1.1 Chevalley groups

Given a reduced root system $\Phi $ (usually assumed irreducible), we denote by $\Phi ^+$ , $\Phi ^{-}$ , and $\Pi =\big \{\alpha _1,\ldots ,\alpha _l\big \}$ the sets of positive, negative, and fundamental roots with respect to a chosen order. Throughout, we denote $N=|\Phi ^+|$ .

For a lattice ${\mathcal P}$ intermediate between the root lattice ${\mathcal Q}(\Phi )$ and the weight lattice ${\mathcal P}(\Phi )$ and any commutative unital ring R with the multiplicative group $R^*$ , we denote by $G=G_{{\mathcal P}}(\Phi ,R)$ the Chevalley group of type $(\Phi ,{\mathcal P})$ over R. In the case ${\mathcal P}={\mathcal P}(\Phi )$ , the group G is called simply connected and is denoted by $G_{\mathrm {sc}}(\Phi ,R)$ . In another extreme case ${\mathcal P}={\mathcal Q}(\Phi )$ , the group G is called adjoint and is denoted by $G_{\mathrm {ad}}(\Phi ,R)$ .

Many results do not depend on the lattice ${\mathcal P}$ , and we often omit any reference to ${\mathcal P}$ in the notation and denote by $G(\Phi ,R)$ any Chevalley group of type $\Phi $ over R. Usually, by default, we assume that $G(\Phi ,R)$ is simply connected, but in some cases, it is convenient to work with the adjoint group, which is then reflected in the notation.

Fixing a split maximal torus $T=T(\Phi ,R)$ in $G=G(\Phi ,R)$ and identifying $\Phi $ with $\Phi (G,T)$ , we denote by $X_{\alpha }$ , $\alpha \in \Phi $ , the unipotent root subgroups in G, elementary with respect to T. We fix maps $x_{\alpha }\colon R\mapsto X_{\alpha }$ , so that $X_{\alpha }=\{x_{\alpha }(\xi )\mid \xi \in R\}$ , and require that these parametrizations are interrelated by the Chevalley commutator formula with integer coefficients; see [Reference CarterCa], [Reference SteinbergSteinb]. The above unipotent elements $x_{\alpha }(\xi )$ , where ${\alpha \in \Phi} $ , $\xi \in R$ , elementary with respect to $T(\Phi ,R)$ , are also called [elementary] unipotent root elements or, for short, simply root unipotents.

Further,

$$ \begin{align*}E(\Phi,R)=\big\langle x_\alpha(\xi),\ \alpha\in\Phi,\ \xi\in R\big\rangle\end{align*} $$

denotes the absolute elementary subgroup of $G(\Phi ,R)$ , spanned by all elementary root unipotents, or, what is the same, by all [elementary] root subgroups $X_{\alpha }$ , $\alpha \in \Phi $ . For $\epsilon \in \{+,-\}$ , denote

$$ \begin{align*}U^\epsilon(\Phi,\,R)=\left\langle x_\alpha(\xi)\mid\alpha\in\Phi^\epsilon,\ \xi\in R\right\rangle\leq\mathrm{E_{sc}}(\Phi,\,R). \end{align*} $$

1.2 Relative groups

Let ${\mathfrak q}\unlhd R$ be an ideal of R, and let $\rho _{\mathfrak q}\colon R\longrightarrow R/{\mathfrak q}$ be the reduction modulo ${\mathfrak q}$ . By functoriality, it defines the reduction homomorphism of Chevalley groups $\rho _{\mathfrak q}\colon G(\Phi ,R)\longrightarrow G(\Phi ,R/{\mathfrak q})$ . The kernel of $\rho _{\mathfrak q}$ is denoted by $G(\Phi ,R,{\mathfrak q})$ and is called the principal congruence subgroup of $G(\Phi ,R)$ of level ${\mathfrak q}$ . We denote by $X_{\alpha }({\mathfrak q})$ the intersection of $X_{\alpha }$ with the principal congruence subgroup $G(\Phi ,R,{\mathfrak q})$ . Clearly, $X_{\alpha }({\mathfrak q})$ consists of all elementary root elements $x_{\alpha }(\xi )$ , $\alpha \in \Phi $ , $\xi \in {\mathfrak q}$ , of level ${\mathfrak q}$ :

$$ \begin{align*}X_{\alpha}({\mathfrak q})=\big\{x_{\alpha}(\xi)\mid\xi\in {\mathfrak q}\big\}.\end{align*} $$

By definition, $E(\Phi ,{\mathfrak q})$ is generated by $X_{\alpha }({\mathfrak q})$ , for all roots $\alpha \in \Phi $ . The same subgroups generate $E(\Phi ,R,{\mathfrak q})$ as a normal subgroup of the absolute elementary group $E(\Phi ,R)$ .

The classical Suslin–Kopeiko–Taddei theorem asserts that for $\mathrm {rk}(\Phi )\ge 2$ , one has $E(\Phi ,R,{\mathfrak q})\unlhd G(\Phi ,R)$ . The quotient

$$ \begin{align*}{\mathrm K}_1(\Phi,R,{\mathfrak q})=G_{\mathrm{sc}}(\Phi,R,{\mathfrak q})/E_{\mathrm{sc}}(\Phi,R,{\mathfrak q})\end{align*} $$

is called the [relative] K $_1$ -functor. The absolute case corresponds to ${\mathfrak q}=R$ ,

$$ \begin{align*}{\mathrm K}_1(\Phi,R)=G_{\mathrm{sc}}(\Phi,R)/E_{\mathrm{sc}}(\Phi,R).\end{align*} $$

Observe

$$ \begin{align*}{\mathrm K}_1(\mathsf A_l,R,{\mathfrak q})=\mathrm{SK}_1(l+1,R,{\mathfrak q}),\end{align*} $$

so that our K $_1$ -functor corresponds rather to the $\mathrm {SK}_1$ of the classical theory.

1.3 Arithmetic case

For a global field K and a finite nonempty set S of places of K (containing all archimedean places when K is a number field), let

$$ \begin{align*}R=\{x\in K\mid v(x)\geq0\ \forall v\not\in S\}. \end{align*} $$

It is a Dedekind domain whose maximal ideals can be canonically identified with the places outside S. Following [Reference Bass, Milnor and SerreBMS], we say that R is the Dedekind ring of arithmetic type defined by the set S (or, for short, an arithmetic ring).

For the arithmetic rings Bass, Milnor, and Serre [Reference Bass, Milnor and SerreBMS] have explicitly calculated $K_1(\Phi ,R,{\mathfrak q})$ , $\Phi =\mathsf A_l,\mathsf C_l$ , $l\ge 2$ , in terms of Mennicke symbols. Namely, they have proven that ${\mathrm K}_1(\mathsf A_l,R,{\mathfrak q})\cong C({\mathfrak q})$ and ${\mathrm K}_1(\mathsf C_l,R,{\mathfrak q})\cong \mathrm {Cp}({\mathfrak q})$ (the universal Mennicke groups), which in turn are then identified via reciprocity laws with certain groups of roots of 1 in R.

The [almost] positive solution of the congruence subgroup problem for these groups amounts to the fact that the congruence kernel

$$ \begin{align*}C(G):=\lim_{\longleftarrow} C({\mathfrak q})\end{align*} $$

taken over all nonzero ideals ${\mathfrak q}\unlhd R$ is finite. Actually, it is trivial, apart from the case when R is the ring of integers $\mathcal O_K$ in a purely imaginary number field K, when $C(G)\cong \mu (K)$ is the group of all roots of 1 in K.

Later, their results were generalized to all Chevalley groups by Hideya Matsumoto [Reference MatsumotoMa]. The following special case of his results [Reference MatsumotoMa, Théorème 12.7] explains why we usually prefer to work with simply connected groups.

Lemma 1.1 Let R be a Dedekind ring of arithmetic type and $\Phi $ a reduced irreducible root system of rank at least $2$ . Then

$$ \begin{align*}\mathrm E_{\mathrm{sc}}(\Phi,\,R)=G_{\mathrm{sc}}(\Phi,\,R). \end{align*} $$

2.1 Reduction to the ring of integers

The following result is a combination of [Reference Bass, Milnor and SerreBMS, Lemma 2.1] and [Reference Bass, Milnor and SerreBMS, Lemma 5.3]. The same proof, with several successful deteriorations, is reproduced on page 685 of [Reference Carter and KellerCK1].

Lemma 2.1 Let R be a Dedekind ring, $s\in R$ , $s\neq 0$ . Then

$$ \begin{align*}\mathrm{SL}\Big(2,R\Big[{\frac 1s}\Big]\Big)= \mathrm{SL}(2,R)E^3\Big(2,R\Big[{\frac 1s}\Big]\Big).\end{align*} $$

In other words, every $2\times 2$ matrix with entries in $R\big [{\frac 1s}\big ]$ can be reduced to a matrix with entries in R by $\le 3$ elementary moves with parameters in $R\big [{\frac 1s}\big ]$ . Since the number of elementary moves during the rank reduction does not depend on the ring R, and the only such dependence occurs at the base of induction, we immediately get the following corollary.

2.2 Arithmetic lemmas

The following lemma is the arithmetic heart of the whole proof. In the number case, it is [Reference Carter and KellerCK1, Lemma 1], and in the function case, it was first proven in full generality in [Reference TrostTr2, Lemma 3.1] (before that, only a special case $R={\mathbb F}_{\!q}[t]$ was established as [Reference Kunyavskiĭ, Plotkin and VavilovKPV, Lemma 6.4]).

For a global function field K with the field of constants ${\mathbb F}_{\!q}$ and $b\in \mathcal O_K$ , $b\neq 0$ , we denote by $\epsilon (b)$ the exponent of the [finite] multiplicative group $(\mathcal O_K/b\mathcal O_K)^*$ and set $\delta (b)=\varepsilon (b)/(q-1)$ . The following result is due to Trost [Reference TrostTr2, Lemma 3.3].

3.1 Tavgen rank reduction theorem

In this section, we prove Theorem B and establish some other useful consequences of Tavgen’s reduction theorem.

The following trick allowing one to reduce the rank of a root system under consideration was invented by Tavgen [Reference TavgenTa2] (and then generalized in [Reference Vavilov, Smolensky and SuryVSS] and [Reference Smolensky, Sury and VavilovSSV]). The following final form is proven in [Reference Kunyavskiĭ, Plotkin and VavilovKPV, Theorem 4.2].

Lemma 3.1 Let $\Phi $ be a reduced irreducible root system of rank $l\geq 2$ , and R be a commutative ring. Let $\Delta _1,\ldots ,\Delta _t$ be some subsystems of $\Phi $ , whose union contains all fundamental roots of $\Phi $ . Suppose that for all $\Delta _i$ , the elementary Chevalley group $\mathrm {E_{sc}}(\Delta _i,\,R)$ admits a unitriangular factorization

$$ \begin{align*}\mathrm{E_{sc}}(\Delta_i,\,R)=U^+(\Delta_i,\,R)\,U^-(\Delta_i,\,R)\,U^+(\Delta_i,\,R)\ldots U^{\pm}(\Delta_i,\,R) \end{align*} $$

of length N (not depending on i). Then the elementary group $\mathrm {E_{sc}}(\Phi ,\,R)$ itself admits unitriangular factorization

$$ \begin{align*}\mathrm{E_{sc}}(\Phi,\,R)=U^+(\Phi,\,R)\,U^-(\Phi,\,R)\,U^+(\Phi,\,R)\ldots U^{\pm}(\Phi,\,R) \end{align*} $$

of the same length N.

It is used below in two cases, when all $\Delta _i$ ’s are $\mathsf {A}_1$ , and when all of them are $\mathsf {A}_2$ .

3.2 The case when $R^*$ is infinite

The case where a Dedekind ring R of arithmetic type has infinitely many units is now completely solved, with very small absolute constant. Here is a brief account of main steps along this route. Vsemirnov [Reference VsemirnovVs] established a first unconditional result of this sort, not depending on the GRH; Morgan, Rapinchuk, and Sury [Reference Morgan, Rapinchuk and SuryMRS] proved that $\mathrm {SL}(2,R)$ is boundedly elementarily generated in number case for an arbitrary R with infinite $R^{\ast }$ . The absolute bound obtained in their paper is $L=9$ .

In the paper presently under way, the first author, Morris, and Rapinchuk [Reference Kunyavskiĭ, Morris and RapinchukKMR] improved the bound to $L=7$ in the number case (which we believe is the best possible and cannot be further improved, in general). A similar result holds in the function case, with the bound $L=8$ (which, we believe, can be further improved to $L=7$ ).

Together with Lemma 3.1, this immediately implies Theorem B: it covers the case $\Delta =\mathsf A_1$ , and all higher rank cases are reduced to rank one by putting $\Delta _i=\mathsf A_1$ in Lemma 3.1.

Thus, the condition $|R^*|=\infty $ makes a huge relief. Essentially, no extra work is needed to treat the general case with not the best possible but still rather plausible bounds (anyway, asymptotically, L cannot be smaller than something like $3N$ to $4N$ ).

So, if we are not interested in actual bounds, but just in uniform boundedness, the rest of the exposition is formally dedicated to the Dedekind rings of arithmetic type with finite multiplicative groups. Thus, in the number case, we restrict our attention to the rings of integers in imaginary quadratic number fields (since $\mathbb Z$ is already covered). In the function case, as discovered by Trost [Reference TrostTr2], for ranks $\ge 2$ , we do not have to distinguish between rings with finite and infinite multiplicative group, so that the rest of this section does not depend on [Reference Kunyavskiĭ, Morris and RapinchukKMR] (but does depend on [Reference Morgan, Rapinchuk and SuryMRS]).

3.3 The simply laced case and $\Phi =\mathsf F_4$

In this section, we prove the statement of Theorem A for the simply laced root systems and also for $\Phi =\mathsf F_4$ .

By a theorem of Carter–Keller–Paige (see [Reference Carter, Keller and PaigeCKP], (2.4)) (rewritten and explained by Morris [Reference MorrisMo]), bounded generation for groups of type $\mathsf A_{l}$ , $l\ge 2$ , holds for all Dedekind rings R in number fields K, with a bound depending on l and also on the degree d of K. But since for all degrees $d\ge 3$ the existence of uniform bound already follows from Theorem B, we only need to take maximum of that, and the universal bound for $d=2$ .

Combining this result with the subsequent work of Trost [Reference TrostTr2] on the function field case, one obtains the following result; see [Reference TrostTr2, Theorem 4.1].

In fact, in the sequel, we only need the special case of the above result pertaining to $\mathrm {SL}(3,R)$ , which corresponds to $\mathsf A_2$ . Indeed, by stability arguments, one has $L(l)\le L(l-1)+3l+1$ for all $l\ge 2$ , so that all $L(l)$ , $l\ge 3$ can be expressed in terms of the constant $L(2)$ . In the function case, Trost [Reference TrostTr2] gave the estimate $L(2)\le 65$ . No such explicit estimate is available in the number case because the uniform boundedness was established by model-theoretic arguments.

Now we are in a position to get a particular case of Theorem A.

Thus, in addition to Theorem B, we obtain another stronger form of Theorem A, now without the assumption that $R^*$ is infinite, but only in the special case of simply laced systems of rank $\ge 2$ and $\mathsf F_4$ . The bound here is very rough, since $L(2)$ is the number of elementary factors; the number of unitriangular ones can be much smaller. Also, the use of stability arguments allows one to get much better bounds, of the type $L=L(2)+M$ , with $3N\le M\le 4N$ , where some multiple of N occurs as a summand, not as a factor.

4.1 The case of $\mathrm {Sp}(2l,R)$ , $l\ge 3$

Thus, we are left with the analysis of the the symplectic groups $\mathrm {Sp}(2l,R)$ , $l\ge 2$ . Quite amazingly, the results of [Reference Kunyavskiĭ, Plotkin and VavilovKPV] and [Reference TrostTr2] allow to reduce $\mathrm {Sp}(6,R)$ to $\mathrm {SL}(3,R)$ as well. As mentioned in [Reference Kunyavskiĭ, Lavrenov, Plotkin and VavilovKLPV], the idea of such a reduction was contained already in Zakiryanov’s thesis; see [Reference ZakiryanovZa]. Of course, as above, the case $\Phi ={\mathsf C}_l$ , $l\ge 3$ is immediately reduced to $\Phi =\mathsf C_3$ by stability.

So, we have two remaining tasks. First of all, we have to prove Theorem A in the $\mathsf C_2=\mathrm {Sp}(4,R)$ case, which is not covered by our previous considerations. Second, we want to make the number $L(2)$ , which is a crucial constituent in all estimates, as small as we can.

5.1 Swindling lemma for $\mathrm {SL}(3,R)$

The following result is essentially [Reference NicaNi, Lemma 4]. Of course, formally Nica assumes that R is a PID, to conclude that all s have the desired factorizations. But calculations with Mennicke symbols [Reference Bass, Milnor and SerreBMS] show that his result holds [at least] for all Dedekind rings. Below, we extract the rationale behind his proof to apply it in the only situation we need. Namely, we stipulate that the desired factorization of s does exist.

Lemma 5.1 Let R be any commutative ring. Assume that

$$ \begin{align*}A=\left(\begin{array}{ccc} a&b&0 \\ sc&d&0 \\ 0&0&1 \end{array}\right) \in\mathrm{SL}(3,R),\end{align*} $$

where s admits factorization $s=s_1s_2$ such that

$$ \begin{align*}a\equiv d\equiv 1\quad\pmod{s_1},\qquad a\equiv d\equiv -1\quad\pmod{s_2}.\end{align*} $$

Then A can be transformed to

$$ \begin{align*}A=\left(\begin{array}{ccc} \pm a&-sb&0 \\ c&\mp d&0\\ 0&0&-1 \end{array}\right) \in\mathrm{SL}(3,R)\end{align*} $$

by $\le 11$ elementary moves.

Proof Let $t_1,t_2\in R$ be such that

$$ \begin{align*}a=1+s_1t_1=-1+s_2t_2.\end{align*} $$

Below, we use programmers’ notation style to describe elementary moves, keeping the letter A to denote all matrices appearing along the way.

$\bullet $ Step 1

$$ \begin{align*}A=At_{31}(s_1)=\left(\begin{array}{ccc} a&b&0 \\ sc&d&0 \\ s_1&0&1 \end{array}\right).\end{align*} $$

$\bullet $ Step 2+3

$$ \begin{align*}A=t_{13}(-t_1)t_{23}(-s_2c)A=\left(\begin{array}{ccc} 1&b&-t_1 \\ 0&d&-s_2c \\ s_1&0&1 \end{array}\right).\end{align*} $$

$\bullet $ Step 4

$$ \begin{align*}A=t_{31}(-s_1)A=\left(\begin{array}{ccc} 1&b&-t_1 \\ 0&d&-s_2c \\ 0&-s_1b&a \end{array}\right).\end{align*} $$

At this stage, we have rolled $s_1$ over the diagonal by simultaneously moving the $2\times 2$ matrix from the NW-corner to the SE-corner. Now we have to roll over $s_2$ by simultaneously returning our $2\times 2$ matrix back to the NW-corner.

$\bullet $ Step 5+6

$$ \begin{align*}A=At_{12}(-b)t_{13}(t_1+s_2)=\left(\begin{array}{ccc} 1&0&s_2 \\ 0&d&-s_2c \\ 0&-s_1b&a \end{array}\right).\end{align*} $$

Now we are in exactly the same position as we were after the first move and can start rolling back.

$\bullet $ Step 7+8

$$ \begin{align*}A=t_{21}(c)t_{31}(-t_2)A=\left(\begin{array}{ccc} 1&0&s_2 \\ c&d&0 \\ -t_2&-s_1b&-1 \end{array}\right).\end{align*} $$

$\bullet $ Step 9

$$ \begin{align*}A=t_{13}(s_2)A=\left(\begin{array}{ccc} -a&-sb&0 \\ c&d&0 \\ -t_2&-s_1b&-1 \end{array}\right).\end{align*} $$

$\bullet $ Step 10+11

$$ \begin{align*}A=At_{31}(-t_2)t_{32}(-s_1b)=\left(\begin{array}{ccc} -a&-sb&0 \\ c&d&0 \\ 0&0&-1 \end{array}\right).\end{align*} $$

For the other choice of signs, one should start rolling over the other way, say, with moving A to $At_{32}(s_2)$ .

5.2 Swindling lemma for $\mathrm {Sp}(4,R)$

In what concerns $\mathrm {Sp}(4,R)$ , we keep the notation and conventions of [Reference Kunyavskiĭ, Plotkin and VavilovKPV, Section 6]. In particular, $\mathrm {Sp}(4,R)$ preserves the symplectic form with Gram matrix

$$ \begin{align*}\begin{pmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&-1&0&0\\ -1&0&0&0\\ \end{pmatrix}.\end{align*} $$

Further, $\alpha $ and $\beta $ are fundamental roots of $\mathsf C_2$ , and the corresponding root unipotents are

$$ \begin{align*}x_{\alpha}(\xi)=\begin{pmatrix} 1&\xi&0&0\\ 0&1&0&0\\ 0&0&1&-\xi\\ 0&0&0&1\\ \end{pmatrix},\qquad x_{\beta}(\xi)= \begin{pmatrix} 1&0&0&0\\ 0&1&\xi&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{pmatrix},\end{align*} $$

while $x_{-\alpha }(\xi )$ and $x_{-\beta }(\xi )$ are their transposes. Together, they generate the elementary symplectic group $\mathrm {Ep}(4,R)$ which for Dedekind rings of arithmetic type coincides with $\mathrm {Sp}(4,R)$ .

There are two natural embeddings of $\mathrm {SL}(2,R)$ into $\mathrm {Sp}(4,R)$ , the short root embedding $\phi _{\alpha }$

$$ \begin{align*}\phi_{\alpha} \begin{pmatrix}1&\xi\\0&1\\ \end{pmatrix} =x_{\alpha}(\xi),\qquad \phi_{\alpha} \begin{pmatrix}1&0\\\xi&1\\ \end{pmatrix} =x_{-\alpha}(\xi),\end{align*} $$

and the long root embedding $\phi _{\beta }$

$$ \begin{align*}\phi_{\beta} \begin{pmatrix}1&\xi\\0&1\\ \end{pmatrix} =x_{\beta}(\xi),\qquad \phi_{\beta} \begin{pmatrix}1&0\\\xi&1\\ \end{pmatrix} =x_{-\beta}(\xi),\end{align*} $$

and unlike groups of other types in the symplectic case, they behave very differently.

The following swindling lemma for the short root embedding that we use in the sequel seems to be stronger than the linear swindling lemma. But this is because, morally, the Mennicke symbol constructed via $\phi _{\alpha }$ is the square root of the Mennicke symbol constructed via $\phi _{\beta }$ . At the same time, stability reduction (see, for instance, [Reference Kunyavskiĭ, Plotkin and VavilovKPV, Lemma 6.1]) reduces a symplectic matrix to an element of $\mathrm {SL}(2,R)$ in the long root embedding. Thus, to be able to use this [seemingly] stronger form of swindling, we should be able to extract square roots of Mennicke symbols anyway.

Lemma 5.3 [Reference Kunyavskiĭ, Plotkin and VavilovKPV]

Let $a,b,c,d,s\in R$ , $ad-bcs=1$ , and, moreover, $a\equiv d\ \ \pmod s$ . Then

$$ \begin{align*}\phi_{\alpha} \begin{pmatrix} a&b\\ cs&d\\ \end{pmatrix}= \begin{pmatrix} a&b&0&0\\ cs&d&0&0\\ 0&0&a&-b\\ 0&0&-cs&d\\ \end{pmatrix}\end{align*} $$

can be moved to

$$ \begin{align*}\phi_{\alpha} \begin{pmatrix} d&c\\ bs&a\\ \end{pmatrix}=\begin{pmatrix} d&c&0&0\\ bs&a&0&0\\ 0&0&d&-c\\ 0&0&-bs&a\\ \end{pmatrix}\end{align*} $$

by $\le 26$ elementary transformations.

We do not use it here, but to put things in the right prospective, let us reproduce the swindling lemma for long roots [Reference Kunyavskiĭ, Plotkin and VavilovKPV, Lemma 6.7], on which the proof of Lemma 5.3 hinges, and which is a true analogue of Lemma 5.1 valid for all commutative rings. The number of moves in Lemma 5.4 seems to be smaller than in Lemma 5.1. The reason is that we do not return the element of $\mathrm {SL}(2,R)$ to the initial position, corresponding to the long root $\beta $ , but leave it in another embedding, corresponding to another long root $2\alpha +\beta $ . This “half-swindling” in the long root embedding was followed by returning to the embedding $\varphi _{\alpha }$ corresponding to the short root $\alpha $ (see [Reference Kunyavskiĭ, Plotkin and VavilovKPV, Lemma 6.9]). This was sufficient for the proof of the key Proposition 6.10 in [Reference Kunyavskiĭ, Plotkin and VavilovKPV]

Lemma 5.4 [Reference Kunyavskiĭ, Plotkin and VavilovKPV, Lemma 6.7]

Let $a,b,c,d,s\in R$ , $ad-bcs^2=1$ , and, moreover, $a\equiv d\equiv 1\ \ \pmod s$ . Then

$$ \begin{align*}\phi_{\beta} \begin{pmatrix} a&b\\ cs^2&d\\ \end{pmatrix} =\begin{pmatrix} 1&0&0&0\\ 0&a&b&0\\ 0&cs^2&d&0\\ 0&0&0&1\\ \end{pmatrix}\end{align*} $$

can be moved to

$$ \begin{align*}\phi_{2\alpha+\beta} \begin{pmatrix} d&-c\\ -bs^2&a\\ \end{pmatrix}= \begin{pmatrix} d&0&0&-c\\ 0&1&0&0\\ 0&0&1&0\\ -bs^2&0&0&a\\ \end{pmatrix}\end{align*} $$

by $\le 8$ elementary transformations.

8.1 Bounded generation of ultrapowers

First, recall that being algebraic groups Chevalley groups themselves commute with direct products:

$$ \begin{align*}G(\Phi,\displaystyle{\prod_{i\in I}R_i})= \prod_{i\in I}G(\Phi,R_i).\end{align*} $$

However, elementary groups do not, in general, commute with direct products, which is due to the lack of the uniform elementary generation. Namely, Wilberd van der Kallen noticed that the quotient

$$ \begin{align*}E(\Phi,R)^{\infty}/E(\Phi,R^{\infty})\end{align*} $$

(countably many copies) is precisely the obstruction to the bounded generation of $E(\Phi ,R)$ . This easily ensues from the following obvious observation. In the case of $\mathrm {SL}(n,R)$ the following result is [Reference Carter, Keller and PaigeCKP, Theorem 2.8]; generalization to all Chevalley groups is immediate.

Lemma 8.2 Let I be any index set and $R_i$ , $i\in I$ be a family of commutative rings. Suppose all $E(\Phi ,R_i)$ have elementary width $\le L$ ,

$$ \begin{align*}E(\Phi,R_i)=E^L(\Phi,R_i).\end{align*} $$

Then the elementary width of

$$ \begin{align*}\prod_{i\in I}E(\Phi,R_i)=E\Big(\Phi,\displaystyle{\prod_{i\in I}R_i}\Big)\end{align*} $$

does not exceed $2LN$ . Conversely, the above equality implies that all $E(\Phi ,R_i)$ are uniformly elementarily boundedly generated.

Proof Take $g_i\in E(\Phi ,R_i)$ , $i\in I$ , and for each $i\in I$ , choose an elementary expression of $g_i$ of length $\le L$ , say

$$ \begin{align*}g_i=x_{\beta(i)_1}(\xi(i)^1)\ldots x_{\beta(i)_L}(\xi(i)^L)=\prod_{j=1}^L x_{\beta(i)^j}(\xi(i)_j)\in E(\Phi,R_i).\end{align*} $$

If an actual expression of $g_i$ is shorter than L, just set the remaining $\beta (i)_j$ to the maximal root of $\Phi $ and $\xi (i)_j$ to $0$ .

Now consider any ordering of roots in $\Phi =\{\gamma _1,\ldots ,\gamma _{2N}\}$ and form products

$$ \begin{align*}u(i)^j=x_{\gamma_1}(\xi(i)^j_{1})\ldots x_{\gamma_{2N}}(\xi(i)^j_{2N})=\prod_{h=1}^{2N} x_{\gamma_h}(\xi(i)^j_{h})\in E(\Phi,R_i),\qquad 1\le j\le L\end{align*} $$

by the following rule:

$$ \begin{align*}\xi(i)^j_h= \begin{cases} \xi(i)^j,&\text{if } \beta(i)_j=\gamma_h,\\ 0,&\text{otherwise.} \end{cases}.\end{align*} $$

Then clearly

$$ \begin{align*}g_i=u(i)^1\ldots u(i)^L\in E(\Phi,R_i),\qquad i\in I.\end{align*} $$

Thus, every element

$$ \begin{align*}g=(g_i)_{i\in I}\in \prod_{i\in I}E(\Phi,R_i)\end{align*} $$

can be expressed as $g=u^1\ldots u^L$ , where each of the L factors

$$ \begin{align*}u^j=(u(i)^j)_{i\in I}\in E(\Phi,\prod_{i\in I}R_i), \qquad 1\le j\le L\end{align*} $$

can be expressed as a product of $2N$ elementary generators

$$ \begin{align*}u^j=x_{\gamma_1}\Big((\xi(i)^j_1)_{i\in I}\Big)\ldots x_{\gamma_{2N}}\Big((\xi(i)^j_{2N})_{i\in I}\Big)\end{align*} $$

with parameters in $\displaystyle {\prod _{i\in I}R_i}$ .

Conversely, an element $g=(g_i)_{i\in I}$ , where $g_i\in E(\Phi ,R_i)$ , such that the length of its components $g_i$ is unbounded, cannot possibly belong to $E(\Phi ,\displaystyle {\prod _{i\in I}R_i})$ .

Since bounded generation is inherited by factors, any ultraproduct $R=\prod _{\mathcal U} R_i$ of rings $R_i$ for which the elementary groups $E(\Phi ,R_i)$ are uniformly boundedly generated enjoys the property that $E(\Phi ,R)$ is boundedly generated. In particular, this applies to ultrapowers ${}^*R$ , also known as nonstandard models of R. In other words, we have the following result.

Lemma 8.3 Bounded elementary generation of the elementary group $E(\Phi ,R)$ is equivalent to the equality

$$ \begin{align*}{}^*E(\Phi,R)=E(\Phi,{}^*R)\end{align*} $$

for all nonstandard models ${}^*R$ of R.

8.2 Congruence subgroup problem for nonstandard models

However, Carter, Keller, and Paige [Reference Carter, Keller and PaigeCKP] made this observation quite a bit more precise. Namely (see [Reference Carter, Keller and PaigeCKP, 2.1] or [Reference MorrisMo, Lemma 2.29]), they established the following fact.

Lemma 8.5 Bounded elementary generation of the elementary group $E(\Phi ,R)$ is equivalent to the condition

$$ \begin{align*} E(\Phi,{}^*R)\quad \text{has a finite index in} \quad {}^*E(\Phi,R), \end{align*} $$

for all nonstandard models ${}^*R$ of R.

In fact, they proved that bounded elementary generation of the elementary group $E(\Phi ,R)$ is equivalent to the almost positive solution of the congruence subgroup problem for all nonstandard models ${}^*R$ of R.

More precisely, [Reference Carter, Keller and PaigeCKP, 2.3] and [Reference MorrisMo] apply the whole machinery not just to the bounded generation of $\mathrm {SL}(n,R)$ but also to the bounded generation of $E(n,R,{\mathfrak q})$ in terms of the conjugates of elementary generators of level ${\mathfrak q}$ . They consider groups $ \mathrm {SL}(n,R,{\mathfrak q})/E(n,R,{\mathfrak q})$ which are isomorphic to quotients of universal Mennicke groups $C({\mathfrak q})$ for all nonzero ideals ${\mathfrak q}\unlhd R$ and restate bounded elementary generation of $E(n,R,{\mathfrak q})$ as the almost positive solution of the congruence subgroup problem for $\mathrm {SL}(n,{}^*R)$ .

Recall that we need a universal bound that depends only on the root system $\Phi $ and the degree of K. Since we reduce the problem to the congruence subgroup problem for $\mathrm {SL}(n,{}^*R)$ , we are very close to that.

8.3 Universal bound

Before going to $\mathrm {Sp} (4,R)$ we shall recall one more principal invention of Carter, Keller, and Paige [Reference Carter, Keller and PaigeCKP]. We need to pass from the ring of integers $R=\mathcal O_K$ in an algebraic number field to the ring ${}^*R$ . The ring ${}^*R$ is a nonstandard model of R; that is, ${}^*R$ equals to an ultrapower $\prod _{\mathcal U} R$ along the ultrafilter $\mathcal U$ . The good point is that thanks to Łoś’s theorem [Reference Chang and KeislerChKe, Theorem 4.1.9] ultrapowers keep first-order properties of structures unchanged. The bad point is that other properties do not survive, and in many senses, ${}^*R$ is far away from the ring of integers R.

With this end, Carter, Keller, and Paige [Reference Carter, Keller and PaigeCKP] introduce arithmetic conditions $\mathrm {Gen}(t,r)$ and $\mathrm {Exp}(t,s)$ on a ring R that depend on natural parameters $r,s$ , and t, which are too technical to describe them here in full. Morally, $\mathrm {Gen}(t,r)$ allows to uniformly bound the number of generators of the abelian groups $C({\mathfrak q})$ , while $\mathrm {Exp}(t,s)$ allows to uniformly bound their exponent. Besides, [Reference Carter, Keller and PaigeCKP] constantly used the fact that the stable rank of R is 1.5.

The importance of these conditions consists in the following pivotal observations. First of all, conditions $\mathrm {Gen}(t,r)$ , $\mathrm {Exp}(t,s)$ , and $\mathrm {sr}(R)=1.5$ are stated in the first-order language of ring theory; see [Reference Carter, Keller and PaigeCKP, 2.2] or [Reference MorrisMo, Sections 3A and 3B]. Hence, the characteristic property of ultraproducts imply the following:

Most importantly, these conditions allow to bound uniformly the universal Mennicke groups $C({\mathfrak q})$ for all ideals ${\mathfrak q}\unlhd R$ – and thus to get the finite congruence kernel of $G(\Phi ,{}^*R)$ . Indeed, the main (difficult!) step in obtaining a uniform bound in the number case is the following result; see [Reference Carter, Keller and PaigeCKP, Theorem 1.8] or [Reference MorrisMo, Theorem 3.11].

Finally, the rings of integers of the number fields of bounded degree $[K:{\mathbb Q}]\le d$ satisfy these conditions for some values of parameters (which depend on d and which we do not wish to specify here). This result depends on [a very strong form of] the Dirichlet theorem on primes in arithmetic progressions. The following lemma is a [weaker form of the] conjunction of [Reference Carter, Keller and PaigeCKP, Lemmas 4.4 and 4.5] or [Reference MorrisMo, Corollary 3.5 and Theorem 3.9]; the proof thereof requires subtle arithmetic properties of norm maps on the groups of units that are based on the aforementioned Dirichlet theorem (see [Reference Carter, Keller and PaigeCKP, Lemma 4.3] or [Reference MorrisMo, Lemma 3.8]).

Hence, one can take $R={}^*R$ in Lemma 8.7 and arrive at the almost positive solution of the congruence problem in $\mathrm {SL}(n,{}^*R)$ , as required.

To use the above fact for $\mathrm {SL}(n,{}^*R)=\mathrm {SL}(n,\prod _{\mathcal U} R)$ is the same as to use the Compactness Theorem as is in [Reference Carter, Keller and PaigeCKP] or [Reference MorrisMo, Theorem 2.7]. In any case (cf. [Reference Carter, Keller and PaigeCKP, Theorem 2.4] or [Reference MorrisMo, Corollary 3.13]), we get bounded elementary generation of $\mathrm {SL}(n,R)$ with the bound that only depends on the root system $\Phi $ and the degree of K.

At this point, [Reference Carter, Keller and PaigeCKP, 2.5] use a standard argument from nonstandard analysis. Since the elementary width $w(G)\in {}^*{\mathbb N}$ of $G=\mathrm {SL}(n,R)$ on this class of rings R is internally defined, everywhere finite, and bounded (by any infinite natural number), it must attain maximal value, which is obviously finite (all of them are!).

The proof of the uniform bounded generation for Chevalley groups will be completed if we cover the case of symplectic groups. The standard reasoning says that it is enough to prove the fact for the rank 2 case – that is, for $\mathrm {Sp}(4,R)$ . The proof basically follows the line depicted above for $G=\mathrm {SL}(n,R)$ .

Passing from $\mathrm {SL}(n,R)$ to $\mathrm {Sp}(2l,R)$ ( $l\ge 2$ ), we have to consider the group $\mathrm {Cp}({\mathfrak q}):=\mathrm {Sp}(2l,{\mathfrak q})/\mathrm {Ep}(2l,{\mathfrak q})$ in place of $C({\mathfrak q})$ (recall that the group $\mathrm {Cp}({\mathfrak q})$ is well defined, finite, and independent of l; see [Reference Bass, Milnor and SerreBMS, Theorem 12.4 and Corollary 12.5]).

The case $\mathrm {Sp}(4,R)$ is similar but much more difficult than the one of $\mathrm {SL}(3,R)$ . The point is that there are two embeddings $\mathsf A_1\to \mathsf C_2$ and $\widetilde {\mathsf {A}}_1\to \mathsf C_2$ , on long and short roots, respectively. This results in a more complicated structure of the universal Mennicke group (cf. [Reference Bass, Milnor and SerreBMS, Lemma 13.3]). The decisive role is played by the following quite technical theorem proven by Trost. It serves as a symplectic analogue of Lemma 8.7.

This parallelism can now be used to conclude that $\mathrm {Sp} (4,R)$ is uniformly elementarily boundedly generated, with a universal bound that only depends on the degree of K (which is equal to 2 in the case under consideration), and is thus an absolute constant (see [Reference TrostTr1, Theorem 3.20 and Section 3.3] for details of the proof). This concludes the proof of Lemma 8.1.