformation (DT) technique MaSa ; GHZ  to obtain the solutions of this
system. Note that the DT technique was applied to the multicomponent YO
systems in the one-dimensional case in Refs. SU2 ; SU3 ; C ; CGSC ; LG .

The paper is organized as follows. The two-dimensional multicomponent YO
system of the general form and the corresponding overdetermined system of
linear equations are given in Section 2. Also, the DT formulas for these
systems are presented here. New type of localized solutions of the two-
dimensional multicomponent YO system on the zero background is considered in
Section 3, and the stability of solutions of this type is discussed.
Concluding remarks are given in Section 4.

II. OVERDETERMINED LINEAR SYSTEM AND DARBOUX TRANSFORMATION

The two-dimensional multicomponent YO system is written in the dimensionless
form as

$\begin{array}[]{c}{\displaystyle\frac{\partial\varphi_{n}}{\partial
t}+\frac{\partial\varphi_{n}}{\partial
y}=i\frac{\partial^{2}\varphi_{n}}{\partial x^{2}}+iu\varphi_{n}\ \
(n=1,\dots,N),}_{\mathstrut}\\\ {\displaystyle\frac{\partial u}{\partial
t}=\frac{\partial}{\partial
x}\sum\limits_{n=1}^{N}\sigma_{n}|\varphi_{n}|^{2}}^{\mathstrut},\end{array}$
(1)

where $\varphi_{n}=\varphi_{n}(x,y,t)$ and $u=u(x,y,t)$ are the $n$th short-
wave and long-wave components, respectively, $\sigma_{n}=\pm 1$
($n=1,\dots,N$). In the case of the YO system of the general form, parameters
$\sigma_{n}$ have different signs.

The two-dimensional multicomponent YO system (1) has infinitely many integrals
of motion. The first few integrals are

$\begin{array}[]{c}{\displaystyle\int\\!\\!\\!\int\\!u\,dx\,dy,\quad\int\\!\\!\\!\int\\!|\varphi_{n}|^{2}\,dx\,dy\
\ (n=1,\dots,N),}_{\mathstrut}\\\
{\displaystyle\int\\!\\!\\!\int\\!\left(u^{2}+i\sum\limits_{n=1}^{N}\sigma_{n}\left[\varphi_{n}\frac{\partial\varphi_{n}^{*}}{\partial
x}-\varphi_{n}^{*}\frac{\partial\varphi_{n}}{\partial
x}\right]\right)dx\,dy.}^{\mathstrut}\end{array}$ (2)

Also, Eqs. (1) are represented as the compatibility condition of the
overdetermined system of linear equations

$\begin{array}[]{c}{\displaystyle\frac{\partial^{2}\psi_{1}}{\partial
x^{2}}=-i\left(\frac{\partial\psi_{1}}{\partial
t}+\frac{\partial\psi_{1}}{\partial y}\right)-u\psi_{1},}_{\mathstrut}\\\
{\displaystyle\frac{\partial\psi_{n+1}}{\partial
x}=\frac{\sigma_{n}}{2}\varphi_{n}^{*}\psi_{1}\ \
(n=1,\dots,N),}^{\mathstrut}\end{array}$ (3)

and

$\begin{array}[]{c}{\displaystyle\frac{\partial\psi_{1}}{\partial
t}=-\sum\limits_{n=1}^{N}\varphi_{n}\psi_{n+1},}_{\mathstrut}\\\
{\displaystyle\frac{\partial\psi_{n+1}}{\partial
t}+\frac{\partial\psi_{n+1}}{\partial
y}=\frac{i\sigma_{n}}{2}\left(\varphi_{n}^{*}\frac{\partial\psi_{1}}{\partial
x}-\frac{\partial\varphi_{n}^{*}}{\partial
x}\psi_{1}\right)}^{\mathstrut}_{\mathstrut}\\\
{(n=1,\dots,N).}^{\mathstrut}\end{array}$ (4)

Here $\psi_{k}=\psi_{k}(x,y,t)$ ($k=1,\dots,N+1$) is the $k$th component of
the solution of Eqs. (3) and (4).

Let $\chi_{k}=\chi_{k}(x,y,t)$ ($k=1,\dots,N+1$) be the $k$th component of a
solution of the overdetermined system (3), (4). Then, the differential 1-form

$d\,\delta(\chi,\psi)=\delta_{x}(\chi,\psi)dx+\delta_{t}(\chi,\psi)dt+\delta_{y}(\chi,\psi)dy,$
(5)

where

$\delta_{x}(\chi,\psi)=\chi_{1}^{*}\psi_{1},\quad\delta_{t}(\chi,\psi)=-2\sum\limits_{n=1}^{N}\sigma_{n}\chi_{n+1}^{*}\psi_{n+1},$
$\delta_{y}(\chi,\psi)=i\left(\chi_{1}^{*}\frac{\partial\psi_{1}}{\partial
x}-\frac{\partial\chi_{1}^{*}}{\partial
x}\psi_{1}\right)-\delta_{t}(\chi,\psi),$

is closed; i.e., for a contour $\Gamma$ connecting the points
$(x_{0},y_{0},t_{0})$ and $(x,y,t)$, integral

$\delta(\chi,\psi)=\int\limits_{\Gamma}d\,\delta(\chi,\psi)+C$ (6)

($C$ is a constant) depends only on initial and final points and is
independent of a specific choice of contour $\Gamma$.

The overdetermined system of linear equations (3), (4) is covariant with
respect to the DT $\psi_{k}\to\psi_{k}[1]$ ($k=1,\dots,N+1$),
$\varphi_{n}\to\varphi_{n}[1]$ ($n=1,\dots,N$), $u\to u[1]$, where the
transformed quantities are defined in the following manner SU4 :

$\psi_{k}[1]=\psi_{k}-\frac{\delta(\chi,\psi)}{\delta(\chi,\chi)}\chi_{k}\ \
(k=1,\dots,N+1),$ (7)
$\varphi_{n}[1]=\varphi_{n}-2\sigma_{n}\frac{\chi_{n+1}^{*}\chi_{1}}{\delta(\chi,\chi)}\
\ (n=1,\,\dots,\,N),$ (8) $u[1]=u+2\frac{\partial^{2}}{\partial
x^{2}}\log\delta(\chi,\chi).$ (9)

Relations (8) and (9) define new solution of the system (1), while expressions
(7) give the components of corresponding solution of the overdetermined system
(3), (4).

III. ROGUE WAVE TYPE SOLUTIONS

Let us assume that the initial solution of the YO system (1) is the zero
background:

$\varphi_{1}=\dots=\varphi_{N}=u=0.$

In this case, we have

$\chi_{n+1}=f_{n}(t-y)\ \ (n=1,\dots,N),$ (10)

where $f_{n}(t-y)$ ($n=1,\dots,N$) are arbitrary functions of their argument.
The complex variants of the source function of the heat equation can be used
to express the component $\chi_{1}$ of solution of the overdetermined system
(3), (4). In the simplest case, this component is written as

$\chi_{1}=\frac{1}{\sqrt{y-\mu}}\,\,\exp\\!\left(\frac{i(x-\lambda)^{2}}{4(y-\mu)}\right),$
(11)

where $\lambda$ and $\mu$ are complex constants. Then, using Eqs. (6), (5),
(10) and (11), we obtain

$\begin{array}[]{c}{\displaystyle\delta=\delta(\chi,\chi)=\sqrt{\frac{\pi}{2\mu_{I}}}\,\exp\\!\left(\frac{\lambda_{I}^{2}}{2\mu_{I}}\right)}_{\mathstrut}\\\
{\displaystyle\mbox{}\times\mbox{\rm
erf}\\!\left(\frac{\lambda_{I}(y-\mu_{R})-\mu_{I}(x-\lambda_{R})}{\sqrt{2\mu_{I}}\,|y-\mu|}\right)}_{\mathstrut}^{\mathstrut}\\\
{\displaystyle\mbox{}+2\\!\\!\int\limits_{t_{0}-y_{0}}^{t-y}\\!\sum\limits_{n=1}^{N}\sigma_{n}|f_{n}(\zeta)|^{2}d\zeta+C_{0},}^{\mathstrut}\end{array}$
(12)

where $\lambda_{R}=\Re(\lambda)$, $\lambda_{I}=\Im(\lambda)$,
$\mu_{R}=\Re(\mu)$, $\mu_{I}=\Im(\mu)>0$, $C_{0}$ is a real constant,
$\mbox{\rm erf}\,(\zeta)$ is the error function.

After substitution of the expressions (10)–(12) into the DT formulas (8), (9),
we find the following solution of the two-dimensional multicomponent YO system
(1):

$\varphi_{n}=-2\sigma_{n}\frac{f_{n}(t-y)^{*}\,{\rm
e}^{\frac{i(x-\lambda)^{2}}{4(y-\mu)}}}{\sqrt{y-\mu}\,\delta}\ \
(n=1,\,\dots,\,N),$ (13) $u=2\frac{\partial^{2}}{\partial x^{2}}\log\delta\,.$
(14)

It is supposed in what follows that the functions $f_{n}(t-y)$ and constant
$C_{0}$ are such that the solution (13), (14) is nonsingular.

Different types of solutions of the two-dimensional YO system (1) are obtained
by choosing the functions $f_{n}(t-y)$ ($n=1,\dots,N$) in Eqs. (12)–(14) in
different manner. If, for example, $f_{n}(t-y)\sim\exp[\varepsilon(t-y)]$
($\varepsilon$ is a constant) or $f_{n}(t-y)\to 0$ at $|t-y|\to\infty$
($n=1,\dots,N$) then solution (13), (14) is localized on the $(x,y)$-plane for
any $t$ and $\varphi_{n}\to 0$ at $|t|\to\infty$.

Consider an interesting case when parameters $\sigma_{n}$ ($n=1,\,\dots,\,N$)
have different signs and $|f_{n}(t-y)|\to\infty$ at $|t-y|\to\infty$
($n=1,\dots,N$). Let us assume for the sake of concreteness that

$f_{n}(t-y)=\alpha_{n}{\rm e}^{\varepsilon_{1}(t-y)}+\beta_{n}{\rm
e}^{\varepsilon_{2}(t-y)}\ \ (n=1,\,\dots,\,N),$ (15)

where $\alpha_{n}$, $\beta_{n}$ ($n=1,\dots,N$), $\varepsilon_{1}$ and
$\varepsilon_{2}$ are complex constants. If
$\Re(\varepsilon_{1})\Re(\varepsilon_{2})<0$, then the solution of YO system
(1), which is obtained after the substitution of expressions (15) into Eqs.
(12)–(14), is localized on the $(x,y)$-plane, and, what is particularly
important, $\varphi_{n}\to 0$ ($n=1,\dots,N$) and $u\to 0$ at $|t|\to\infty$.
So, we have localized solution having zero temporal asymptotics. Such kind of
the dynamics resembles that of rogue waves.

It is supposed here that the YO system (1) is of general form. In the opposite
case, when all parameters $\sigma_{n}$ ($n=1,\,\dots,\,N$) have the same sign,
using expressions (15) leads to the singular solution of the YO system.

To illustrate the dynamics of the solutions discussed above we consider the
simplest case $N=2$, $\sigma_{1}=1$ and $\sigma_{2}=-1$. Eqs. (12)–(15) give
us the following expressions for the solution of the two-component YO system:

$\varphi_{n}=-2\sigma_{n}\frac{\alpha_{n}^{*}{\rm
e}^{\varepsilon_{1}^{*}(t-y)}+\beta_{n}^{*}{\rm
e}^{\varepsilon_{2}^{*}(t-y)}}{\sqrt{y-\mu}\,\Delta}\,{\rm
e}^{\frac{i(x-\lambda)^{2}}{4(y-\mu)}}\ \ (n=1,\,2),$ (16)
$u=2\frac{\partial^{2}}{\partial x^{2}}\log\Delta\,,$ (17)

where

$\begin{array}[]{c}{\displaystyle\Delta=\sqrt{\frac{\pi}{2\mu_{I}}}\,{\rm
e}^{\frac{\lambda_{I}^{2}}{2\mu_{I}}}\mbox{\rm
erf}\\!\left(\frac{\lambda_{I}(y-\mu_{R})-\mu_{I}(x-\lambda_{R})}{\sqrt{2\mu_{I}}\,|y-\mu|}\right)}_{\mathstrut}\\\
{\displaystyle\mbox{}+2\\!\\!\int\limits_{t_{0}-y_{0}}^{t-y}\\!\sum\limits_{n=1}^{2}\sigma_{n}\left|\alpha_{n}{\rm
e}^{\varepsilon_{1}\zeta}+\beta_{n}{\rm
e}^{\varepsilon_{2}\zeta}\right|^{2}d\zeta+C_{0}.}^{\mathstrut}\end{array}$

The profiles of the absolute value of component $\varphi_{1}$ and component
$u$ of solution (16), (17) for different values of variable $t$ and for the
parameter values $\lambda=i$, $\mu=2i$, $y_{0}=t_{0}=0$, $\alpha_{1}=1$,
$\beta_{1}=2$, $\alpha_{2}=2$, $\beta_{2}=1$, $\varepsilon_{1}=-1$,
$\varepsilon_{2}=1$ and $C_{0}=6$ are presented in Figs. 1 and 2.

Figure 1: Profiles of $|\varphi_{1}|$ for $t=-16$ (a), $t=-4$ (b), $t=0$ (c),
$t=4$ (d) and $t=16$ (e).

The complete dynamics is given in the files SM1.gif and SM2.gif in
Supplemental Material SM . It is seen that this solution has form of the
solitary wave, and all its components are localized on the $(x,y)$-plane for
any $t$. In the limit $|t|\to\infty$, the amplitudes of components
$\varphi_{1}$ and $\varphi_{2}$ tend to zero as $1/\sqrt{|t|}$ (see Fig. 1).
The length $l_{y}$ of the wave along axis $y$ can be estimated as
$l_{y}\sim|\Re(\varepsilon_{1})|^{-1}+|\Re(\varepsilon_{2})|^{-1}$. For
$|t|\gg\sqrt{|\mu|^{2}+|\lambda|^{2}}$, the length $l_{x}$ along axis $x$
exceeds $l_{y}$ and can be estimated as

$l_{x}\sim
2|t|\left(\sqrt{\lambda_{I}^{2}+4\mu_{I}}-|\lambda_{I}|\right)/\mu_{I}.$

The decrease of long-wave component $u$ as $|t|\to\infty$ occurs faster than
the short-wave ones (see Fig. 2).

Figure 2: Profiles of $u$ for $t=-16$ (a), $t=-4$ (b), $t=0$ (c), $t=4$ (d)
and $t=16$ (e).

Thus, we see that the dynamics of solitary wave (16), (17) matches with that
of rogue waves AAT . There is, however, an important distinction. Whereas the
phenomenon of rogue wave develops on the background waves, the solitary wave
(16), (17) evolves on the zero background.

The height of rogue wave has to be more than about twice the significant
height of background waves DKM ; ORBMA ; ORB . The waves, whose height exceeds
the background value more than five times, are sometimes called super rogue
waves CHOA ; SPSCHOA . Here the background waves are absent. The maximum
values of amplitudes of $\varphi_{1}$, $\varphi_{2}$ and $u$ of the solitary
wave (16), (17) depend on its parameters and can be arbitrary large.

Solitary waves having similar dynamics exist for arbitrary number $N>1$ of the
short-wave components of system (1). The functions $f_{n}(t-y)$
($n=1,\dots,N$) in Eqs. (12)–(14) have to satisfy conditions
$|f_{n}(t-y)|\to\infty$ at $t-y\to\pm\infty$ in this case. For example, these
functions can be chosen in accordance with Eqs. (15). Different signs among
$\sigma_{n}$ ($n=1,\,\dots,\,N$) are necessary to obtain the nonsingular
solutions in that case.

The generalizations of rogue wave of the form (16), (17) can be obtained if
some generalizations of the complex variant of the source function (11) are
used as component $\chi_{1}$ in the DT formulas. In particular, component
$\chi_{1}$ can be chosen in the following manner:

$\chi_{1}=\sum\limits_{m=1}^{M}\sum\limits_{l=1}^{L}\frac{\nu_{lm}}{\sqrt{y-\mu_{m}}}\exp\left(\frac{i(x-\lambda_{l})^{2}}{4(y-\mu_{m})}\right),$
(18)

where $\nu_{lm}$, $\lambda_{l}$ and $\mu_{m}$ ($l=1,\,\dots,\,L$;
$m=1,\,\dots,\,M$) are complex constants, $\Im(\mu_{m})>0$. Also, we can put

$\chi_{1}=\left(c_{1}\frac{\partial}{\partial\lambda}+c_{2}\frac{\partial}{\partial\mu}\right)\frac{1}{\sqrt{y-\mu}}\exp\left(\frac{i(x-\lambda)^{2}}{4(y-\mu)}\right),$
(19)

where $c_{1}$ and $c_{2}$ are constants. The study of such generalizations of
rogue wave (16), (17) (multi- and higher-order waves) and their interaction
with waves of other types requires a separate consideration.

Note that the stability of rogue wave (16), (17) with respect to the
perturbations of a special kind can be established within the frameworks of
the DT technique. Indeed, let us take the solution of the overdetermined
system (3), (4) in the form

$\chi_{1}=\frac{1}{\sqrt{y-\mu}}\,\,\exp\\!\left(\frac{i(x-\lambda)^{2}}{4(y-\mu)}\right)+\kappa\tilde{\chi}_{1},$
(20) $\chi_{n+1}=\alpha_{n}{\rm e}^{\varepsilon_{1}(t-y)}+\beta_{n}{\rm
e}^{\varepsilon_{2}(t-y)}+\kappa F_{n}(t-y)\ \ (n=1,\,2),$ (21)

where $\kappa$ is parameter considered to be small, $\tilde{\chi}_{1}$ is
defined as $\chi_{1}$ in Eq. (18) or in Eq. (19), $F_{1,2}(t-y)$ are the
functions of their argument such that $|F_{1,2}(t-y)|<1$. The substitution of
expressions (20), (21) into the DT formulas (8), (9) gives us the perturbed
solution of the two-dimensional YO system (1). This solution coincides with
rogue wave (16), (17) in the case $\kappa=0$. It is important that the
difference between the perturbed solution and rogue wave (16), (17) will be
insignificant during the time evolution if $|\kappa|\ll 1$. This indicates the
stability of rogue wave considered with respect to the perturbations of
special form.

The existence of integrals of motion (2) is important in the investigation of
stability of rogue wave (16), (17) in the general case and in the numerical
simulations. Also, the integrals of motion can be helpful under the study of
blowing up of solution (13), (14) that takes place for some values of its
parameters.

IV. CONCLUSION

In this paper, the new type of rogue waves for the two-dimensional
multicomponent Yajima–Oikawa system is presented. The waves of this type are
distinguished by the fact that their dynamics occur on the zero background.
This implies that rogue waves presented here are formed solely due to the
nonlinear focusing. It seems very important to extend this type of rogue waves
to other models of various physical contexts describing the wave interactions.

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# Strong boundedness of ${\rm SL}_{2}(R)$ for rings of S-algebraic integers
with infinitely many units

Alexander A. Trost  Fakultät für Mathematik, Ruhr Universität Bochum, D-44780
Bochum, Germany<EMAIL_ADDRESS>
###### Abstract.

A group is called strongly bounded, if the speed with which it is generated by
finitely many conjugacy classes has a positive, lower bound only dependent on
the number of the conjugacy classes in question rather than the actual
conjugacy classes. Earlier papers [5, 15] have shown that this is a property
common to split Chevalley groups defined using an irreducible root system of
rank at least $2$ and the ring of all S-algebraic integers and that the
situation is dependent on the number theory of $R$ for ${\rm Sp}_{4}$ and
$G_{2}.$ In this paper, we will show that ${\rm SL}_{2}(R)$ is also strongly
bounded for $R$ the ring of all S-algebraic integers in a number field $K$
with $R$ having infinitely many units and will give a complete account of the
existence of small conjugacy classes generating ${\rm SL}_{2}(R)$ in terms of
the prime factorization of the rational primes $2$ and $3$ in $R.$

##  1\. Introduction

It is a common belief that the group ${\rm SL}_{2}(R)$ for $R$ the ring of
S-algebraic integers in a number field $K$ such that $R$ has infinitely many
units, behaves very similar to S-arithmetic split Chevalley groups of higher
rank and there have been various results in favor of this philosophy:
Classically, there are the results by Serre proving the Congruence Subgroup
Property for ${\rm SL}_{2}(R)$ [11], which show that the Congruence kernel for
${\rm SL}_{2}(R)$ and ${\rm SL}_{n}(R)$ for $n\geq 3$ is identical. Further,
it was shown by Carter, Keller and Paige and written up by Morris [8] that
bounded generation not only occurs for higher rank, S-arithmetic Chevalley
groups as proven by Carter and Keller [3] and Tavgen [12], but also in the
case of ${\rm SL}_{2}(R).$ In this paper, we will show that a different
property, called strong boundedness, previously only shown for S-arithmetic,
split Chevalley groups of higher rank, also holds in the case of ${\rm
SL}_{2}(R)$ for $R$ the ring of S-algebraic integers in a number field $K$
such that $R$ has infinitely many units. Roughly speaking, strong boundedness
is the property of a group $G$ to admit for each natural number $k$ a global
bound $\Delta_{k}(G)$ such that the word norm $\|\cdot\|_{T}$ induced by $k$
generating conjugacy classes $T$ of $G$ has diameter at most $\Delta_{k}(G)$
independent of the specific $T$ in question. This property has first been
described by Kedra, Libman and Martin:

######  Theorem 1.1.

[5, Theorem 6.1] Let $R$ be a principal ideal domain and let ${\rm SL}_{n}(R)$
be boundedly generated by elementary matrices for $n\geq 3$ with the diameter
$\|{\rm SL}_{n}(R)\|_{EL(n)}$ satisfying $\|{\rm SL}_{n}(R)\|_{EL(n)}\leq
C_{n}$ for some $C_{n}\in\mathbb{N}.$ Then ${\rm SL}_{n}(R)$ is normally
generated by the single element $E_{1,n}(1)$ and

  1. (1)

for all finite, normally generating subsets $T$ of $G$, it holds $\|{\rm
SL}_{n}(R)\|_{T}\leq C_{n}(4n+4)|T|.$

  2. (2)

if $R$ has infinitely many maximal ideals, then for each $k\in\mathbb{N}$
there is a finite, normally generating subset $T_{k}$ of $G$ with $|T_{k}|=k$
and $\|{\rm SL}_{n}(R)\|_{T_{k}}\geq k$.

So, in particular ${\rm SL}_{n}(R)$ is strongly bounded for rings of
S-algebraic integers $R$ with class number $1.$ This is due to the fact that
according to Carter and Keller [12], the group ${\rm SL}_{n}(R)$ is boundedly
generated by elementary matrices. In our previous paper [15], we have shown
that this type of statement generalizes to all other S-arithmetic, split
Chevalley groups $G(\Phi,R)$ defined using an irreducible root system $\Phi$
of rank at least $2$ and the ring of S-algebraic integers $R$. However
surprisingly, it turned out that the behavior of the rank two examples ${\rm
Sp}_{4}(R)$ and $G_{2}(R)$ are markedly different from the other cases.
Namely, to describe the behavior of the $\Delta_{k}$ for ${\rm Sp}_{4}(R)$ and
$G_{2}(R)$, it is necessary to understand how the rational prime $2$ factors
in the ring of integers $R$ and the situation is generally more involved in
this case. So it is unsurprising that the investigation for the case of ${\rm
SL}_{2}(R)$ is even more complicated: The strategies used by Kedra, Libman and
Martin to prove Theorem 1.1 and the similar ones used to derive our results in
[15] rely ultimately on the presence of an irreducible, rank $2$ root
subsystem of $\Phi$ and the nice interaction of different root elements in
$G(\Phi,R)$. So clearly this approach will not work anymore for the example of
${\rm SL}_{2}(R)$. However, using so-called self-reproducing elements $C(x)$
of $SL_{2}(R)$ for $x\in R$ in addition to root elements, one can still prove
strong boundedness for ${\rm SL}_{2}(R)$ assuming $R$ has infinitely many
units:

######  Theorem 1.2.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers in a number
field $K$ such that $R$ has infinitely many units. Then there is a constant
$C(R)\in\mathbb{N}$ such that $\Delta_{k}({\rm SL}_{2}(R))\leq C(R)k$ holds
for all $k\in\mathbb{N}.$

To prove this we will reformulate a description of normal subgroups of ${\rm
SL}_{2}(R)$ for $R={\mathcal{O}}_{S}$ a ring of S-algebraic integers with
infinitely many units by Costa and Keller [4] in terms of a first order theory
and then invoke a compactness argument together with the afore-mentioned
bounded generation result by Carter, Keller and Paige [8].

Further, the behavior of $\Delta_{k}$ for ${\rm Sp}_{4}(R)$ and $G_{2}(R)$
depends on the number theory of $R.$ Namely, our theorem [15, Theorem 6.3]
states that the size of the smallest possible normally generating subset is
controlled by the number of certain problematic prime factors of the rational
prime $2.$ This dependence on the number theory of $R$ still exists for ${\rm
SL}_{2}(R)$, but the situation is overall more complicated. Not only are there
other problematic prime divisors besides certain divisors of $2$, but to a
certain extent it is even relevant how the prime divisors of $2$ ramify in
$R$:

######  Theorem 1.3.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers in a number
field $K$ such that $R$ has infinitely many units. Set

$\displaystyle r_{1}(R):=|\\{{\mathcal{P}}\mid{\mathcal{P}}\text{ is an
unramified prime divisor of }2R\text{ with
}R/{\mathcal{P}}=\mathbb{F}_{2}\\}|,$ $\displaystyle
r_{2}(R):=|\\{{\mathcal{P}}\mid{\mathcal{P}}\text{ is a ramified prime divisor
of }2R\text{ with }R/{\mathcal{P}}=\mathbb{F}_{2}\\}|\text{ and}$
$\displaystyle q(R):=|\\{{\mathcal{P}}\mid{\mathcal{P}}\text{ is a prime
divisor of }3R\text{ with }R/{\mathcal{P}}=\mathbb{F}_{3}\\}|.$

Then define $v(R):=\max\\{2r_{2}(R)+r_{1}(R),q(R)\\}.$ Then

  1. (1)

$\Delta_{k}({\rm SL}_{2}(R))\geq 2k$ holds for all $k\geq v(R)$ and

  2. (2)

$\Delta_{k}({\rm SL}_{2}(R))=-\infty$ holds for all $k<v(R).$

This is proven by extracting necessary and sufficient conditions on a set $T$
to normally generate ${\rm SL}_{2}(R)$ from the proof of Theorem 1.2 to
subsequently use the congruence subgroup property to describe the problem
posed by the prime ideals mentioned in Theorem 1.3. Of particular interest in
this context is an epimorphism ${\rm
SL}_{2}(R/{\mathcal{P}}^{2})\to\mathbb{F}_{2}\oplus\mathbb{F}_{2}=(R/{\mathcal{P}}^{2},+)$
we construct in the proof of Proposition 5.5 for a ramified prime divisor
${\mathcal{P}}$ of $2$ with $R/{\mathcal{P}}=\mathbb{F}_{2}$ highlighting the
difference between ramified and unramified prime divisors of $2$ in $R.$ At
the end of the paper, we also apply Theorem 1.3 to the more explicit example
of rings of quadratic integers to obtain Corollary 5.13.

The paper is structured as follows: In the second section, we introduce the
used notation. The third section explains how to derive Theorem 1.2 from
various technical results and in the fourth section we explain how to deduce
these technical results from the structure theory of normal subgroups of ${\rm
SL}_{2}.$ In the fifth section, we will prove Theorem 1.3 using the congruence
subgroup property and a careful analysis of groups of the form ${\rm
SL}_{2}(R/{\mathcal{P}}^{L})$ for prime ideals ${\mathcal{P}}$ and natural
numbers $L\in\mathbb{N}.$

##  2\. Basic definitions and notions

First, recall the definition of S-algebraic integers:

######  Definition 2.1.

Let ${\mathcal{O}}_{K}$ be the ring of integers in the number field $K$ and
let $S$ be a finite set of non-zero prime ideals in ${\mathcal{O}}_{K}.$ Then
the ring

${\mathcal{O}}_{S}:=\left\\{a/b\mid a,b\in{\mathcal{O}}_{K},b\neq 0,\\{\text{
prime divisors of }b{\mathcal{O}}_{K}\\}\subset S\right\\}$

is called the ring of S-algebraic integers in $K$.

######  Remark 2.2.

In the rest of the paper, we will for the sake of brevity, usually not mention
the choice of the finite set of non-zero prime divisors $S$ in
${\mathcal{O}}_{K},$ when talking about the ring of S-algebraic integers
${\mathcal{O}}_{S}$ in a number field $K.$

We also recall the word norms studied in this paper:

######  Definition 2.3.

Let $G$ be a group.

  1. (1)

We define $A^{B}:=B^{-1}AB$ for $A,B\in G$.

  2. (2)

For $T\subset G$, we define $\langle\langle T\rangle\rangle$ as the smallest
normal subgroup of $G$ containing $T.$

  3. (3)

A subset $T\subset G$ is called a normally generating set of $G$, if
$\langle\langle T\rangle\rangle=G$.

  4. (4)

For $k\in\mathbb{N}$ and $T\subset G$ denote by

$B_{T}(k):=\bigcup_{1\leq i\leq k}\\{x_{1}\cdots x_{i}\mid\forall j\leq
i:x_{j}\text{ conjugate to }A\text{ or }A^{-1}\text{ and }A\in
T\\}\cup\\{1\\}.$

Further set $B_{T}(0):=\\{1\\}.$ If $T$ only contains the single element $A$,
then we write $B_{A}(k)$ instead of $B_{\\{A\\}}(k)$.

  5. (5)

Define for a set $T\subset G$ the conjugation invariant word norm
$\|\cdot\|_{T}:G\to\mathbb{N}_{0}\cup\\{+\infty\\}$ by
$\|A\|_{T}:=\min\\{k\in\mathbb{N}_{0}|A\in B_{T}(k)\\}$ for
$A\in\langle\langle T\rangle\rangle$ and by $\|A\|_{T}:=+\infty$ for
$A\notin\langle\langle T\rangle\rangle.$ The diameter $\|G\|_{T}={\rm
diam}(\|\cdot\|_{T})$ of $G$ is defined as the minimal $N\in\mathbb{N}$ such
that $\|A\|_{T}\leq N$ for all $A\in G$ or as $+\infty$ if there is no such
$N$.

  6. (6)

Define for $k\in\mathbb{N}$ the invariant

$\Delta_{k}(G):=\sup\\{{\rm diam}(\|\cdot\|_{T})|\ T\subset G\text{ with
}|T|\leq k,\langle\langle
T\rangle\rangle=G\\}\in\mathbb{N}_{0}\cup\\{\pm\infty\\}$

with $\Delta_{k}(G)$ defined as $-\infty$, if there is no normally generating
set $T\subset G$ with $|T|\leq k.$

  7. (7)

The group $G$ is called strongly bounded, if $\Delta_{k}(G)<+\infty$ for all
$k\in\mathbb{N}$.

Let us next recall the definition of ${\rm SL}_{2}:$

######  Definition 2.4.

Let $R$ be a commutative ring with $1$. Then ${\rm SL}_{2}(R):=\\{A\in
R^{2\times 2}\mid a_{11}a_{22}-a_{12}a_{21}=1\\}.$

Obviously for any commutative ring $R$ and any $x\in R,$ the matrices

$E_{12}(x)=\begin{pmatrix}1&x\\\ 0&1\end{pmatrix}\text{ and
}E_{21}(x)=\begin{pmatrix}1&0\\\ x&1\end{pmatrix}$

are elements of ${\rm SL}_{2}(R).$ They are called elementary matrices and we
denote the set of elementary matrices $\\{E_{12}(x),E_{21}(x)\mid x\in R\\}$
by ${\rm EL}.$ The subgroup of ${\rm SL}_{2}(R)$ generated by ${\rm EL}$ is
denoted by $E(2,R).$ Furthermore, for an ideal $I\unlhd R,$ we denote by
$E(2,R,I)$ the normal subgroup of $E(2,R)$ generated by the
$E(2,R)$-conjugates of elements of $\\{E_{12}(x)\mid x\in I\\}.$ Further, we
define the reduction homomorphism $\pi_{I}:{\rm SL}_{2}(R)\to{\rm
SL}_{2}(R/I)$ as the group homomorphism induced by the quotient homomorphism
$R\to R/I$ and the subgroup ${\rm SL}_{2}(R,I)$ of ${\rm SL}_{2}(R)$ as the
kernel of the homomorphism $\pi_{I}.$ Further, for a unit $u\in R^{*}$ the
element

$h(u)=\begin{pmatrix}u&0\\\ 0&u^{-1}\end{pmatrix}$

is also an element of ${\rm SL}_{2}(R).$ We also introduce the concept of
self-reproducing elements:

######  Definition 2.5.

Let $R$ be a commutative ring with $1$ and let $x\in R$ be given. Then
$C(x):=E_{21}(x)\cdot E_{12}(x)$ is called a self-reproducing element.
Further, for $X\subset R,$ one defines $C(X)$ as the subgroup of $E(2,R)$
normally generated by the set $\\{C(x)\mid x\in X\\}.$

######  Remark 2.6.

The elements $C(x)$ are called self-reproducing, because the subgroup $C(X)$
normally generated by the self-reproducing elements $\\{C(x)\mid x\in X\\}$
has the property

$C(X)=[E(2,R),C(X)]$

according to [4, Theorem 1.1.1(i)].

Self-reproducing elements have the following useful property:

######  Lemma 2.7.

Let $R$ be a commutative ring with $1$, let $T\subset{\rm
SL}_{2}(R),x_{1},x_{2}\in R$ and $k_{1},k_{2}\in\mathbb{N}$ be given. Assume
further that $C(x_{1})\in B_{T}(k_{1})$ and $C(x_{2})\in B_{T}(k_{2}).$ Then
$C(x_{1}+x_{2})\in B_{T}(k_{1}+k_{2}).$

###### Proof.

First, note for $x,y\in R$ that

$\displaystyle C(x)^{-1}\cdot C(y)$ $\displaystyle=[E_{21}(x)\cdot
E_{12}(x)]^{-1}\cdot[E_{21}(y)\cdot E_{12}(y)]$ $\displaystyle=E_{12}(-x)\cdot
E_{21}(-x)\cdot E_{21}(y)\cdot E_{12}(y)$ $\displaystyle=E_{12}(-x)\cdot
E_{21}(y-x)\cdot E_{12}(y-x)\cdot E_{12}(x)$
$\displaystyle=C(y-x)^{E_{12}(x)}.$

This equation implies in particular that
$C(-x_{1})=[C(x_{1})^{-1}]^{E_{12}(-x_{1})}\in B_{T}(k_{1})$ and so
$C(x_{1}+x_{2})=(C(-x_{1})^{-1}\cdot C(x_{2}))^{E_{12}(x_{1})}\in
B_{T}(k_{1}+k_{2})$ follows. ∎

Next, we recall the definition of level ideals:

######  Definition 2.8.

Let $R$ be a commutative ring with $1$ and let

$A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in{\rm SL}_{2}(R)$

be given. The level ideal $l(A)$ is defined as the ideal $(a-d,b,c)\unlhd R.$
For a subset $T\subset{\rm SL}_{2}(R),$ we set $l(T):=\sum_{A\in T}l(A).$

##  3\. The proof of Theorem 1.2

The proof of the main theorem operates by extracting increasingly more generic
elements from a given collection of conjugacy classes $T$ generating ${\rm
SL}_{2}(R)$ and use them to construct a congruence subgroup. First, we
introduce the class of rings used in the investigation:

######  Definition 3.1.

Let $R$ be a commutative ring with $1$ satisfying the following two
properties:

  1. (1)

For each $c\in R-(\\{0\\}\cup R^{*})$ the ring $R/cR$ has (Bass) stable range
$1$.

  2. (2)

For each $x\in R-\\{0\\},$ there is a unit $u\in R$ such that $u^{4}\neq 1$
and $u^{2}\equiv 1\text{ mod }xR.$

Then we call $R$ a ring with many units.

######  Remark 3.2.

  1. (1)

A ring $R$ having stable range $1$ is defined as follows: For all $a,b\in R$
with $(a,b)=R$, there is an $x\in R$ such that $a+bx$ is a unit in $R.$

  2. (2)

A ring satisfying only Definition 3.1(i) is called a ring of stable range at
most $3/2.$

  3. (3)

Observe that $R$ being a ring with many units is a property describable in
first order terms.

Next, we note the following important intermediate step that we will prove in
the next section:

######  Theorem 3.3.

Let $R$ be a ring with many units. Then there is a natural number
$L_{1}\in\mathbb{N}$ independent of $R$ such that for each $A\in{\rm
SL}_{2}(R)$ and $x\in l(A)$, one has

$\\{C(bx^{3})\mid b\in R\\}\subset B_{A}(L_{1}).$

Before continuing, we note the following:

######  Proposition 3.4.

Let $R$ be a commutative ring with $1$ and let $T\subset{\rm SL}_{2}(R)$ be a
normally generating subset of ${\rm SL}_{2}(R).$ Then $l(T)=\sum_{A\in
T}l(A)=R.$

We will omit a proof, because it is virtually identical to the proof of [15,
Lemma 3.3] and rather straightforward.

######  Remark 3.5.

For later use, we remark that the set of non-zero prime ideals ${\mathcal{P}}$
with $\pi_{{\mathcal{P}}}(T)$ consisting only of scalar matrices in ${\rm
SL}_{2}(R/{\mathcal{P}})$ is denoted by $\Pi(T)$ and that $\Pi(T)=\emptyset$
is equivalent to $\sum_{A\in T}l(A)=R.$

Next, we note the following:

######  Theorem 3.6.

Let $R$ be a commutative ring with $1$ and a unit $u\in R$. Then there is a
natural number $L_{2}$ independent of $R$ and $u$ such that
$\\{E_{12}(x\cdot(u^{4}-1))\mid x\in R\\}\subset B_{C(1)}(L_{2}).$

Again, we postpone the proof until the next section. From this, we can deduce:

######  Theorem 3.7.

Let $R$ be a ring with many units and $T\subset{\rm SL}_{2}(R)$ finite with
$\sum_{A\in T}l(A)=R.$ Then there is a unit $u\in R$ with $u^{4}\neq 1$
independent of $T$ and a natural number $L_{3}$ independent of $R,u$ and $T$
such that $\\{E_{12}(x(u^{4}-1))\mid x\in R\\}\subset B_{T}(L_{3}\cdot|T|).$

###### Proof.

Observe that as $T$ has the property $\sum_{A\in T}l(A)=R$, there are elements
$x_{A}\in l(A)$ for $A\in T$ with $(x_{A}\mid A\in T)=R.$ So in particular, we
have $(x_{A}^{3}\mid A\in T)=R$. Thus for each $A\in T$ there are $b_{A}\in R$
with $1=\sum_{A\in T}b_{A}\cdot x_{A}^{3}.$ Observe that Theorem 3.3 then
implies that $C(b_{A}\cdot x_{A}^{3})\in B_{A}(L_{1})\subset B_{T}(L_{1})$ for
each $A\in T.$ Hence using Lemma 2.7 repeatedly, we obtain

$C(1)=C(\sum_{A\in T}b_{A}\cdot x_{A}^{3})\in B_{T}(L_{1}\cdot|T|).$

But $R$ has many units and so there is a unit $u\in R$ with $u^{4}\neq 1.$ But
then Theorem 3.6 implies that

$\\{E_{12}(x\cdot(u^{4}-1))\mid x\in R\\}\subset B_{C(1)}(L_{2})\subset
B_{T}(L_{1}\cdot L_{2}\cdot|T|)$

and so setting $L_{3}:=L_{1}\cdot L_{2}$, we are done. ∎

Next, we note the following:

######  Lemma 3.8.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers in a number
field $K$ such that $R$ has infinitely many units. Then $R$ is a ring with
many units.

###### Proof.

Rings with many units are defined by two properties so showing both properties
is enough to prove the lemma. First, to show that $R$ has stable range at most
$3/2$ let $c\in R-(\\{0\\}\cup R^{*})$ be given. Then the ring $R/cR$ is
finite, so in particular it is semi-local and hence has stable range $1$
according to [1, Lemma 6.4, Corollary 6.5]. For the second property, let $x\in
R-(\\{0\\}\cup R^{*})$ be given. According to Dirichlet’s Unit Theorem [9,
Corollary 11.7], there is a unit $v\in R$ of infinite order. But observe that
$R/xR$ is a finite ring, so there must be a non-trivial power $v^{k}$ of $v$
such that $v^{k}+xR=1+xR.$ Setting $u:=v^{k}$, we obtain then that $u^{4}\neq
1$ and $u^{2}\equiv 1\text{ mod }xR.$ ∎

To finish the proof of Theorem 1.2, we also need

######  Lemma 3.9.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers in a number
field $K$ such that $R$ has infinitely many units. Further, let $I\unlhd R$ be
a non-trivial ideal in $R.$ Also define $Q_{I}:=\\{E_{12}(x)\mid x\in I\\}$
and $N_{I}:=\langle\langle Q_{I}\rangle\rangle$ and let
$\|\cdot\|_{Q_{I}}:N_{I}\to\mathbb{N}\cup\\{+\infty\\}$ be the conjugation
invariant word norm on $N_{I}$ defined by $Q_{I}.$

  1. (1)

Then the group ${\rm SL}_{2}(R)/N_{I}$ is finite.

  2. (2)

Then there is a $K(I,R)\in\mathbb{N}$ such that $\|N_{I}\|_{Q_{I}}\leq
K(I,R)$.

###### Proof.

First, note that as $R$ is a ring of S-algebraic integers with infinitely many
units, the group ${\rm SL}_{2}(R)$ is boundedly generated by elementary
matrices according to [8, Theorem 1.2]. But then the lemma follows in
essentially the same way as [15, Lemma 3.4] by replacing the ideal $2R$ by the
ideal $I$. ∎

We have accumulated the tools to prove Theorem 1.2 now:

###### Proof.

According to Lemma 3.8, the ring $R$ of S-algebraic integers is a ring with
many units. Let $T$ be a finite, normally generating subset of ${\rm
SL}_{2}(R).$ Then according to Lemma 3.4, we have $\sum_{A\in T}l(A)=R$. So
Theorem 3.7 implies that there is a unit $u\in R$ independent of $T$ with
$u^{4}\neq 1$ and

$\\{E_{12}(x(u^{4}-1))\mid x\in R\\}\subset B_{T}(L_{3}\cdot|T|)$

with $L_{3}$ independent of $R,u$ and $T.$ Define $I:=(u^{4}-1)\unlhd R$ and
consider the group $G:={\rm SL}_{2}(R)/N_{I}$ together with the quotient map
$\pi:{\rm SL}_{2}(R)\to G.$ Then consider the set

$E(G):=\\{\bar{S}\subset G\mid\bar{S}\text{ normally generates }G\\}.$

But according to Lemma 3.9(i), the group $G$ is finite and hence so is $E(G).$
Thus there is a constant $M:=M(I)\in\mathbb{N}$ such that for each
$S\subset{\rm SL}_{2}(R)$ with $\pi(S)\in E(G)$ and each $A\in{\rm
SL}_{2}(R)$, there are $s_{1},\dots,s_{M(I)}\in S\cup S^{-1}\cup\\{I_{2}\\}$
and $X_{1},\dots,X_{M(I)}\in{\rm SL}_{2}(R)$ with

$\pi(A)=\pi\left(\prod_{j=1}^{M(I)}s_{j}^{X_{j}}\right).$

Hence $A(\prod_{j=1}^{M(I)}s_{j}^{X_{j}})^{-1}\in N_{I}$ and so setting
$\|N_{I}\|_{T}:={\rm sup}\\{\|g\|_{T}|g\in N\\}\in\mathbb{N}\cup\\{+\infty\\}$
implies

$\|A\|_{T}\leq\|\prod_{j=1}^{M(I)}s_{j}^{X_{j}}\|_{T}+\|N_{I}\|_{T}\leq
M(I)\cdot\max\\{\|s\|_{T}|\ s\in S\\}+\|N_{I}\|_{T}.$

This implies $\|{\rm SL}_{2}(R)\|_{T}\leq M(I)\max\\{\|s\|_{T}|\ s\in
S\\}+\|N_{I}\|_{T}$ for all $S\subset{\rm SL}_{2}(R)$ with $\pi(S)\in E(G).$
Next, note that as $T$ normally generates ${\rm SL}_{2}(R)$, we obtain
$\pi(T)\in E(G)$ and for $t\in T$, one clearly has $\|t\|_{T}\leq 1$ and thus
$\|{\rm SL}_{2}(R)\|_{T}\leq M(I)+\|N_{I}\|_{T}.$ But $M(I)$ does not not
depend on $T$ and hence it suffices to bound $\|N_{I}\|_{T}$ linearly in $|T|$
to finish the proof of Theorem 1.2. But note that according to Theorem 3.7,
there is an $L_{3}\in\mathbb{N}$ such that for all $q\in Q_{I}$, one has
$\|q\|_{T}\leq L_{3}|T|.$ Further by Lemma 3.9(ii), there is a
$K(I,R)\in\mathbb{N}$ such that $\|N_{I}\|_{Q_{I}}\leq K(I,R)$. This implies
$\|N_{I}\|_{T}\leq K(I,R)L_{3}|T|$ and this finishes the proof. ∎

Next, we want to derive a criterion for a subset $T$ to normally generate
${\rm SL}_{2}(R)$ from the previous discussion. First, let $T$ be a subset of
${\rm SL}_{2}(R)$ with $\Pi(T)=\emptyset$ and let $N$ be the normal subgroup
generated by $T.$ Note that we derived in the proof of Theorem 3.7 that $C(1)$
is an element of $N.$ However, the argument can equally well be used to derive
that $\\{C(x)\mid x\in R\\}$ is a subset of $N.$ However, note that as
$C(x)=E_{21}(x)E_{12}(x)$, this implies that $E_{21}(-x)N=E_{12}(x)N$ holds in
${\rm SL}_{2}(R)/N$ for all $x\in R.$ But ${\rm SL}_{2}(R)$ is generated by
the set ${\rm EL}$ of elementary matrices. Hence ${\rm SL}_{2}(R)/N$ is in
fact an abelian group. Thus $T$ normally generates ${\rm SL}_{2}(R)$ if it has
the additional property that $T$ maps to a generating set of the
abelianization $H_{1}({\rm SL_{2}}(R)).$ But clearly if on the other hand $T$
is a normally generating subset of ${\rm SL}_{2}(R),$ then $\Pi(T)=\emptyset$
and $T$ mapping onto a generating subset of $H_{1}({\rm SL}_{2}(R))$ must
hold. Thus we have proven:

######  Corollary 3.10.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers of a number
field $K$ such that $R$ has infinitely many units and let $T\subset{\rm
SL}_{2}(R)$ be given. Then $T$ normally generates ${\rm SL}_{2}(R)$ if and
only if $\Pi(T)=\emptyset$ and $T$ maps onto a generating set of the
abelianization of ${\rm SL}_{2}(R).$

##  4\. Compactness arguments, radices and the proofs of Theorem 3.3 and
Theorem 3.6

The strategy to prove Theorem 3.3 is to interpret it as a consequence of the
inconsistency of a certain first order theory. In order to do this we need
some more terminology.

######  Remark 4.1.

Much of the terminology and the main statements in this subsection are due to
the paper [4] by Costa and Keller and we highly encourage to read it.

First, we introduce the concept of radices defined in [4]:

######  Definition 4.2.

Let $R$ be a commutative ring with $1$ and $P\subset R$ a subgroup of $(R,+).$
Then $P$ is called a radix, if it satisfies the following two conditions:

  1. (1)

$(a^{3}-a)x^{2}+a^{2}x$ is an element of $P$ for all $x\in P$ and $a\in R.$

  2. (2)

$ax^{3}$ is an element of $P$ for all $x\in P$ and $a\in R.$

Next, we define a special map called $\rho:$

######  Definition 4.3.

Let $R$ be a commutative ring with $1$. Then the map $\rho:{\rm SL}_{2}(R)\to
R$ is defined by

$\rho\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=a^{2}-1+ab.$

Similarly, the maps $\rho_{T},\rho_{-1}$ and $\rho_{-T}:{\rm SL}_{2}(R)\to R$
are defined as follows for $A\in{\rm SL}_{2}(R)$:

  1. (1)

$\rho_{T}(A):=\rho(A^{T}),$

  2. (2)

$\rho_{-1}(A):=\rho(A^{-1})$ and

  3. (3)

$\rho_{-T}(A):=\rho(A^{-T}).$

We also define the following ideal ${\rm vn}_{2}(R),$ the so-called
booleanizing ideal of $R$:

######  Definition 4.4.

Let $R$ be a commutative ring with $1.$ Then ${\rm vn}_{2}(R)$ is defined as
the ideal

$(x^{2}-x\mid x\in R).$

Using the map $\rho$ and the ideal ${\rm vn}_{2}(R)$, one can define the
following subgroups:

######  Definition 4.5.

Let $R$ be a commutative ring with $1$ and $P\subset R$ a radix. Then the
subgroup $G(P)$ of ${\rm SL}_{2}(R)$ is defined as:

$\displaystyle G(P):=\\{A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in{\rm
SL}_{2}(R)\mid$ $\displaystyle\rho(A),\rho_{T}(A),\rho_{-1}(A),\rho_{-T}(A)\in
P\text{ and}$ $\displaystyle(a^{2}-1)\cdot{\rm vn}_{2}(R)+(d^{2}-1)\cdot{\rm
vn}_{2}(R)\subset P\\}.$

Further, for $J\unlhd R$ a non-trivial ideal and a subgroup $U$ of
$(R/J)^{*}$, slightly abusing notation, the subgroup $G(J,U)$ is defined as

$G(J,U):=\\{A\in{\rm SL}_{2}(R)\mid A\equiv uI_{2}\text{ mod }J\text{ for some
}u\in U\\}$

Last, define $G(J,U,P):=G(P)\cap G(J,U).$

Next, we define another subgroup associated to a normal subgroup $N$ of ${\rm
SL}_{2}(R):$

######  Definition 4.6.

Let $R$ be a commutative ring with $1$ and let $N$ be a normal subgroup of
${\rm SL}_{2}(R).$ Then define $\rho(N)$ as the subgroup of $(R,+)$ generated
by the set $\\{\rho(A)\mid A\in N\\}$ and $U(N)$ as the subgroup of
$(R/l(N))^{*}$ generated by all $u\in(R/l(N))^{*}$ such that (slightly abusing
notation) there is an $A\in N$ with

$A\equiv uI_{2}\text{ mod }l(N).$

Then $G(N)$ is defined as $G(l(N),U(N),\rho(N)).$

######  Remark 4.7.

Implicit in this definition is the fact that $\rho(N)$ is a radix [4, Theorem
2.2.2].

Finally, we can state the following version of [4, Theorem 3.1.3]:

######  Theorem 4.8.

Let $R$ be a ring with many units and $N$ a normal subgroup of ${\rm
SL}_{2}(R)$ with $l(N)\neq(0).$ Then the following chain of inclusions holds.

$[E(2,A),G(N)]\subset N\subset G(N).$

###### Proof.

We will only sketch the proof, because it is largely identical to the one of
[4, Theorem 3.1.3] with the main difference being technical details. Costa and
Keller prove their theorem by showing that $N$ and $R$ satisfy the assumptions
of a different, technical theorem [4, Theorem 3.1.2] which states that a chain
of inclusions as required exists for any commutative ring $R$ and any normal
subgroup $N$ of ${\rm SL}_{2}(R)$ under the assumption that there is an ideal
$J_{0}\unlhd R$ not equal to $R$ satisfying two properties:

  1. (1)

$[E(2,R),{\rm SL}_{2}(R,J_{0})]\subset C(J_{0})\subset N$ and

  2. (2)

$R/J_{0}$ has stable range $1.$

However, note that as $l(N)\neq 0,$ there is an element

$A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$

of $N$ such that $b\neq 0.$ But $R$ has many units by assumption and hence
there is a unit $u\in R$ such that $u^{4}\neq 1$ and $u^{2}\equiv 1\text{ mod
}bR.$ But then [8, Lemma 6.5] implies that $E(2,R,(u^{4}-1)R)$ is a subgroup
of $N.$ Setting $J_{0}:=(u^{4}-1)R,$ it is clear that $J_{0}$ is a non-zero
ideal in $R$ and we may assume wlog that $J_{0}\neq R.$ However, the ring $R$
has many units and so the ring $R/J_{0}$ has stable range $1$.This settles the
second condition. Next, note that $C(J_{0})\subset E(2,R,J_{0})$ holds. But
note that $C(J_{0})$ itself also has the property that $l(C(J_{0}))\neq(0),$
because $C(u^{4}-1)$ is not a scalar element. Thus using the property of $R$
to have many units together with [8, Lemma 6.5] again, we can find another
non-trivial ideal $J$ such that $E(2,R,J)\subset C(J_{0}).$ But then [2,
p.240] immediately implies that

${\rm SL}_{2}(R,J_{0})=E(2,R,J_{0})\cdot{\rm SL}_{2}(R,J).$

But this then implies

$[E(2,R),{\rm SL}_{2}(R,J_{0})]=[E(2,R),E(2,R,J_{0})\cdot{\rm
SL}_{2}(R,J)]=[E(2,R),E(2,R,J_{0})]\cdot[E(2,R),{\rm SL}_{2}(R,J)].$

However, note that $[E(2,A),E(2,R,J_{0})]=C(J_{0})$ holds according to [4,
Theorem 1.1.1]. Hence it suffices to show that $[E(2,R),{\rm
SL}_{2}(R,J)]\subset C(J_{0})$ holds to finish the proof of the first
condition above and so the proof of the theorem. But already

$[E(2,R),{\rm SL}_{2}(R,J)]\subset E(2,R,J)$

holds according to [8, Corollary 5.23] and this finishes the proof together
with the aforementioned inclusion $E(2,R,J)\subset C(J_{0}).$ ∎

This implies the following:

######  Corollary 4.9.

Let $R$ be a ring with many units and $N$ a normal subgroup of ${\rm
SL}_{2}(R)$. Then for each $x\in l(N)$ and $b\in R$, the element $C(x^{3}b)$
is an element of $N.$

###### Proof.

Let $x\in l(N)$ and $b\in R$ be given. Then we first show that
$E_{12}(x^{3}b)$ is an element of $G(N).$ First, it is clear that
$E_{12}(x^{3}b)$ is an element of $G(l(N),U(N)),$ because $x$ is an element of
$l(N).$ Second, we observe that

$\displaystyle\rho(E_{12}(x^{3}b))=1^{2}-1+1\cdot
x^{3}b=x^{3}b,\rho_{-1}(E_{12}(x^{3}b))=-x^{3}b$
$\displaystyle\rho_{T}(E_{12}(x^{3}b))=\rho(E_{21}(x^{3}b))=1^{2}-1+1\cdot
0=0\text{ and }\rho_{-T}(E_{12}(x^{3}b))=0.$

and that $(1^{2}-1)\cdot{\rm vn}_{2}(R)+(1^{2}-1)\cdot{\rm vn}_{2}(R)=(0).$
Thus $E_{12}(x^{3}b)$ being an element of $G(\rho(N))$ is equivalent to
$x^{3}b$ being an element of $\rho(N).$ But this follows from [4, Theorem
2.2.2]. Furthermore, [4, Theorem 1.1.1] implies that $C(x^{3}b)$ is an element
of $[E(2,R),\langle\langle E_{12}(x^{3}b)\rangle\rangle]$ and finishes the
proof. ∎

Using Corollary 4.9, we can finally prove Theorem 3.3:

###### Proof.

Let a language ${\mathcal{L}}$ with the relation symbols, constants and
function symbols

$({\mathcal{R}},0,1,+,\times,(a_{i,j})_{1\leq i,j\leq 2},t,u,v,w,s)$

and a further function symbol $\cdot^{-1}:{\mathcal{R}}^{2\times
2}\to{\mathcal{R}}^{2\times 2}$ be given. Note that we use capital letters to
denote matrices of variables (or constants) in the language in the following.
For example the symbol ${\mathcal{A}}$ denotes the $2\times 2$-matrix of
constants $(a_{i,j})$ and $X$ commonly refers to matrices of $2\times 2$
variables in ${\mathcal{L}}$. We also use the notation $X^{-1}:=^{-1}(X)$.
However this is only a way to simplify notation, because first order sentences
about matrices can always be reduced to first order sentences about their
entries. Furthermore, we use the function symbol $P:{\mathcal{R}}^{2\times
2}\to{\mathcal{R}}$ defined in the language ${\mathcal{L}}$ by

$P(X)=x_{11}\times x_{44}-x_{12}\times x_{21}-1.$

Next, we define the following first order theory ${\mathcal{T}}$:

  1. (1)

Sentences forcing the universe $R:={\mathcal{R}}^{{\mathcal{M}}}$ of each
model ${\mathcal{M}}$ of ${\mathcal{T}}_{kl}$ to be a commutative ring with
respect to the functions $+^{{\mathcal{M}}},\times^{{\mathcal{M}}}$ and with
$0^{{\mathcal{M}}},1^{{\mathcal{M}}}$ being $0$ and $1$.

  2. (2)

The sentence $\forall a,b,c:(\exists x,y,z:a\times x+b\times y+c\times
z=1)\rightarrow(\exists x,y,z:(a+b\times x)\times y+c\times z=1).$

  3. (3)

The equation of constants $t=u\times(a_{11}-a_{44})+v\times a_{12}+w\times
a_{21}.$

  4. (4)

The sentence $\forall x\neq 0:\exists c\exists y\exists z:(c^{4}\neq
0)\wedge(c^{2}-1=y\times x)\wedge(c\times z=1).$

  5. (5)

The sentence $P({\mathcal{A}})=0$.

  6. (6)

The sentence $\forall X:(P(X)=0)\rightarrow(XX^{-1}=I_{2}),$ where $I_{2}$
denotes the unit matrix in ${\mathcal{R}}^{2\times 2}$ with entries the
constant symbols $0,1$ as appropriate.

  7. (7)

A family of sentences $(\theta_{r})_{r\in\mathbb{N}}$ as follows:

$\displaystyle\theta_{r}:$ $\displaystyle\bigwedge_{1\leq k\leq r}\forall
X_{1},\dots,X_{r},\forall e_{1},\dots,e_{r}\in\\{0,1,-1\\}:$
$\displaystyle((P(X_{1})=\cdots=P(X_{r})=0)\rightarrow(C(t^{3}\times
s))\neq({\mathcal{A}}^{e_{1}})^{X_{1}}\cdots({\mathcal{A}}^{e_{r}})^{X_{r}})$

Here ${\mathcal{A}}^{1}:={\mathcal{A}},{\mathcal{A}}^{-1}:={\mathcal{A}}^{-1}$
and ${\mathcal{A}}^{0}:=I_{2}.$

Next, we will show that this theory ${\mathcal{T}}$ is inconsistent. In order
to see this let ${\mathcal{M}}$ be a model of the sentences in (1) through
(6). We denote the ring ${\mathcal{R}}^{{\mathcal{M}}}$ by $R$ and the element
${\mathcal{A}}^{{\mathcal{M}}}$ by $A.$ Note that the sentences (2) and (4)
imply that $R$ is a ring with many units. Note that sentence (5) forces $A$ to
be an element of ${\rm SL}_{2}(R)$ and that sentence (3) forces
$t^{{\mathcal{M}}}$ to be an element of $l(A).$ Abusing notation, we will
denote $t^{{\mathcal{M}}}$ by $t.$ Next, consider the normal subgroup $N$ of
${\rm SL}_{2}(R)$ normally generated by $A.$ But then Corollary 4.9 implies
that $C(t^{3}\times s^{{\mathcal{M}}})$ is an element of $N$ and hence there
are $X_{1},\dots,X_{r}\in{\rm SL}_{2}(R)$ and
$e_{1}\dots,e_{r}\in\\{0,1,-1\\}$ such that

$C(t^{3}\times
s^{{\mathcal{M}}})=(A^{e_{1}})^{X_{1}}\cdots(A^{e_{r}})^{X_{r}}.$

But this contradicts the statement $\theta_{r}^{{\mathcal{M}}}.$ Hence the
theory ${\mathcal{T}}$ is inconsistent and thus according to Gödel’s
Compactness Theorem [10, Satz 3.2] already a finite sub-theory
${\mathcal{T}}_{0}$ is inconsistent. Next, let $L_{1}$ be the largest natural
number $r\in\mathbb{N}$ such that $\theta_{r}\in{\mathcal{T}}_{0}.$ But note
that $\\{(1)-(6),\theta_{r+1}\\}\vdash\theta_{r}$ holds for all natural
numbers $r\in\mathbb{N}$ and so already the theory ${\mathcal{T}}_{1}$
consisting of all the sentences in (1) through (6) and the single sentence
$\theta_{L_{1}}$ is inconsistent.

However, if $R$ is any ring with many units, $A$ an element of ${\rm
SL}_{2}(R)$, $t$ an element of $l(A)$ and $b$ an arbitrary element of $R$,
then this setup gives a model ${\mathcal{M}}$ for all the sentences in (1)
through (6) and hence this setup must violate the statement
$\theta_{L_{1}}^{{\mathcal{M}}}.$ Thus we obtain $C(x^{3}b)$ is an element of
$B_{A}(L_{1}).$ This finishes the proof. ∎

We finish this section with the proof of Theorem 3.6. Again, this proof is
done with a model-theoretic compactness argument similar to the proof of
Theorem 3.3. For this reason, we will only give a brief sketch.

###### Proof.

Again, one defines a language ${\mathcal{L}}$ with the relation symbols,
constants and function symbols $({\mathcal{R}},0,1,+,\times,t,u)$ and then
defines a theory ${\mathcal{T}}$, which enforces that for each model
${\mathcal{M}},$ the universe ${\mathcal{R}}^{{\mathcal{M}}}$ is a commutative
ring. Additionally the theory ${\mathcal{T}}$ should contain sentences that
ensure that for each $k\in\mathbb{N},$ the element $E_{12}(t(u^{4}-1))$ can
not be written as a product of $k$ conjugates of $C(1)$ or $C(1)^{-1}$ in
${\rm SL}_{2}(R).$ However, one then easily obtains that the theory
${\mathcal{T}}$ is inconsistent, because $E(R,2,(u^{4}-1)R)$ is contained in
the subgroup of ${\rm SL}_{2}(R)$ normally generated by $C(1)$ according to
[8, Lemma 6.5]. Then using compactness this implies in particular that there
is an $L_{2}\in\mathbb{N}$ such that $E_{12}(t(u^{4}-1))$ can be written as a
product of at most $L_{2}$ conjugates of $C(1)$ or $C(1)^{-1}$ for any $t\in
R.$ ∎

##  5\. Normally generating subsets of ${\rm SL}_{2}(R)$ and the proof of
Theorem 1.3

To prove Theorem 1.3, we will use Corollary 3.10. Note that it requires the
study of the abelianization of ${\rm SL}_{2}(R).$ Hence in the first
subsection, we will determine the abelianization of ${\rm SL}_{2}(R)$ and use
it to prove the second part of Theorem1.3. Then in the second subsection, we
will use the description of the abelianization to prove the first part of
Theorem 1.3.

###  5.1. The abelianization of ${\rm SL}_{2}(R)$

Note that the commutator subgroup of ${\rm SL}_{2}(R)$ is a non-central
subgroup of ${\rm SL}_{2}(R).$ Assuming wlog. that ${\rm SL}_{2}(R)$ is not
perfect, Serre’s solution of the congruence subgroup problem for ${\rm
SL}_{2}(R)$ [11], implies the existence of a maximal, non-trivial finite index
ideal $I$ of $R$ such that the abelianization map of ${\rm SL}_{2}(R)$ factors
through ${\rm SL}_{2}(R/I).$ Let $I$ factor into prime ideals as
$I=\prod_{j=1}^{n}{\mathcal{X}}_{j}^{y_{j}}$ for prime ideals
${\mathcal{X}}_{j}$ and positive integers $y_{j}.$ Then using the Chinese
Remainder Theorem ${\rm SL}_{2}(R/I)$ factors as $\prod_{j=1}^{n}{\rm
SL}_{2}(R/{\mathcal{X}}_{j}^{y_{j}}).$ Hence to determine the abelianization
of ${\rm SL}_{2}(R)$ it suffices to determine the abelianization of the
various ${\rm SL}_{2}(R/{\mathcal{X}}_{j}^{y_{j}}).$ Most of the prime ideals
however give rise to perfect ${\rm SL}_{2}$ according to the following lemma:

######  Lemma 5.1.

Let $R$ be a ring of integers and ${\mathcal{X}}$ a non-zero prime ideal of
$R$ with $|R/{\mathcal{X}}|\geq 4$ and $L$ a natural number. Than ${\rm
SL}_{2}(R/{\mathcal{P}}^{L})$ is a perfect group.

###### Proof.

First, note that there has to be a unit $\bar{u}\in
R/{\mathcal{X}}^{L}=:\bar{R}$ such that $\bar{u}^{2}-1$ is also a unit. This
is the case, because an element of $R/{\mathcal{X}}^{L}$, which is not a unit
must be the image of an element of ${\mathcal{X}}\subset R.$ Hence if there is
no unit $\bar{u}\in\bar{R}$ as required, then $y(y^{2}-1)=y(y-1)(y+1)$ would
have to be an element of ${\mathcal{X}}$ for each element $y$ of $R.$ Phrased
differently, the field $R/{\mathcal{X}}$ would have at most three elements,
which contradicts $|R/{\mathcal{X}}|\geq 4.$ Then for any $\bar{x}\in\bar{R},$
consider the element $(h(\bar{u}),E_{12}(\bar{x}\cdot(\bar{u}^{2}-1)^{-1}))$
of the commutator subgroup of ${\rm SL}_{2}(\bar{R}).$ We obtain

$\displaystyle(h(\bar{u}),E_{12}(\bar{x}\cdot(\bar{u}^{2}-1)^{-1})))$
$\displaystyle=h(\bar{u})\cdot E_{12}(\bar{x}\cdot(\bar{u}^{2}-1)^{-1}))\cdot
h(\bar{u})^{-1}\cdot E_{12}(-\bar{x}\cdot(\bar{u}^{2}-1)^{-1})$
$\displaystyle=E_{12}(\bar{u}^{2}\bar{x}\cdot(\bar{u}^{2}-1)^{-1})\cdot
E_{12}(-\bar{x}\cdot(\bar{u}^{2}-1)^{-1}))=E_{12}(\bar{x}).$

So each elementary matrix of ${\rm SL}_{2}(\bar{R})$ is contained in the
commutator subgroup of ${\rm SL}_{2}(\bar{R}).$ But by [2, p.240], the set of
elementary matrices generates ${\rm SL}_{2}(\bar{R})$ and we are done. ∎

So we can assume that the aforementioned ideal $I$ only has prime divisors
${\mathcal{X}}$ such that $R/{\mathcal{X}}$ has at most three elements. Next,
set $r_{1}:=r_{1}(R),r_{2}:=r_{2}(R)$ and $q:=q(R)$ and

  1. (1)

let ${\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r_{2}}$ be the ramified prime
divisors of $2R$ with $R/{\mathcal{P}}_{j}=\mathbb{F}_{2}$,

  2. (2)

let ${\mathcal{Q}}_{1},\dots,{\mathcal{Q}}_{r_{1}}$ be the unramified prime
divisors of $2R$ with $R/{\mathcal{Q}}_{i}=\mathbb{F}_{2}$ and

  3. (3)

let ${\mathcal{K}}_{1},\dots,{\mathcal{K}}_{q}$ be the prime divisors of $3R$
with $R/{\mathcal{K}}_{l}=\mathbb{F}_{3}$.

By the previous discussion, there are natural numbers
$p_{1},\dots,p_{r_{2}},q_{1},\dots,q_{r_{2}}$ and $k_{1},\dots,k_{q}$ such
that
$I=(\prod_{j=1}^{r_{2}}{\mathcal{P}}_{j}^{p_{j}})\cdot(\prod_{i=1}^{r_{1}}{\mathcal{Q}}_{i}^{q_{i}})\cdot(\prod_{l=1}^{q}{\mathcal{K}}_{l}^{k_{l}}).$
First, we are going to determine the abelianization of ${\rm
SL}_{2}(R/{\mathcal{Q}}_{i}^{q_{i}}).$ To find this abelianization, we note:

######  Lemma 5.2.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers of a number
field $K$, ${\mathcal{Q}}$ be an unramified prime divisor of $2R$ with
$R/{\mathcal{Q}}=\mathbb{F}_{2}$ and $L\in\mathbb{N}.$ Then
$\mathbb{Z}_{2^{L}}\to R/{\mathcal{Q}}^{L},n+2^{L}\mathbb{Z}\mapsto
n+{\mathcal{Q}}^{L}$ defines an isomorphism.

###### Proof.

First, note that $2\in{\mathcal{Q}}$ and hence $2^{L}\mathbb{Z}$ is a subset
of ${\mathcal{Q}}^{L}.$ This implies that the inclusion $\mathbb{Z}\to R$
induces the ring homomorphism $q:\mathbb{Z}_{2^{L}}\to
R/{\mathcal{Q}}^{L},n+2^{L}\mathbb{Z}\mapsto n+{\mathcal{Q}}^{L}.$ However,
note that both $\mathbb{Z}_{2^{L}}$ and $R/{\mathcal{Q}}^{L}$ have
$2^{L}=|R/{\mathcal{Q}}|^{L}$ many elements. Thus to prove that $q$ is an
isomorphism it suffices to prove that the order of $1=1+{\mathcal{Q}}^{L}$ in
the group $(R/{\mathcal{Q}}^{L},+)$ is $2^{L}$. To this end, let $j$ be the
order of $1$ as a group element in $(R/{\mathcal{Q}}^{L},+).$ Note that $j$
must be a power of $2,$ say $j=2^{i}.$ This implies
$2^{i}+{\mathcal{Q}}^{L}=0$ and so $2^{i}$ is an element of
${\mathcal{Q}}^{L}.$ However, ${\mathcal{Q}}$ is an unramified prime divisor
of $2R$ and hence ${\mathcal{Q}}$ can divide $2^{i}$ at most $i$ times. Thus
$j=2^{L}$ must hold and this finishes the proof. ∎

But using this lemma one can identify the abelianization of ${\rm
SL}_{2}(R/{\mathcal{Q}}_{i}^{q_{i}}):$

######  Proposition 5.3.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers of a number
field $K$, ${\mathcal{Q}}$ be an unramified prime divisor of $2R$ with
$R/{\mathcal{Q}}=\mathbb{F}_{2}$ and $L\geq 2.$ Then the abelianization map of
${\rm SL}_{2}(R/{\mathcal{Q}}^{L})$ is the unique epimorphism ${\rm
SL}_{2}(R/{\mathcal{Q}}^{L})\to R/{\mathcal{Q}}^{2}$ mapping
$E_{12}(x+{\mathcal{Q}}^{L})$ to $x+{\mathcal{Q}}^{2}$ for all $x\in R.$

###### Proof.

According to Lemma 5.2, the additive group of the ring $R/{\mathcal{Q}}^{L}$
is generated by the unit $1+{\mathcal{Q}}^{L}.$ However, an observation by Jun
Morita [7, Fact 2] states that the abelianization of ${\rm SL}_{2}(S)$ is a
quotient of $\mathbb{Z}_{12}\oplus\mathbb{Z}_{6}$ for any ring $S$ whose
additive group is generated by two units of $S.$ Thus the abelianization of
${\rm SL}_{2}(R/{\mathcal{Q}}^{L})$ is a quotient of
$\mathbb{Z}_{12}\oplus\mathbb{Z}_{6}.$ But on the other hand, the group ${\rm
SL}_{2}(R/{\mathcal{Q}}^{L})$ is generated by the set
$\\{E_{12}(\bar{x}),E_{21}(\bar{x})\mid\bar{x}\in R/{\mathcal{Q}}^{L}\\}$
according to [2, p.240] and

$\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\cdot
E_{12}(\bar{x})\cdot\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}=E_{21}(\bar{x})$

holds for all $\bar{x}\in R/{\mathcal{Q}}^{L}.$ Hence the abelianization of
${\rm SL}_{2}(R/{\mathcal{Q}}^{L})$ is a quotient of $(R/{\mathcal{Q}}^{L},+)$
and so is in particular a cyclic group with at most four elements according to
Lemma 5.2. But together with the abelianization being a quotient of
$\mathbb{Z}_{12}\oplus\mathbb{Z}_{6}$, this implies that the abelianization is
the cyclic group with either $1,2$ or $4$ elements. But consider the reduction
homomorphism ${\rm SL}_{2}(R/{\mathcal{Q}}^{L})\to{\rm
SL}_{2}(R/{\mathcal{Q}}^{2})={\rm SL}_{2}(\mathbb{Z}_{4})$. We leave it as an
exercise to the reader to check that there is an epimorphism ${\rm
SL}_{2}(\mathbb{Z}_{4})\to\mathbb{Z}_{4}$ mapping $E_{12}(n+4\mathbb{Z})$ to
$n+4\mathbb{Z}$. Using the isomorphism $\mathbb{Z}_{4}\to
R/{\mathcal{Q}}^{2},n+4\mathbb{Z}\mapsto n+{\mathcal{Q}}^{2}$ from Lemma 5.2,
this finishes the proof. ∎

Next, we are going to determine the abelianization of ${\rm
SL}_{2}(R/{\mathcal{P}}_{j}^{p_{j}}).$ We note:

######  Lemma 5.4.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers of a number
field $K$, let ${\mathcal{P}}$ be a ramified prime divisor of $2R$ with
$R/{\mathcal{P}}=\mathbb{F}_{2}$ and let $L\geq 2$ be a natural number. Then
the abelianization homomorphism of ${\rm SL}_{2}(R/{\mathcal{P}}^{L})$ factors
through ${\rm SL}_{2}(R/{\mathcal{P}}^{2})$. In particular, the abelianization
of ${\rm SL}_{2}(R/{\mathcal{P}}^{L})$ is a quotient of $R/{\mathcal{P}}^{2}$
and so has at most $|R/{\mathcal{P}}^{2}|=4$ elements.

###### Proof.

Choose an element $x$ of ${\mathcal{P}}-{\mathcal{P}}^{2}$ and let $\bar{x}$
be its image in $\bar{R}:=R/{\mathcal{P}}^{L}.$ Then $1+\bar{x}$ is a unit in
$\bar{R}.$ Further, let $\bar{a}\in\bar{R}$ be arbitrary. Then

$[h(1+\bar{x}),E_{12}(\bar{a})]=E_{12}((1+\bar{x})^{2}\bar{a})\cdot
E_{12}(-\bar{a})=E_{12}((2\bar{x}+\bar{x}^{2})\cdot\bar{a})$

is an element of the commutator subgroup $[{\rm SL}_{2}(\bar{R}),{\rm
SL}_{2}(\bar{R})].$ But note that $2x+x^{2}$ is an element of
${\mathcal{P}}^{2},$ but not of ${\mathcal{P}}^{3}.$ This is the case, because
${\mathcal{P}}$ is a ramified prime divisor of $2R$ and so
$2x\in{\mathcal{P}}^{3}$ holds. Thus $2x+x^{2}\in{\mathcal{P}}^{3}$ would
imply $x^{2}\in{\mathcal{P}}^{3}$ and so $x\in{\mathcal{P}}^{2},$ a
contradiction. But this implies that $2\bar{x}+\bar{x}^{2}$ generates the
principal ideal $\bar{{\mathcal{P}}}^{2}$ for $\bar{{\mathcal{P}}}$ the image
of ${\mathcal{P}}$ in $\bar{R}.$ So in particular, the commutator subgroup of
${\rm SL}_{2}(\bar{R})$ contains the normal subgroup $N:=\langle\langle
E_{12}(\bar{a})\mid\bar{a}\in\bar{{\mathcal{P}}}^{2}\rangle\rangle.$ But
according to [2, p.240], this normal subgroup is the kernel of the epimorphism
$\pi_{\bar{{\mathcal{P}}^{2}}}:{\rm SL}_{2}(\bar{R})\to{\rm
SL}_{2}(\bar{R}/\bar{{\mathcal{P}}}^{2})={\rm SL}_{2}(R/{\mathcal{P}}^{2}).$
Thus the abelianization homomorphism of ${\rm SL}_{2}(\bar{R})$ indeed factors
through ${\rm SL}_{2}(R/{\mathcal{P}}^{2})$ and we may assume $L=2.$ But as in
the proof of Proposition 5.3, this implies that the abelianization of ${\rm
SL}_{2}(\bar{R})$ is a quotient of $(\bar{R},+)=(R/{\mathcal{P}}^{2},+)$. But
$|R/{\mathcal{P}}^{2}|=|R/{\mathcal{P}}|^{2}=4.$ ∎

To determine the abelianization of ${\rm SL}_{2}(R/{\mathcal{P}}^{2})$, we
however need a technical proposition. To state it, let $R={\mathcal{O}}_{S}$
be the ring of S-algebraic integers of a number field $K$ and ${\mathcal{P}}$
a ramified prime divisor of $2R$ with $R/{\mathcal{P}}=\mathbb{F}_{2}.$ Then
set $\bar{R}:=R/{\mathcal{P}}^{2}$ and $\bar{{\mathcal{P}}}$ the image of
${\mathcal{P}}$ in $\bar{R}.$ Then consider the group ${\rm SL}_{2}(\bar{R})$
and note that

$K:=\left\\{\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in{\rm SL}_{2}(\bar{R})\mid
a-1,d-1,b,c\in\bar{{\mathcal{P}}}\right\\}$

is the kernel of the reduction homomorphism

$\pi_{\bar{{\mathcal{P}}}}:{\rm SL}_{2}(\bar{R})\to{\rm
SL}_{2}(\bar{R}/\bar{{\mathcal{P}}})={\rm SL}_{2}(R/{\mathcal{P}})={\rm
SL}_{2}(\mathbb{F}_{2}).$

This yields a short exact sequence

$1\longrightarrow K\longrightarrow{\rm
SL}_{2}(\bar{R})\xrightarrow{\pi_{\bar{{\mathcal{P}}}}}{\rm
SL}_{2}(\mathbb{F}_{2})\longrightarrow 1.$

However, this short exact sequence gives rise to a splitting of ${\rm
SL}_{2}(\bar{R})$ as a semi-direct product $K\rtimes{\rm
SL}_{2}(\mathbb{F}_{2}):$ Namely, as ${\mathcal{P}}$ is a ramified prime
divisor of $2R$, it follows that $2$ is an element of ${\mathcal{P}}^{2}.$ So
in particular, we obtain that $2=0$ holds in $\bar{R}.$ But this then implies
that $\mathbb{F}_{2}$ is a subring of $\bar{R}$ and so ${\rm
SL}_{2}(\mathbb{F}_{2})$ is a subgroup of ${\rm SL}_{2}(\bar{R}).$ Then the
inclusion $j:{\rm SL}_{2}(\mathbb{F}_{2})\to{\rm SL}_{2}(\bar{R})$ has the
property $\pi_{\bar{{\mathcal{P}}}}\circ j={\rm id}_{{\rm
SL}_{2}(\mathbb{F}_{2})}$ and this gives us the desired splitting. Next, we
state the required proposition:

######  Proposition 5.5.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers of a number
field $K$ and ${\mathcal{P}}$ a ramified prime divisor of $2R$ with
$R/{\mathcal{P}}=\mathbb{F}_{2}.$ Then for the splitting ${\rm
SL}_{2}(\bar{R})=K\rtimes{\rm SL}_{2}(\mathbb{F}_{2})$ described above, the
map

$K\rtimes{\rm SL}_{2}(\mathbb{F}_{2})\ni\left[\begin{pmatrix}a&b\\\
c&d\end{pmatrix},A\right]\mapsto a-1+b+c$

defines a group homomorphism $q:{\rm SL}_{2}(\bar{R})\to\bar{{\mathcal{P}}}.$

###### Proof.

We split the proof into two parts: We first show that the three maps

$q_{1}:K\to\bar{{\mathcal{P}}},\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\mapsto
b,q_{2}:K\to\bar{{\mathcal{P}}},\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\mapsto
c\text{ and }q_{3}:K\to\bar{{\mathcal{P}}},\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto a-1$

all individually define group homomorphisms. This yields that

$q^{\prime}:K\to\bar{{\mathcal{P}}},\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto a-1+b+c$

is a group homomorphism. Then secondly, we will show that the map $q^{\prime}$
is invariant under conjugation by ${\rm SL}_{2}(\mathbb{F}_{2})$ and this will
yield that $q^{\prime}$ extends to the required group homomorphism $q:{\rm
SL}_{2}(\bar{R})\to\bar{{\mathcal{P}}}.$ For the first step, let

$A_{1}=\begin{pmatrix}a_{1}&b_{1}\\\
c_{1}&d_{1}\end{pmatrix},A_{2}=\begin{pmatrix}a_{2}&b_{2}\\\
c_{2}&d_{2}\end{pmatrix}$

be elements of $K$ and note that

$A_{1}\cdot A_{2}=\begin{pmatrix}a_{1}\cdot a_{2}+b_{1}\cdot c_{2}&a_{1}\cdot
b_{2}+b_{1}\cdot d_{2}\\\ c_{1}\cdot a_{2}+d_{1}\cdot c_{2}&c_{1}\cdot
b_{2}+d_{1}\cdot d_{2}\end{pmatrix}.$

But note that this implies $q_{1}(A_{1}\cdot A_{2})=a_{1}\cdot
b_{2}+b_{1}\cdot d_{2}.$ However, $a_{1}-1$ is an element of
$\bar{{\mathcal{P}}}$ and so is $b_{2}.$ Thus $(a_{1}-1)\cdot b_{2}$ is an
element of $\bar{{\mathcal{P}}}^{2}=0$ in $\bar{R}.$ But this yields

$a_{1}\cdot b_{2}=(a_{1}-1+1)\cdot b_{2}=(a_{1}-1)\cdot b_{2}+b_{2}=b_{2}.$

Similarly, we obtain $b_{1}\cdot d_{2}=b_{1}$ and hence

$q_{1}(A_{1}\cdot A_{2})=a_{1}\cdot b_{2}+b_{1}\cdot
d_{2}=b_{2}+b_{1}=q_{1}(A_{2})+q_{1}(A_{1}).$

holds. This shows that $q_{1}$ is indeed a homomorphism. The claim for $q_{2}$
follows in essentially the same manner. To show the claim for $q_{3}$ note
that $q_{3}(A_{1}\cdot A_{2})=a_{1}\cdot a_{2}+b_{1}\cdot c_{2}-1.$ But again
$b_{1}$ and $c_{2}$ are elements of $\bar{{\mathcal{P}}}$ and hence
$b_{1}\cdot c_{2}=0.$ Thus $q_{3}(A_{1}\cdot A_{2})=a_{1}\cdot a_{2}-1.$ Next,

$\displaystyle q_{3}(A_{1}\cdot A_{2})$ $\displaystyle=a_{1}\cdot
a_{2}-1=a_{1}\cdot(a_{2}-1)+a_{1}-1=(a_{1}-1)\cdot(a_{2}-1)+(a_{2}-1)+(a_{1}-1)$
$\displaystyle=(a_{1}-1)\cdot(a_{2}-1)+q_{3}(A_{2})+q_{3}(A_{1}).$

However, as $a_{1}-1$ and $a_{2}-1$ are both elements of
$\bar{{\mathcal{P}}},$ we obtain $(a_{1}-1)\cdot(a_{2}-1)=0$ and so the claim
for $q_{3}$ follows. Thus the map $q^{\prime}:K\to\bar{{\mathcal{P}}}$ indeed
defines a homomorphism. For the second claim, that is, that $q^{\prime}$ is
invariant under conjugation by ${\rm SL}_{2}(\mathbb{F}_{2}),$ we leave it as
an exercise to the reader to check that ${\rm SL}_{2}(\mathbb{F}_{2})$ is
generated by the two matrices

$X:=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\text{ and }Y:=\begin{pmatrix}1&1\\\
1&0\end{pmatrix}.$

Thus it suffices to check that $q^{\prime}$ is invariant under conjugation by
$X$ and $Y.$ To this end, note for $A_{1}\in K$ as above, that

$XA_{1}X^{-1}=\begin{pmatrix}d_{1}&c_{1}\\\ b_{1}&a_{1}\end{pmatrix}.$

This implies $q^{\prime}(XA_{1}X^{-1})=c_{1}+b_{1}+d_{1}-1.$ Hence to show
that $q^{\prime}(XA_{1}X^{-1})=q^{\prime}(A_{1}),$ it suffices to show that
$d_{1}-1=a_{1}-1.$ To this end, note that both $a_{1}-1$ and $d_{1}-1$ are
elements of $\bar{{\mathcal{P}}}$ but the ideal $\bar{{\mathcal{P}}}$ only has
a single non-trivial element $\bar{t}$. But assume wlog. that $a_{1}-1$ is $0$
and $d_{1}-1$ is $\bar{t}.$ Then note that as $A_{1}$ is an element of ${\rm
SL}_{2}(\bar{R})$, we obtain

$1=a_{1}\cdot d_{1}-b_{1}\cdot c_{1}=1\cdot(1+\bar{t})-b_{1}\cdot
c_{1}=1+\bar{t}-b_{1}\cdot c_{1}.$

However, $b_{1}\cdot c_{1}=0$ holds and thus $\bar{t}=0$ follows. But this is
impossible. Thus $a_{1}-1$ and $d_{1}-1$ must necessarily be the same element
of $\bar{{\mathcal{P}}}$ and hence $q^{\prime}$ is invariant under conjugation
by $X.$ For the invariance under $Y$, note that

$YA_{1}Y^{-1}=\begin{pmatrix}1&1\\\
1&0\end{pmatrix}\cdot\begin{pmatrix}a_{1}&b_{1}\\\
c_{1}&d_{1}\end{pmatrix}\cdot Y^{-1}=\begin{pmatrix}a_{1}+c_{1}&b_{1}+d_{1}\\\
a_{1}&b_{1}\end{pmatrix}\cdot\begin{pmatrix}0&1\\\
1&1\end{pmatrix}=\begin{pmatrix}b_{1}+d_{1}&a_{1}+b_{1}+c_{1}+d_{1}\\\
b_{1}&a_{1}+b_{1}\end{pmatrix}.$

But this implies

$\displaystyle q^{\prime}(YA_{1}Y^{-1})$
$\displaystyle=(a_{1}+b_{1}+c_{1}+d_{1})+b_{1}+(b_{1}+d_{1}-1)=a_{1}+3b_{1}+c_{1}+2d_{1}-1=b_{1}+c_{1}+a_{1}-1$
$\displaystyle=q^{\prime}(A_{1}).$

Here, we used the aforementioned fact that $2=0$ holds in $\bar{R}.$ This
finishes the proof. ∎

######  Remark 5.6.

One can actually show that $\bar{R}=\mathbb{F}_{2}[T]/(T^{2})$ and that ${\rm
SL}_{2}(\bar{R})=\mathbb{Z}_{2}^{3}\rtimes S_{3}$ with the permutation group
$S_{3}$ operating on $\mathbb{Z}_{2}^{3}$ by permutation of components. Then
$q:{\rm SL}_{2}(\bar{R})\to\mathbb{Z}_{2}$ has the form
$q(x,\sigma)=x_{1}+x_{2}+x_{3}$ for $x\in\mathbb{Z}_{2}^{3}$ and $\sigma\in
S_{3}.$

Note, that this implies:

######  Corollary 5.7.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers of a number
field $K$ and ${\mathcal{P}}$ a ramified prime divisor of $2R$ with
$R/{\mathcal{P}}=\mathbb{F}_{2}$ and let $L\geq 2$ be given. Then there is a
unique epimorphism ${\rm SL}_{2}(R/{\mathcal{P}}^{L})\to R/{\mathcal{P}}^{2}$
mapping $E_{12}(x+{\mathcal{P}}^{L})$ to $x+{\mathcal{P}}^{2}$ for all $x\in
R.$ This unique epimorphism is the abelianization map of ${\rm
SL}_{2}(R/{\mathcal{P}}^{L}).$

###### Proof.

Due to Lemma 5.4, we may assume $L=2.$ Next, note that according to the afore-
mentioned splitting ${\rm SL}_{2}(R/{\mathcal{P}}^{2})=K\rtimes{\rm
SL}_{2}(\mathbb{F}_{2}),$ there is an epimorphism ${\rm
SL}_{2}(R/{\mathcal{P}}^{2})\to{\rm SL}_{2}(\mathbb{F}_{2}).$ Then one can
easily check that

$\displaystyle{\rm SL}_{2}(\mathbb{F}_{2})\ni E_{12}(1)\mapsto
1\in\mathbb{F}_{2}\text{ and }E_{21}(1)\mapsto 1$

extends to an epimorphism ${\rm SL}_{2}(\mathbb{F}_{2})\to\mathbb{F}_{2}$ and
so there is an epimorphism $h:{\rm
SL}_{2}(R/{\mathcal{P}}^{2})\to\mathbb{F}_{2}$ with $h(E_{12}(1))=1.$ Second,
consider the homomorphism $q:{\rm
SL}_{2}(R/{\mathcal{P}}^{2})\to{\mathcal{P}}/{\mathcal{P}}^{2}=\mathbb{F}_{2}$
from Proposition 5.5 and consider the homomorphism $h\oplus q:{\rm
SL}_{2}(R/{\mathcal{P}}^{2})\to\mathbb{F}_{2}\oplus\mathbb{F}_{2}.$ Note that
$h\oplus q(E_{12}(1))=(1,0)$. For $\bar{t}$ the unique non-trivial element of
${\mathcal{P}}/{\mathcal{P}}^{2},$ we have further $h\oplus
q(E_{12}(\bar{t}))=(0,1).$ Thus $h\oplus q$ is an epimorphism onto
$\mathbb{F}_{2}\oplus\mathbb{F}_{2}$. But we know from Lemma 5.4 that the
abelianization of ${\rm SL}_{2}(R/{\mathcal{P}}^{L})$ has at most four
elements and is a quotient of $R/{\mathcal{P}}^{2}.$ This finishes the proof
using the isomorphism $\mathbb{F}_{2}\oplus\mathbb{F}_{2}=R/{\mathcal{P}}^{2}$
sending $(1,0)$ to $1+{\mathcal{P}}^{2}$ and $(0,1)$ to $\bar{t}.$ ∎

Last, we will determine the abelianization of ${\rm
SL}_{2}(R/{\mathcal{K}}_{l}^{k_{l}}).$ To this end, note

######  Lemma 5.8.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers of a number
field $K$, let ${\mathcal{K}}$ be a prime divisor of $3R$ with
$R/{\mathcal{K}}=\mathbb{F}_{3}$ and let $L$ be a natural number. Then the
abelianization homomorphism of ${\rm SL}_{2}(R/{\mathcal{K}}^{L})$ factors
through ${\rm SL}_{2}(R/{\mathcal{K}})={\rm SL}_{2}(\mathbb{F}_{3})$.

###### Proof.

Choose an element $x\in{\mathcal{K}}$ and let $\bar{x}$ be its image in
$\bar{R}:=R/{\mathcal{K}}^{L}.$ Then $1+\bar{x}$ is a unit in $\bar{R}.$
Further, let $\bar{a}\in\bar{R}$ be arbitrary. Then

$[h(1+\bar{x}),E_{12}(\bar{a})]=E_{12}((1+\bar{x})^{2}\bar{a})\cdot
E_{12}(-\bar{a})=E_{12}((2\bar{x}+\bar{x}^{2})\cdot\bar{a})=E_{12}(\bar{x}(2+\bar{x})\cdot\bar{a})$

is an element of the commutator subgroup $[{\rm SL}_{2}(\bar{R}),{\rm
SL}_{2}(\bar{R})].$ But note that $2\notin{\mathcal{K}}$ and so $2+\bar{x}$ is
a unit of $\bar{R}.$ So in particular, the commutator subgroup of ${\rm
SL}_{2}(\bar{R})$ contains the normal subgroup $N:=\langle\langle
E_{12}(\bar{a})\mid\bar{a}\in\bar{{\mathcal{K}}}\rangle\rangle$ for
$\bar{{\mathcal{K}}}$ the image of ${\mathcal{K}}$ in $\bar{R}.$ But according
to [2, p.240], this normal subgroup is the kernel of the epimorphism
$\pi_{\bar{{\mathcal{P}}^{2}}}:{\rm SL}_{2}(\bar{R})\to{\rm
SL}_{2}(\bar{R}/\bar{{\mathcal{K}}})={\rm SL}_{2}(R/{\mathcal{K}})={\rm
SL}_{2}(\mathbb{F}_{3}).$ Thus the abelianization homomorphism of ${\rm
SL}_{2}(\bar{R})$ indeed factors through ${\rm SL}_{2}(R/{\mathcal{K}})$. ∎

But we have the following lemma:

######  Lemma 5.9.

There is an epimorphism ${\rm SL}_{2}(\mathbb{F}_{3})\to\mathbb{F}_{3}$
mapping $E_{12}(1)$ to $1$.

We skip the proof, because it is similar to the proof of [15, Lemma 6.5]. But
similar to the proof of Proposition 5.3 or Corollary 5.7, one sees that the
abelianization of ${\rm SL}_{2}(\mathbb{F}_{3})$ is a cyclic group with at
most three elements and so the abelianization homomorphism of ${\rm
SL}_{2}(R/{\mathcal{K}}_{l}^{k_{l}})$ is the unique epimorphism ${\rm
SL}_{2}(R/{\mathcal{K}}_{l}^{k_{l}})\to R/{\mathcal{K}}_{l}$ mapping
$E_{12}(x+{\mathcal{K}}_{l}^{k_{l}})$ to $x+{\mathcal{K}}_{l}$ for all $x\in
R.$ To summarize the discussion:

######  Corollary 5.10.

Let $R={\mathcal{O}}_{S}$ be the ring of S-algebraic integers of a number
field $K$ such that $R$ has infinitely many units and

  1. (1)

let ${\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r_{2}(R)}$ be the ramified prime
divisors of $2R$ with $R/{\mathcal{P}}_{j}=\mathbb{F}_{2}$,

  2. (2)

let ${\mathcal{Q}}_{1},\dots,{\mathcal{Q}}_{r_{1}(R)}$ be the unramified prime
divisors of $2R$ with $R/{\mathcal{Q}}_{i}=\mathbb{F}_{2}$ and

  3. (3)

let ${\mathcal{K}}_{1},\dots,{\mathcal{K}}_{q(R)}$ be the prime divisors of
$3R$ with $R/{\mathcal{K}}_{l}=\mathbb{F}_{3}$.

Then the abelianization homomorphism of ${\rm SL}_{2}(R)$ is the unique
epimorphism

${\rm
SL}_{2}(R)\to(\prod_{j=1}^{r_{2}(R)}R/{\mathcal{P}}_{j}^{2})\times(\prod_{i=1}^{r_{1}(R)}R/{\mathcal{Q}}_{i}^{2})\times(\prod_{l=1}^{q(R)}R/{\mathcal{K}}_{l})=\mathbb{F}_{2}^{2r_{2}(R)}\times\mathbb{Z}_{4}^{r_{1}(R)}\times\mathbb{F}_{3}^{q(R)}$

which maps $E_{12}(x)$ to
$(\prod_{j=1}^{r_{2}(R)}(x+{\mathcal{P}}_{j}^{2}),\prod_{i=1}^{r_{1}(R)}(x+{\mathcal{Q}}_{i}^{2}),\prod_{l=1}^{q(R)}(x+{\mathcal{K}}_{l}))$
for all $x\in R.$

###### Proof.

This follows immediately from the determination of the abelianization maps of
${\rm SL}_{2}(R/{\mathcal{Q}}_{i}^{q_{i}})$, ${\rm
SL}_{2}(R/{\mathcal{P}}_{j}^{p_{j}})$ and ${\rm
SL}_{2}(R/{\mathcal{K}}_{l}^{k_{L}})$ done above. ∎

######  Remark 5.11.

Note in particular, that the abelianization of ${\rm
SL}_{2}(R/{\mathcal{P}}_{i}^{2})$ and ${\rm SL}_{2}(R/{\mathcal{Q}}_{j}^{2})$
both have $4$ elements, whereas the abelianization of $S_{3}={\rm
SL}_{2}(\mathbb{F}_{2})={\rm SL}_{2}(R/{\mathcal{P}}_{i})={\rm
SL}_{2}(R/{\mathcal{Q}}_{j})$ only has $2$ elements. Thus the multiplicities
$p_{i}$ and $q_{j}$ in the ideal $I$ mentioned before can not be chosen
smaller as $2.$

But we can prove the second part of Theorem 1.3 now:

###### Proof.

Assume for contradiction that there is a finite, normally generating subset
$T$ of ${\rm SL}_{2}(R)$ with $|T|\leq v(R)-1.$ But $T$ must map onto a
generating set of the abelianization
$\mathbb{F}_{2}^{2r_{2}(R)}\times\mathbb{Z}_{4}^{r_{1}(R)}\times\mathbb{F}_{3}^{q(R)}$
of ${\rm SL}_{2}(R).$ However, the rank of the abelian group
$\mathbb{F}_{2}^{2r_{2}(R)}\times\mathbb{Z}_{4}^{r_{1}(R)}\times\mathbb{F}_{3}^{q(R)}$
is $\max\\{2r_{2}(R)+r_{1}(R),q(R)\\}=v(R)$ as $2$ and $3$ are distinct
primes. This is a contradiction. ∎

###  5.2. The proof of the first part of Theorem 1.3

For the sake of a more uniform proof, we assume in the following that for
$r_{1}:=r_{1}(R),r_{2}:=r_{2}(R)$ and $q:=q(R)$, one has $2r_{2}+r_{1}=q=v(R)$
and $r_{1},q\geq 1$. Next, due to the Chinese Remainder Theorem there is an
epimorphism

$\displaystyle
R\to\left(\prod_{j=1}^{r_{2}}R/{\mathcal{P}}_{j}^{2}\right)\times\left(\prod_{i=1}^{r_{1}}R/{\mathcal{Q}}_{i}^{2}\right)\times\left(\prod_{l=1}^{q}R/{\mathcal{K}}_{l}\right),$
$\displaystyle
x\mapsto\left[\left(\prod_{j=1}^{r_{2}}x+{\mathcal{P}}_{j}^{2}\right),\left(\prod_{i=1}^{r_{1}}x+{\mathcal{Q}}_{i}^{2}\right),\left(\prod_{l=1}^{q}x+{\mathcal{K}}_{l}\right)\right]$

Next, choose elements $c_{1},\dots,c_{r_{2}}\in R$ which under the epimorphism
given by the Chinese Remainder Theorem map as follows: For each $1\leq u\leq
r_{2}$, we can choose $c_{u}^{\prime}\in R$ which maps to an element of
${\mathcal{P}}_{u}/{\mathcal{P}}_{u}^{2},$ which is not $0.$ Then we choose
the $c_{u}\in R$ as mapping to

$(c^{\prime}_{u}+{\mathcal{P}}_{u}^{2},0+\prod_{j=1,j\neq
u}^{r_{2}}{\mathcal{P}}_{j}^{2},0+\prod_{i=1}^{r_{1}}{\mathcal{Q}}_{i}^{2},1+{\mathcal{K}}_{u+r_{1}+r_{2}},0+\prod_{l=1,l\neq
u+r_{1}+r_{2}}^{q}{\mathcal{K}}_{l}).$

Also for $k\geq\max\\{2r_{2}+r_{1},q\\}=2r_{2}+r_{1}=q=v(R),$ choose elements
$y_{1},\dots,y_{k}$ of $R$ with distinct prime divisors
${\mathcal{Y}}_{1},\dots,{\mathcal{Y}}_{k}$, that is for $1\leq f\leq k$ the
only prime divisor of the ideal $y_{f}R$ is ${\mathcal{Y}}_{f}$ and such that
none of the ${\mathcal{Y}}_{f}$ agree with any of the prime divisors of the
elements $c_{1},\dots,c_{r_{2}},$ each other or the prime ideals
${\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r_{2}},{\mathcal{Q}}_{1},\dots,{\mathcal{Q}}_{r_{1}},{\mathcal{K}}_{1},\dots,{\mathcal{K}}_{q}.$
Next, let $C$ be the class number of $R$.

  1. (1)

For each $1\leq u\leq r_{1}$, choose a generator $s_{u}$ of the principal
ideal
$({\mathcal{Y}}_{1}\cdots\hat{{\mathcal{Y}}_{u}}\cdots{\mathcal{Y}}_{k}\cdot{\mathcal{P}}_{1}\cdots{\mathcal{P}}_{r_{2}}\cdot{\mathcal{Q}}_{1}\cdots\hat{{\mathcal{Q}}_{u}}\cdots{\mathcal{Q}}_{r_{1}}\cdot{\mathcal{K}}_{1}\cdots\hat{{\mathcal{K}}_{u}}\cdots{\mathcal{K}}_{q})^{C}$
with the hats denoting the omission of the corresponding prime factors.

  2. (2)

For $r_{1}+1\leq u\leq r_{1}+r_{2}$, choose a generator $s_{u}$ of the
principal ideal
$({\mathcal{Y}}_{1}\cdots\hat{{\mathcal{Y}}_{u}}\cdots{\mathcal{Y}}_{k}\cdot{\mathcal{P}}_{1}\cdots\hat{{\mathcal{P}}_{u}}\cdots{\mathcal{P}}_{r_{2}}\cdot{\mathcal{Q}}_{1}\cdots{\mathcal{Q}}_{r_{1}}\cdot{\mathcal{K}}_{1}\cdots\hat{{\mathcal{K}}_{u}}\cdots{\mathcal{K}}_{q})^{C}.$

  3. (3)

For $r_{2}+r_{1}+1\leq u\leq r_{1}+2r_{2}$, consider a generator $t_{u}$ of
the principal ideal
$({\mathcal{Y}}_{1}\cdots\hat{{\mathcal{Y}}_{u}}\cdots{\mathcal{Y}}_{k}\cdot{\mathcal{P}}_{1}\cdots{\mathcal{P}}_{r_{2}}\cdot{\mathcal{Q}}_{1}\cdots{\mathcal{Q}}_{r_{1}}\cdot{\mathcal{K}}_{1}\cdots\hat{{\mathcal{K}}_{u}}\cdots{\mathcal{K}}_{q})^{C}$
and set $s_{u}:=c_{u-r_{1}-r_{2}}\cdot t_{u}$.

  4. (4)

For $2r_{2}+r_{1}+1\leq u\leq k$, set $s_{u}:=\prod_{z=1,z\neq u}^{k}y_{z}$.

Finally, consider the set $T:=\\{E_{12}(s_{u})\mid 1\leq u\leq k\\}.$ Next, we
are going to check that $T$ satisfies the two necessary conditions for a
normally generating subset of ${\rm SL}_{2}(R)$ stated in Corollary 3.10.
First, to prove $\Pi(T)=\emptyset,$ note

$\displaystyle\Pi(\\{E_{12}(s_{u})\\})=\\{{\mathcal{Y}}_{f}\mid f\neq
u\\}\cup\\{{\mathcal{P}}_{f}\\}\cup\\{{\mathcal{Q}}_{f}\mid f\neq
u\\}\cup\\{{\mathcal{K}}_{f}\mid f\neq u\\}\text{ for }1\leq u\leq r_{1},$
$\displaystyle\Pi(\\{E_{12}(s_{u})\\})=\\{{\mathcal{Y}}_{f}\mid f\neq
u\\}\cup\\{{\mathcal{P}}_{f}\mid f\neq
u-r_{1}\\}\cup\\{{\mathcal{Q}}_{f}\\}\cup\\{{\mathcal{K}}_{f}\mid f\neq
u\\}\text{ for }r_{1}+1\leq u\leq r_{1}+r_{2},$
$\displaystyle\Pi(\\{E_{12}(s_{u})\\})=\\{{\mathcal{Y}}_{f}\mid f\neq
u\\}\cup\\{{\mathcal{K}}_{f}\mid f\neq u\\}\cup\\{\text{prime divisors of
}c_{u-r_{1}-r_{2}}R\\}$ $\displaystyle\text{ for }r_{1}+r_{2}+1\leq u\leq
r_{1}+2r_{2}\text{ and}$
$\displaystyle\Pi(\\{E_{12}(s_{u})\\})=\\{{\mathcal{Y}}_{f}\mid f\neq
u\\}\text{ for }r_{1}+2r_{2}+1\leq u\leq k.$

But we have chosen all the ${\mathcal{Y}}_{f}$ to be distinct from each other
and from the prime divisors of the elements $c_{1},\dots,c_{r_{2}}.$ Thus the
only prime divisors of $c_{u}R$ for $1\leq u\leq r_{2}$ that can possibly be
contained in $\Pi(T)=\bigcap_{i=1}^{k}\Pi(\\{E_{12}(s_{i})\\})$ are elements
of

$\\{{\mathcal{P}}_{1},\dots,{\mathcal{P}}_{r_{2}}\\}\cup\\{{\mathcal{Q}}_{1},\dots,{\mathcal{Q}}_{r_{1}}\\}\cup\\{{\mathcal{K}}_{f}\mid
f\neq u+r_{1}+r_{2}\\}.$

Thus we obtain $\Pi(T)=\bigcap_{i=1}^{k}\Pi(\\{E_{12}(s_{i})\\})=\emptyset.$

Next, we have to show that $T$ maps onto a generating set of the
abelianization of ${\rm SL}_{2}(R).$ Note that the abelianization map of ${\rm
SL}_{2}(R)$ factors through the quotient

$(\prod_{j=1}^{r_{2}}{\rm
SL}_{2}(R/{\mathcal{P}}_{j}^{2}))\times(\prod_{i=1}^{r_{1}}{\rm
SL}_{2}(R/{\mathcal{Q}}_{i}^{2}))\times(\prod_{l=1}^{q}{\rm
SL}_{2}(R/{\mathcal{K}}_{l}))=:G.$

First, observe that for each $u\leq 2r_{2}+r_{1}=q,$ the only non-trivial
component of the image of $E_{12}(s_{u})^{4}=E_{12}(4\cdot s_{u})$ in $G$ is
the ${\rm SL}_{2}(R/{\mathcal{K}}_{u})$-component. However, both $4$ and
$s_{u}$ map to units in $R/{\mathcal{K}}_{u}=\mathbb{F}_{3}$. Hence
$E_{12}(s_{u})^{4}$ maps onto a generator of the
$R/{\mathcal{K}}_{u}=\mathbb{F}_{3}$-component of the abelianization of ${\rm
SL}_{2}(R)$ according to Corollary 5.10. Hence we may assume for simplicity
that there are no prime ideals of the type ${\mathcal{K}}_{u}$ anymore.

Second, consider an element $E_{12}(s_{u})$ for $1\leq u\leq r_{1}.$ Now, the
only non-trivial component of $E_{12}(s_{u})$ after mapping to $G$ is the
${\rm SL}_{2}(R/{\mathcal{Q}}_{u}^{2})$-component. This component is
$E_{12}(s_{u}+{\mathcal{Q}}_{u}^{2})$ and due to the choice of $s_{u}$ the
element $s_{u}+{\mathcal{Q}}_{u}^{2}$ is a unit in
$R/{\mathcal{Q}}_{u}^{2}=\mathbb{Z}_{4}.$ However, we know from Corollary 5.10
that $E_{12}(s_{u})$ gets mapped to the element $s_{u}+{\mathcal{Q}}_{u}^{2}$
of the $R/{\mathcal{Q}}_{u}^{2}$-component of the abelianization of ${\rm
SL}_{2}(R)$ and clearly units in $\mathbb{Z}_{4}$ generate
$(\mathbb{Z}_{4},+).$

Last, observe that for $r_{1}+1\leq u\leq r_{1}+r_{2}$ the only non-trivial
components of $E_{12}(s_{u})$ and $E_{12}(s_{u+r_{2}})$ after mapping to $G$
are the ${\rm SL}_{2}(R/{\mathcal{P}}_{u-r_{1}}^{2})$-component. Due to the
choice of $s_{u}$ and $s_{u+r_{2}},$ we know that
$s_{u}+{\mathcal{P}}_{u-r_{1}}^{2}$ is a unit of
$R/{\mathcal{P}}_{u-r_{1}}^{2}$ and $s_{u+r_{2}}+{\mathcal{P}}_{u-r_{1}}^{2}$
is a non-trivial element of
${\mathcal{P}}_{u-r_{1}}/{\mathcal{P}}_{u-r_{1}}^{2}$. But these two elements
generate $(R/{\mathcal{P}}_{u-r_{1}}^{2},+)$ and according to Corollary 5.10
the elements $E_{12}(s_{u})$ and $E_{12}(s_{u+r_{2}})$ map to the elements
$s_{u}+{\mathcal{P}}_{u-r_{1}}^{2}$ and
$s_{u+r_{2}}+{\mathcal{P}}_{u-r_{1}}^{2}$ in the
$R/{\mathcal{P}}_{u-r_{1}}^{2}$-component of the abelianization of ${\rm
SL}_{2}(R)$ respectively. So to summarize, $T$ maps onto a generating set of
the abelianization of ${\rm SL}_{2}(R).$

So we have proven that $T$ does in fact normally generate ${\rm SL}_{2}(R).$
But $T$ has $k$ elements and so $\Delta_{k}({\rm SL}_{2}(R))\geq\|{\rm
SL}_{2}(R)\|_{T}$ holds. But similar to the proofs of [15, Theorem 6.3] or [5,
Corollary 6.2], it is now easy to check that indeed $\|{\rm
SL}_{2}(R)\|_{T}\geq 2k$ holds. Thus we have shown $\Delta_{k}({\rm
SL}_{2}(R))\geq 2k$ for all $k\geq\max\\{2r_{2}+r_{1},q\\}=v(R)$ at least
assuming that $r_{1},q\geq 1$ and $r_{1}+2r_{2}=q$. The other cases work in
essentially the same way though with merely slight differences concerning the
precise definition of the various $s_{u}.$ This finishes the proof of the
first part of Theorem 1.3.

######  Remark 5.12.

In [14, Theorem 3], we showed that the presence of bad primes of a ring of
S-algebraic integers $R$ resulted in the better lower bound
$4k+r_{1}(R)+r_{2}(R)$ on $\Delta_{k}({\rm Sp}_{4}(R))$, essentially because
the conjugacy diameter of ${\rm Sp}_{4}(\mathbb{F}_{2})=S_{6}$ is $5$, rather
than $4.$ One can use a similar argument to improve the lower bound on
$\Delta_{k}({\rm SL}_{2}(R))$ for $k\geq v(R)$ from $2k$ to $2k+r_{2}(R).$
This is due to $\Delta_{2}({\rm
SL}_{2}(\mathbb{F}_{2}[T]/(T^{2})))=\Delta_{2}(\mathbb{F}_{2}^{3}\rtimes
S_{3})\geq 3.$ Curiously, neither the presence of unramified prime divisors
${\mathcal{Q}}$ of $2R$ with $R/{\mathcal{Q}}=\mathbb{F}_{2}$ nor of prime
ideals ${\mathcal{K}}$ with $R/{\mathcal{K}}=\mathbb{F}_{3}$ results in better
lower bounds on $\Delta_{k}({\rm SL}_{2}(R))$, because both $G_{1}:={\rm
SL}_{2}(R/{\mathcal{Q}})={\rm SL}_{2}(\mathbb{F}_{2})=S_{3}$ and $G_{2}:={\rm
SL}_{2}(R/{\mathcal{K}})={\rm SL}_{2}(\mathbb{F}_{3})$ satisfy
$\Delta_{1}(G_{i})=2$ for $i=1,2.$

We also want to note the following:

######  Corollary 5.13.

Let $D$ be a positive, square-free integer and $R$ the ring of algebraic
integers in $\mathbb{Q}[\sqrt{D}].$ Then the value of
$v(R)=\max\\{2r_{2}(R)+r_{1}(R),q(R)\\}$ is

  1. (1)

$v(R)=0$ precisely if $D\equiv 5\text{ mod }8$ and $D\equiv 2\text{ mod }3$,
so $\Delta_{1}({\rm SL}_{2}(R))>-\infty.$

  2. (2)

$v(R)=1$ precisely if $D\equiv 5\text{ mod }8$ and $D\equiv 0\text{ mod }3$,
so $\Delta_{1}({\rm SL}_{2}(R))>-\infty.$

  3. (3)

$v(R)=2$ precisely if $D\equiv 1,2,3,6,7\text{ mod }8$ or $D\equiv 1\text{ mod
}3,$ so $\Delta_{1}({\rm SL}_{2}(R))=-\infty.$

###### Proof.

We obtain from [6, Theorem 25] that the ideal $2R$ splits and ramifies in $R$
as follows:

  1. (1)

$2R$ is inert precisely if $D\equiv 5\text{ mod }8.$

  2. (2)

$2R$ ramifies precisely if $D\equiv 2,3,6,7\text{ mod }8.$

  3. (3)

$2R$ splits precisely if $D\equiv 1\text{ mod }8.$

We first deal with the case of $2R$ not inert. If $2R$ ramifies, then
$r_{2}(R)=1,r_{1}(R)=0$ and $q(R)$ can not be bigger than $2$ and so $v(R)=2$
holds. Similarly, the splitting of $2R$ implies that $v(R)=r_{1}(R)=2.$
However, if $2R$ is inert, then the field of residue $R/2R$ is
$\mathbb{F}_{4}$ and so $r_{1}(R)=r_{2}(R)=0$ and hence $v(R)=q(R).$ Then we
have to distinguish two cases: Either $3$ is a divisor of $D$ or not. If $3$
divides $D,$ then [6, Theorem 25] implies further that $3R$ is a ramified
prime in $R$ and so $q(R)=1$ and $v(R)=1$ hold in this case. If $3$ is not a
prime divisor of $D,$ then [6, Theorem 25] implies two further sub-cases: $3R$
splits in $R$ if and only if $D$ is a quadratic residue in $\mathbb{F}_{3}$
and otherwise $3R$ is inert in $R.$ However, considering that $3$ does not
divide $D,$ one can check that $D$ is a quadratic residue precisely if
$D\equiv 1\text{ mod }3.$ So $v(R)=q(R)=2$ if $D\equiv 1\text{ mod }3$ and
$v(R)=q(R)=0$ if $D\equiv 2\text{ mod }3.$ Finally, the claim of the corollary
follows from Theorem 1.3 together with the determined values of $v(R).$ ∎

##  6\. Closing remarks

This paper finishes the qualitative discussion of strong boundedness of
S-arithmetic, split Chevalley groups. However, the question remains how
precisely $\Delta_{k}(G(\Phi,R))$ for $R$ the ring of S-algebraic integers in
a number field $K$ and $\Phi$ an irreducible root system depends on $R$ and
$\Phi.$ The previous papers [5] and [13] showed that $\Delta_{k}(G(\Phi,R))$
has a lower bound proportional to $k\cdot{\rm rank}(\Phi)$ and an upper bound
proportional to $k\cdot{\rm rank}(\Phi)^{2}$ with proportionality factors
independent of $R$ at least in the case of $R$ having infinitely many units.
However, as mentioned in [15, Remark 5.14(2)] it seems likely to us that
$\Delta_{k}(G(\Phi,R))$ also has an upper bound proportional to $k\cdot{\rm
rank}(\Phi)$ and in future research we will study this further.

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