The strong approximation theorem and computing with linear groups

TL;DR

This paper develops algorithms to compute all congruence quotients of finitely generated Zariski dense subgroups of special linear groups, providing a computational realization of the strong approximation theorem.

Contribution

It introduces a method to explicitly compute congruence quotients for Zariski dense subgroups of SL(n, Z) and SL(n, Q), extending the practical application of the strong approximation theorem.

Findings

Algorithms successfully compute all congruence quotients for specified groups.

The approach applies to groups in SL(n, Z) for n ≥ 2 and in SL(n, Q) for n > 2.

Provides a computational framework for strong approximation in linear groups.

Abstract

We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n \geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n, \mathbb{Q})$ for $n > 2$.

Tables

Table 1. Table 1. Order bounds in small degrees

Degree	Geometric	Group Order	Element order	Exponent lcm
$3𝒮$	$1,2,3,6,8$	$1080$	$21$	$1260$
$4𝒮$	$1,2,3,6,8$	$103680$	$36$	$2520$
$5𝒮$	$1,2,3,6,8$	$129600$	$60$	$3960$
$6$	$1,2,3,4,8$	$39191040$	$60$	$2520$
$7$	$1,2,3,6,8$	$115248$	$56$	$168$
$7𝒮$		$115248$	$84$	$168$
$8$	$1,2,3,4,6,8$	$743178240$	$120$	$5040$
$9$	$1,2,3,6,7,8$	$37791360$	$90$	$360$
$10$	$1,2,3,4,8$	$4435200$	$120$	$9240$
$11$	$1,2,3,6,8$	$244823040$	$198$	$637560$
$11𝒮$		$244823040$	$253$	$637560$
$12$	$1,2,3,4,8$	$5380145971200$	$156$	$360360$

Table 2. Table 2. Experimental data for the groups H ​ ( k ) , F ​ ( k ) ≤ SL ​ ( 4 , ℤ ) 𝐻 𝑘 𝐹 𝑘 SL 4 ℤ H(k),F(k)\leq\mathrm{SL}(4,\mathbb{Z})

$H$	$M$	Index	$t_A$	$t_I$	$t_B$	$t_M$
$H\left(1\right)$	$2^5{7}^2$	$2^{41}{3}^3{5}^3{7}^{6}19$	$63$	$7-69;27$	$4$	$7$
$H\left(2\right)$	$2^{3}313$	$2^{17}{3}^2{5}^{2}13\cdot 97\cdot 101\cdot {181}^2$	$54$	$10-104;30$	$7$	$1373$
$H\left(3\right)$	$2^{5}7\cdot 199$	$2^{43}{3}^6{5}^{3}7\cdot 11\cdot 19\cdot 13267\cdot 19801$	$62$	$9-90;43$	$7$	$334$
$H\left(4\right)$	$2^{3}7\cdot 607$	$2^{21}{3}^5{5}^{5}7\cdot 13\cdot 19\cdot 101\cdot 7369\cdot 9463$	$90$	$22-65;37$	$19$	$5938$
$H\left(5\right)$	$2^5{5}^{2}409$	$2^{44}{3}^3{5}^{6}17\cdot 31\cdot 55897\cdot 83641$	$73$	$13-107;48$	$11$	$2883$
$H\left(6\right)$	$2^{3}7\cdot 31\cdot 97$	$2^{27}{3}^7{5}^{5}7\cdot 13\cdot 19\cdot 37$	$85$	$14-144;63$	$7$	$308$
		$\cdot 331\cdot 941\cdot 3169$
$H\left(10\right)$	$2^3{5}^{2}7\cdot 919$	$2^{26}{3}^8{5}^8{7}^{2}13\cdot 17\cdot {19}^{2}31$	$93$	$67-390;235$	$14$	$30382$
		$\cdot 37\cdot 101\cdot 113\cdot 163$
$F\left(1\right)$	$2^5{3}^2{7}^2$	$2^{53}{3}^8{5}^4{7}^{6}19$	$77$	$595-707;645$	$3$	$16$
$F\left(2\right)$	$2^4{3}^{2}7\cdot 13\cdot 313$	$2^{38}{3}^9{5}^{6}7\cdot 13\cdot 17\cdot 97\cdot 101\cdot {181}^2$	$78$	$689-831;750$	$11$	$5986$
$F\left(3\right)$	$2^5{3}^{2}7\cdot 29$	$2^{62}{3}^{15}{5}^6{7}^{3}11\cdot 19\cdot 67\cdot 137$	$106$	$718-851;769$	$10$	$10094$
	$\cdot 37\cdot 199$	$\cdot 421\cdot 13267\cdot 19801$
$F\left(4\right)$	$2^4{3}^{3}7\cdot 59\cdot 607$	$2^{37}{3}^{15}{5}^{7}7\cdot 13\cdot 19\cdot 29\cdot 101$	$102$	$719-899;798$	$19$	$74079$
		$\cdot 1741\cdot 7369\cdot 9463$
$F\left(5\right)$	$2^5{3}^3{5}^{2}7$	$2^{66}{3}^{15}{5}^{10}7\cdot 17\cdot 31\cdot 2521$	$139$	$700-1010;881$	$27$	$129470$
	$\cdot 71\cdot 409$	$\cdot 55897\cdot 83641$

Table 3. Table 3. Experimental data for the groups K ​ ( a , b , m ) 𝐾 𝑎 𝑏 𝑚 K(a,b,m) and C ​ ( p , q ) 𝐶 𝑝 𝑞 C(p,q)

Group	Degree	Primes	$M$	$t_A$	$t_I$	$t_M$
$K(2,2,275)$	$4$	$5,11$	$5^{2}11$	$101$	$1-3;1$	$8$
$K(2,3,441)$	$6$	$3,7$	$3^3{7}^2$	$37951$	$4-47;17$	$107$
$K(3,2,8959)$	$6$	$17,31$	${17}^{2}31$	$39873$	$8-43;28$	$3946$
$K(2,4,100)$	$8$	$2,5$	$2^4{5}^2$		$17-96;53$	$956$
$K(3,3,11979)$	$9$	$3,11$	$3^3{11}^3$		$81-246;180$	$4283$
$C(x^4-x+1,x^4+5x^3-x^2+1)$	$4$	$11,61$	$11\cdot 61$	$58$	$3-26;8$	$2131$
$C(x^6+2x^4+x+1,x^6-x^2+1)$	$6$	$7,23$			$12-305;73$
$C(x^8+x+1,x^8-x+1)$	$8$	$2$	$2^2$		$52-368;150$	$10$
$C(x^8+2x+1,x^8+x^4+1)$	$8$	$2,3,5$	$2^{4}3\cdot 5$		$33-1982;505$	$35813$

Equations

$$\mathfrak{sl}(n,\mathbb{F})=\{x\in \mathrm{Mat}(n,\mathbb{F})\mid \mathrm{trace}(x)=0\}$$

$$\rho_{k}(a)=\left(\begin{array}[]{cccc}k(3-4k+4k^{2})&-1-4k-8k^{2}+16k^{3}-16k^{4}&0&0\\ \vspace{-8pt}\hfil\\ 1-k+k^{2}&-1-3k+4k^{2}-4k^{3}&0&0\\ \vspace{-8pt}\hfil\\ k(1-2k+2k^{2})&-3-4k-2k^{2}+8k^{3}-8k^{4}&1&0\\ \vspace{-8pt}\hfil\\ 2(1-k+k^{2})&-2(1+2k-4k^{2}+4k^{3})&0&1\end{array}\right),$$

$$\rho_{k}(b)=\left(\begin{array}[]{ccrr}1&\,0&-4&0\\ 0&\,1&0&-1\\ 0&\,0&-1&-1\\ 0&\,0&1&0\end{array}\right).$$

Full text

1. Introduction

The strong approximation theorem (SAT) is a milestone of linear group theory and its applications [17, Window 9]. It has come to play a similarly important role in computing with linear groups [4].

Let $H$ be a finitely generated subgroup of $\mathrm{SL}(n,\mathbb{Z})$ that is Zariski dense in $\mathrm{SL}(n,\mathbb{C})$. Then SAT asserts that $H$ is congruent to $\mathrm{SL}(n,p)$ for all but a finite number of primes $p\in\mathbb{Z}$. Therefore, we can describe the congruence quotients of $H$ modulo all primes. Moreover, we can describe the congruence quotients of $H$ modulo all positive integers if $n>2$ (see [4, Section 4.1]).

The congruence quotients of $H$ provide important information about $H$; especially when $H$ is arithmetic, i.e., of finite index in $\mathrm{SL}(n,\mathbb{Z})$. In that case, the set $\Pi(H)$ of all primes $p$ such that $H\not\equiv\mathrm{SL}(n,p)$ modulo $p$ is (apart from some exceptions for $p=2$ and $n\leq 4$) the set of primes dividing the level of $H$, defined to be the level of the unique maximal principal congruence subgroup in $H$ [5, Section 2]. If $H$ is thin, i.e., dense but of infinite index in $\mathrm{SL}(n,\mathbb{Z})$, then we consider the arithmetic closure $\mathrm{cl}(H)$ of $H$: this is the intersection of all arithmetic groups in $\mathrm{SL}(n,\mathbb{Z})$ containing $H$ [5, Section 3]. Note that $\Pi(H)=\Pi(\mathrm{cl}(H))$ determines the level of $\mathrm{cl}(H)$ just as it does when $H$ is arithmetic. The level is a key component of subsequent algorithms for computing with arithmetic subgroups, such as membership testing and orbit-stabilizer algorithms [7].

In [5, Section 3.2] and [4], we developed algorithms to compute $\Pi(H)$ when $n$ is prime or $H$ has a known transvection. This paper presents a complete solution: practical algorithms to compute $\Pi(H)$ for arbitrary finitely generated dense $H\leq\mathrm{SL}(n,\mathbb{Z})$, $n\geq 2$. We also give a characterization of density that allows us to compute $\Pi(H)$ without preliminary testing of density (although this can certainly be done; see [5, Section 5] and [6]). Our methods extend in a straightforward manner to handle input $H\leq\mathrm{SL}(n,\mathbb{Q})$.

As in [4], we rely on the classification of maximal subgroups of $\mathrm{SL}(n,p)$. Specifically, we follow the proof of SAT in [17, Window 9, Theorem 10], which credits C. R. Matthews, L. N. Vaserstein, and B. Weisfeiler. In Section 2 we prove results about maximal subgroups of $\mathrm{SL}(n,p)$ that are needed for the main algorithms. Then Section 3 provides methods to compute $\Pi(H)$ for dense $H\leq\mathrm{SL}(n,\mathbb{Q})$. In Section 4 we outline the algorithms, and in Section 5 demonstrate their practicality.

We now fix some basic terms and notation. Let $S=\{g_{1},\ldots,g_{r}\}\subseteq\mathrm{SL}(n,\mathbb{Q})$ and $H=\langle S\rangle$. Then $R$ is the ring (localization) $\frac{1}{\mu}\mathbb{Z}$ generated by the entries of the $g_{i}$ and $g_{i}^{-1}$; here $\mu$ is a positive integer. Note that $R$ depends only on $H$, not on the choice of generating set $S$ for $H$. For $m$ coprime to $\mu$, the congruence homomorphism $\varphi_{m}$ induced by natural surjection $\mathbb{Z}\rightarrow\mathbb{Z}_{m}=\mathbb{Z}/m\mathbb{Z}$ maps $\mathrm{SL}(n,R)$ onto $\mathrm{SL}(n,\mathbb{Z}_{m})$. Let $\Pi(H)$ be the set of all primes $p$ (not dividing $\mu$) such that $\varphi_{p}(H)\neq\mathrm{SL}(n,p)$. Overlining will denote the image modulo a prime $p$ of an element of $R$ or a matrix or set of matrices over $R$. In particular, $\bar{H}=\langle\bar{S}\rangle=\varphi_{p}(H)$. If $\bar{h}\in\bar{H}$ is given as a word $\Pi_{i}{\bar{g}_{j_{i}}^{e_{i}}}$ in $\bar{S}$, then the ‘lift’ of $\bar{h}$ is its preimage $h=\Pi_{i}{g_{j_{i}}^{e_{i}}}$.

Throughout, $\mathbb{F}$ is a field, $\mathbb{F}_{p}$ is the field of size $p$, $\mathrm{Mat}(n,\mathbb{F})$ is the $\mathbb{F}$-algebra of $n\times n$ matrices over $\mathbb{F}$, and $1_{n}\in\mathrm{Mat}(n,\mathbb{F})$ is the identity matrix. We write $\langle G\rangle_{D}$ for the enveloping algebra of $G\leq\mathrm{GL}(n,\mathbb{F})$ over a subring $D\subseteq\mathbb{F}$.

2. Maximality of subgroups in $\mathrm{SL}(n,p)$

Let $G\leq\mathrm{SL}(n,p)$. We show how to recognize when $G$ is not in any maximal subgroup of $\mathrm{SL}(n,p)$, i.e., when $G=\mathrm{SL}(n,p)$. Our approach, which characterizes maximal subgroups by means of the adjoint representation, is motivated by [17, Window 9, Section 2].

We identify the adjoint module for $\mathrm{SL}(n,\mathbb{F})$ with the $\mathbb{F}$-space

[TABLE]

of dimension $n^{2}-1$ on which $\mathrm{SL}(n,\mathbb{F})$ acts by conjugation. Let $\mathrm{ad}:\mathrm{SL}(n,\mathbb{F})\rightarrow\mathrm{GL}(n^{2}-1,\mathbb{F})$ be the corresponding linear representation.

The set of maximal subgroups of $\mathrm{SL}(n,p)$ is the union of Aschbacher classes $\mathscr{C}_{1},\ldots,\mathscr{C}_{8},\mathscr{S}$ (see [1] and [17, p. 397]). The classes $\mathscr{C}_{4}$ and $\mathscr{C}_{7}$ involve tensor products, for which we adopt the following convention. If $H_{1}\leq\mathrm{GL}(a,\mathbb{F})$ and $H_{2}\leq\mathrm{GL}(b,\mathbb{F})$ then $H_{1}\times H_{2}$ acts on $\mathbb{F}^{a}\otimes\mathbb{F}^{b}$. The associated matrix representation of degree $ab$ has $(h_{1},h_{2})\in H_{1}\times H_{2}$ acting as the matrix Kronecker product $h_{1}\dot{\times}h_{2}$. The group generated by these Kronecker products is denoted $H_{1}\otimes H_{2}$.

Proposition 2.1**.**

Let $G$ be a proper absolutely irreducible subgroup of $\mathrm{SL}(n,p)$ such that $\mathrm{ad}(G)$ is irreducible. Then $G$ lies in a maximal subgroup in ${\mathscr{C}}_{6}\cup\mathscr{S}$.

Proof.

Since $G$ is absolutely irreducible, it cannot be in a subgroup in ${\mathscr{C}}_{1}$. Class ${\mathscr{C}}_{5}$ is irrelevant over a field of prime size. For each of the remaining Aschbacher classes other than ${\mathscr{C}}_{6}$ or $\mathscr{S}$, we identify a proper submodule $T$ of the adjoint module $A$ for $\mathrm{SL}(n,p)$.

• ${\mathscr{C}}_{2}$.

A maximal subgroup lies in $W=\mathrm{GL}(a,p)\wr S_{b}$ with $n=ab$. Let $T\leq A$ be the subspace spanned by block matrices with $b$ blocks from $\{1_{a},0_{a},-1_{a}\}$ and zero trace. Clearly $T$ is preserved under conjugation by $W$ and has dimension $b-1$.

• ${\mathscr{C}}_{3}$.

A maximal subgroup here has a normal subgroup $N\cong\mathrm{SL}(a,p^{b})$ with $n=ab$, $1<a,b<n$. Each ‘entry’ of $N$ is a $b\times b$ submatrix. The set of matrices in the center of $N$ with trace [math] is a proper submodule of $A$.

• ${\mathscr{C}}_{4}$.

A maximal subgroup $L$ is $\mathrm{SL}(a,p)\otimes\mathrm{SL}(b,p)$ for some $a,b<n$ such that $n=ab$. If $x\in\mathfrak{sl}(a,p)$ and $y=x\dot{\times}1_{b}$ then $\mathrm{trace}(y)=0$ and thus $y\in A$. Let $T$ be the space spanned by all such products. Then $L$ acts on $T$ by the adjoint action of the $\mathrm{SL}(a,p)$-part of elements on the $x$-components of such products. Thus $T\leq A$ is invariant under $L$, so is a proper submodule of $A$.

• ${\mathscr{C}}_{7}$.

We use an argument similar to the preceding one. Here a maximal subgroup is generated by $\mathrm{Sym}(b)$ and $\mathrm{SL}(a,p)\otimes\cdots\otimes\mathrm{SL}(a,p)$ with $b$ factors, where $n=a^{b}$ and $1<a,b<n$. Let $T$ be the subspace of $A$ spanned by all Kronecker products of length $b$ with every factor $1_{a}$ except for one, drawn from the adjoint module of $\mathrm{SL}(a,p)$. Then $T$ is invariant under action by the maximal subgroup.

• ${\mathscr{C}}_{8}$.

A maximal subgroup that stabilizes a form preserves its own adjoint module (see, e.g., [17, p. 398] or [11, Section 1.4.3]), which cannot be $A$. ∎

Remark 2.2*.*

(Cf. [17, p. 392].) Even if $\mathrm{ad}(G)$ is absolutely irreducible, $G$ could still be in a maximal subgroup in $\mathscr{C}_{6}$. For example, $\mathrm{SL}(8,5)$ contains the maximal subgroup $4\circ 2^{1+6}.\mathrm{Sp}_{6}(2)\in\mathscr{C}_{6}$ which acts absolutely irreducibly on $A$; see [3, p. 399].

Theorem 2.3**.**

There exists a function $f$, depending only on the degree $n$, such that $|G|\leq f(n)$ for any proper absolutely irreducible subgroup $G$ of $\mathrm{SL}(n,p)$ such that $\mathrm{ad}(G)$ is irreducible.

Proof.

(Cf. [17, p. 398]). By [3, Section 2.2.6], $L\leq\mathrm{SL}(n,p)$ in $\mathscr{C}_{6}$ has order bounded by a function of $n$ only. By Proposition 2.1, then, let $L\in\mathscr{S}$. That is, $L=N_{\mathrm{SL}(n,p)}(K)$ with $K\leq\mathrm{SL}(n,p)$ simple non-abelian and $C_{L}(K)=\langle 1_{n}\rangle$. As $L$ is embedded in $\mathrm{Aut}(K)$, a bound on $|K|$ implies a bound on $|L|$.

By the classification of finite simple groups, $K$ can be alternating, or of Lie type, or sporadic. Sporadic groups are of course bounded in order.

If $K\cong\mathrm{Alt}(k)$ then [9, Theorem 5.7A, corrected] shows that $n\geq\frac{2k-6}{3}$; i.e., for fixed $n$, the permutation degree $k$ and hence $|K|$ is bounded.

Now let $K=Y_{l}(r^{e})$ for a Lie class $Y$, Lie rank $l$, and $r$ prime. If $r\not=p$ then [22, Table 1] gives lower bounds for the smallest coprime degree $n$ in which $K$ has a faithful projective representation. These bounds are functions $a(l,r^{e})$, independent of $p$, such that $a(l,r^{e})\to\infty$ as $\l\to\infty$ or $r^{e}\to\infty$. Thus, in bounded degree $n$, only a finite number (up to isomorphism) of groups $Y_{l}(r^{e})$ are candidates for $K$.

If $r=p$ then [17, p. 398] shows that $K$ and $L$ must be in a proper connected algebraic subgroup, and so do not act irreducibly on the adjoint module $A$. ∎

Corollary 2.4**.**

Let $G\leq\mathrm{SL}(n,p)$, and let $f(n)$ be as in Theorem 2.3. If $\mathrm{ad}(G)$ is absolutely irreducible and $|G|>f(n)$ then $G=\mathrm{SL}(n,p)$.

Proof.

Working over the algebraic closure of $\mathbb{F}_{p}$, suppose that $G$ is block upper triangular with main diagonal $(G_{1},G_{2})$ where $G_{i}$ has degree $n_{i}<n$. Then $\mathrm{ad}(G)$ leaves invariant the subspace of the adjoint module consisting of all block upper triangular matrices with main diagonal $(x,0_{n_{2}})$, where $\mathrm{trace}(x)=0$. Hence $G$ must be absolutely irreducible. By Theorem 2.3, $G=\mathrm{SL}(n,p)$. ∎

Remark 2.5*.*

Theorem 2.3 and Corollary 2.4 remain valid if we let $f(n)$ be a bound on $\mathrm{exp}(G)$, or a bound on the largest order of an element of $G$.

Using the formulae for the smallest representation degree of alternating groups, and of Lie-type groups in cross-characteristic, it would be possible to give a rough upper estimate of $f(n)$. We do not attempt this. In Section 5.1, we instead use the tables of [3, Chapter 8] to give tight values for $f(n)$ in degrees $n\leq 12$, extending the values in [4, Remark 3.3].

3. Realizing strong approximation computationally

Let $H$ be a dense subgroup of $\mathrm{SL}(n,R)$, $R=\frac{1}{\mu}\mathbb{Z}$. By Corollary 2.4 and Remark 2.5, if $\mathrm{ad}(\varphi_{p}(H))$ is absolutely irreducible and $f(n)$ is exceeded by $\varphi_{p}(H)$, then $\varphi_{p}(H)=\mathrm{SL}(n,p)$. This result, and a well-known equivalent statement of density, comprise the background for our main algorithm.

Input groups for all the algorithms are finitely generated. Sometimes we write input as a finite generating set, or as the group itself.

3.1. Preliminaries

We start by giving two auxiliary procedures.

3.1.1. Bounded order test

The first auxiliary procedure is a slight generalization of the one in [4, Section 2.1].

Lemma 3.1**.**

If $k$ is a positive integer and $H\leq\mathrm{GL}(n,R)$ is infinite, then $\varphi_{p}(H)$ has an element of order greater than $k$ for almost all primes $p$.

Proof.

The proof is the same as in [4, Section 2.1]. ∎

Lemma 3.2**.**

Suppose that $H\leq\mathrm{SL}(n,R)$ and $\varphi_{p}(H)=\mathrm{SL}(n,p)$ for some prime $p$. If $n\geq 3$ or $p>2$ then $H$ is infinite.

Proof.

See [4, Lemma 2.1]; a finite subgroup of $\mathrm{SL}(n,R)$ can be conjugated into $\mathrm{SL}(n,\mathbb{Z})$. ∎

The procedure PrimesForOrder$(H,k)$ accepts an infinite subgroup $H\leq\mathrm{GL}(n,R)$ and a positive integer $k$, and returns the finite set of all primes $p$ such that $\varphi_{p}(H)$ has maximal element order at most $k$. This output obviously contains all primes $p$ such that $|\varphi_{p}(H)|\leq k$.

3.1.2. Testing absolute irreducibility

For this subsection, we refer to [8, p. 401] and [5, Section 3.2].

Let $N$ be the normal closure $\langle X\rangle^{H}$ where $X$ is a finite subset of a finitely generated group $H\leq\mathrm{GL}(n,\mathbb{F})$. The procedure BasisAlgebraClosure$(X,S)$ computes a basis $\{A_{1},\ldots,A_{m}\}$ of $\langle N\rangle_{\mathbb{F}}$, thereby deciding whether $N$ is absolutely irreducible, i.e., whether $m=n^{2}$.

The procedure PrimesForAbsIrreducible from [4, Section 2.2] will operate in the same way for absolutely irreducible $H\leq\mathrm{GL}(n,R)$: it accepts a generating set $S$ of $H$, and returns the (finite) set of primes $p$ such that $\varphi_{p}(H)$ is not absolutely irreducible. The first step is to compute a basis of $\langle H\rangle_{\mathbb{Q}}$. By making a small adjustment, we get PrimesForAbsIrreducible$(X,S)$; for absolutely irreducible $N=\langle X\rangle^{H}$, it returns the primes $p$ such that $\varphi_{p}(N)$ is not absolutely irreducible.

If $\bar{H}=\varphi_{p}(H)$ is absolutely irreducible (e.g., $\bar{H}=\mathrm{SL}(n,p)$) and $\{\bar{A}_{1},\ldots,\bar{A}_{n^{2}}\}$ is a basis of $\langle\bar{H}\rangle_{\mathbb{F}_{p}}$, then $H$ is absolutely irreducible and $\{A_{1},\ldots,A_{n^{2}}\}$ is a basis of $\langle H\rangle_{\mathbb{Q}}$. Thus, we can simplify PrimesForAbsIrreducible by computing a basis of the enveloping algebra over a finite field and then lifting it to a basis of $\langle H\rangle_{\mathbb{Q}}$ (cf. [4, Section 2.2]).

3.2. Density and strong approximation

Now we give elementary proofs of some properties of dense groups, including strong approximation (cf. [16], [17, Theorem 9, p. 396], and [4, Corollary 3.10]).

The following is fundamental.

Proposition 3.3** ([21, p. 22]).**

A subgroup $H$ of $\mathrm{SL}(n,\mathbb{C})$ is dense if and only if $H$ is infinite and $\mathrm{ad}(H)$ is absolutely irreducible.

Let $H$ be a finitely generated subgroup of $\mathrm{SL}(n,R)$.

Lemma 3.4**.**

$\varphi_{p}(\mathrm{ad}(H))=\mathrm{ad}(\varphi_{p}(H))$* for all primes $p$ (coprime to $\mu$).*

Corollary 3.5**.**

If $\mathrm{ad}(H)$ is absolutely irreducible then $\mathrm{ad}(\varphi_{p}(H))$ is absolutely irreducible for almost all primes $p$.

Lemma 3.6**.**

If $\varphi_{p}(H)=\mathrm{SL}(n,p)$ then $\mathrm{ad}(H)$ is absolutely irreducible.

Proof.

By Lemma 3.4, $\varphi_{p}(\mathrm{ad}(H))=\mathrm{ad}(\mathrm{SL}(n,p))$. Since the latter is absolutely irreducible, its preimage $\mathrm{ad}(H)$ is too. ∎

Proposition 3.7**.**

The following are equivalent.

• (i)

$H$* is dense.*

• (ii)

$H$* surjects onto $\mathrm{SL}(n,p)$ for almost all primes $p$.*

• (iii)

$H$* surjects onto $\mathrm{SL}(n,p)$ for some prime $p>2$.*

Proof.

Suppose that (i) holds. Then by Lemma 3.1, Proposition 3.3, and Corollary 3.5, $\mathrm{ad}(\varphi_{p}(H))$ is absolutely irreducible and $|\varphi_{p}(H)|>f(n)$ for almost all primes $p$. By Corollary 2.4, $\varphi_{p}(H)=\mathrm{SL}(n,p)$ for such $p$.

Suppose that (iii) holds. By Lemma 3.6, $\mathrm{ad}(H)$ is absolutely irreducible, and by Lemma 3.2, $H$ is infinite. Therefore $H$ is dense by Proposition 3.3. ∎

4. The main algorithms

In this section we combine results from Sections 2 and 3 to obtain the promised algorithms to compute $\Pi(H)$ for dense groups $H$. These consist of the main procedure, a variation aimed at improved performance, and an alternative that could be preferable in certain degrees.

Our main procedure, based on Corollary 2.4, follows.

PrimesNonSurjectiveSL

Input: a finite generating set of a dense group $H\leq\mathrm{SL}(n,R)$.

Output: $\Pi(H)$.

1. $\mathcal{P}:=$ PrimesForOrder$(H,f(n))\,\cup$ PrimesForAbsIrreducible$(\mathrm{ad}(H))$.

2. Return $\{p\in\mathcal{P}\mid\varphi_{p}(H)\not=\mathrm{SL}(n,p)\}$.

Step 2 is performed via standard methods for matrix groups over finite fields (e.g., as in [18]).

Proposition 4.1**.**

PrimesNonSurjectiveSL* returns $\Pi(H)$ for dense input $H$.*

Proof.

Proposition 3.3 implies that Step 1 terminates. Then $\varphi_{p}(H)=\mathrm{SL}(n,p)$ for any $p\notin\mathcal{P}$ by Corollary 2.4 and Lemma 3.4. ∎

4.1. Testing irreducibility

Testing absolute irreducibility of $\mathrm{ad}(H)$ for $H$ of degree $n$ entails computation in degree about $n^{4}$, which is comparatively expensive. However, Theorem 2.3 offers a way to bypass this test. That is, we adapt Meataxe ideas [13, 20] to determine all primes modulo which the adjoint representation is merely reducible. For simplicity, the discussion will be restricted to $R=\mathbb{Z}$.

Recall the following special case of Norton’s criterion for the natural module $V$ of a matrix algebra $\mathcal{A}$.

Suppose that $B\in\mathcal{A}$ has rank $\mathrm{rk}(B)=n-1$. Assume that $v\mathcal{A}=V$ for some non-zero $v$ in the nullspace of $B$, and $\mathcal{A}w=V^{\perp}$ and for some non-zero $w^{\top}$ in the nullspace of $B^{\top}$. Then $V$ is irreducible.

Now let $\mathcal{A}\subseteq\mathrm{Mat}(n,\mathbb{Q})$ be a $\mathbb{Z}$-algebra, and suppose that the following hold.

1. (1)

We have found $B\in\mathcal{A}$ such that $\mathrm{rk}(B)=n-1$. 2. (2)

For a non-zero $v$ in the nullspace of $B$, the $\mathbb{Z}$-span $v\mathcal{A}$ contains $n$ linearly independent vectors $v_{1},\ldots,v_{n}$. 3. (3)

For a non-zero $w^{\top}$ in the nullspace of $B^{\top}$, there are $n$ linearly independent vectors $w_{1},\ldots,\allowbreak w_{n}\in\mathcal{A}w$.

Norton’s criterion, applied to the above configuration modulo $p$, shows that $\varphi_{p}(\mathcal{A})$ is irreducible unless

• $\mathrm{rk}(\varphi_{p}(B))<n-1$, or
• $\varphi_{p}(v_{1}),\ldots,\varphi_{p}(v_{n})$ are linearly dependent, or
• $\varphi_{p}(w_{1}),\ldots,\varphi_{p}(w_{n})$ are linearly dependent.

To find (a finite superset of) the set of primes $p$ for which $\varphi_{p}(\mathcal{A})$ is reducible, we form the union of three sets, namely the prime divisors of $\det(M_{1})$, $\det(M_{2})$, and $\det(M_{3})$, where

• $M_{1}$ is a full rank $(n-1)\times(n-1)$ minor of $B$ (modulo other primes, $B$ has rank $n-1$),
• $M_{2}$ is the matrix with rows $v_{1},\ldots,v_{n}$ (modulo other primes, $v$ spans the whole module),
• $M_{3}$ is the matrix with rows $w_{1},\ldots,w_{n}$.

To make this into a concrete test PrimesForIrreducible, let $\mathcal{A}=\langle\mathrm{ad}(H)\rangle_{\mathbb{Z}}$. Take a small number (say, $100$) of random $\mathbb{Z}$-linear combinations $B\in\mathcal{A}$ until a $B$ of rank $n-1$ is detected. Although we do not have a justification that such elements occur with sufficient frequency, they seem to (as observed in [19]); in every experiment so far we found such a $B$. (Also note that there are irreducible $H$ such that $\langle H\rangle_{\mathbb{Q}}$ does not have an element of rank $n-1$; but if $H$ is absolutely irreducible then such elements always exist.)

We now state a version of PrimesNonSurjectiveSL that may have improved performance in many situations (see Section 5).

PrimesNonSurjectiveSL, modified.

1. If PrimesForIrreducible confirms that $\mathrm{ad}(H)$ is irreducible then

    $\mathcal{P}:=$ PrimesForOrder$(H,f(n))\,\cup$ PrimesForAbsIrreducible$(H)$

           $\cup$ PrimesForIrreducible$(\mathrm{ad}(H))$;

else

    $\mathcal{P}:=$ PrimesForOrder$(H,f(n))\,\cup$ PrimesForAbsIrreducible$(\mathrm{ad}(H))$.

2. Return $\{p\in\mathcal{P}\mid\varphi_{p}(H)\not=\mathrm{SL}(n,p)\}$.

Proposition 4.2**.**

The above modification of PrimesNonSurjectiveSL terminates, returning $\Pi(H)$ for input dense $H$.

Proof.

This follows from Theorem 2.3 and Proposition 4.1. ∎

Remark 4.3*.*

Suppose that PrimesForIrreducible completes, i.e., $\mathrm{ad}(H)$ is confirmed to be irreducible. Then $H$ is dense if it is infinite and absolutely irreducible. This gives a more efficient density test than the procedure IsDenseIR2 in [6].

4.2. Individual Aschbacher classes

Some Aschbacher classes may not occur in a given degree. For example, the tensor product classes $\mathscr{C}_{4}$ and $\mathscr{C}_{7}$ are empty in degree $4$. Consonant with the approach of [4], we show how to determine the primes $p$ such that $\varphi_{p}(H)$ lies in a group in $\mathscr{C}_{i}\not\in\{\mathscr{C}_{4},\mathscr{C}_{7},\mathscr{S}\}$, using tests that do not involve $\mathrm{ad}(H)$. The following is vital.

Lemma 4.4**.**

Let $H\leq\mathrm{SL}(n,\mathbb{Q})$ be dense. If $N\unlhd H$ is non-scalar then $N$ is dense, thus absolutely irreducible.

Proof.

This follows from Proposition 3.7: since $N$ is non-scalar, $\varphi_{p}(N)$ is a normal non-scalar subgroup of $\mathrm{SL}(n,p)$ for almost all primes $p$. ∎

4.2.1. Testing imprimitivity

Suppose that $H\leq\mathrm{GL}(n,\mathbb{F})$ is imprimitive, so $H\leq\mathrm{GL}(a,\mathbb{F})\wr\mathrm{Sym}(b)$ for some $a$, $b>1$ such that $n=ab$. If $\mathrm{Sym}(b)$ has exponent $k$ then $\langle h^{k}:h\in H\rangle\leq\mathrm{GL}(a,\mathbb{F})^{b}$ is reducible. Hence we have the following procedure.

PrimesForPrimitive

Input: dense $H=\langle S\rangle\leq\mathrm{SL}(n,\mathbb{Q})$.

Output: the set of primes $p$ for which $\varphi_{p}(H)$ is imprimitive.

1. Select $h\in H$ such that $h^{e}$ is non-scalar, where $e=\exp(\mathrm{Sym}(n))$.

2. $\mathcal{P}:={\tt PrimesForAbsIrreducible}(h^{e},S)$.

3. Return all $p\in\mathcal{P}$ such that $\varphi_{p}(H)$ is imprimitive.

Once more [18] is used in implementing the last step. Lemma 4.4 guarantees termination and correctness of the output.

If we happen to know a prime $p$ such that $\varphi_{p}(H)=\mathrm{SL}(n,p)$, then PrimesForPrimitive simplifies in the familiar way (i.e., by computing in a congruence image and then lifting).

PrimesForPrimitive, modified.

1. Let $p$ be a prime for which $\varphi_{p}(H)=\mathrm{SL}(n,p)$.

2. Find $n^{2}$ elements $h_{i}\in H$ such that the $\varphi_{p}(h_{i}^{k})$ span $\mathrm{Mat}(n,\mathbb{F}_{p})$, where $k:=\exp(\mathrm{Sym}(n))$.

3. Return all $p\in$ PrimesForAbsIrreducible$(h_{1}^{k},\ldots,h_{n^{2}}^{k})$ such that $\varphi_{p}(H)$ is

     imprimitive.

The $h_{i}$ exist by Step 1 and Lemma 4.4.

4.2.2. Testing for field extensions

The second derived subgroup $G^{(2)}$ of $G\in\mathscr{C}_{3}$ is quasisimple and reducible ([3, p. 66] and [14, §4.3]). Accordingly, PrimesForReducibleSecondDerived selects a non-scalar double commutator $g$ in the dense group $H$ then returns PrimesForAbs-Irreducible$(g,S)$. By Lemma 4.4, this will yield all primes modulo which $H$ is in a group in $\mathscr{C}_{3}$.

If we know a prime $p$ such that $\varphi_{p}(H)=\mathrm{SL}(n,p)$ then PrimesForReducibleSecond-Derived can be modified along the lines of our modification of PrimesForPrimitive. We search for double commutators (rather than $k$th powers) in $\varphi_{p}(H)$ that span $\mathrm{Mat}(n,\mathbb{F}_{p})$; these exist because $\varphi_{p}(H)=\mathrm{SL}(n,p)$ is perfect (if $n>2$ or $p>3$).

4.2.3. Excluding classes

For prime $n$ or $n=4$, the results of Sections 4.2.1 and 4.2.2, together with those of [4], enable us to avoid $\mathrm{ad}(H)$ in computing $\Pi(H)$. We use the procedures below to rule out individual Aschbacher classes in those degrees.

$\mathscr{C}_{1}$**: **

PrimesForAbsIrreducible.

$\mathscr{C}_{2}$**: **

PrimesForPrimitive.

$\mathscr{C}_{3}$**: **

PrimesForReducibleSecondDerived.

$\mathscr{C}_{6}$**, $\mathscr{S}$: **

PrimesForOrder.

$\mathscr{C}_{8}$**: **

PrimesForSimilarity, as in [4, Section 2.5].

5. Experiments

Our algorithms have been implemented in GAP [10], enhancing previous functionality for computing with dense groups [6]. The software can be accessed at

http://www.math.colostate.edu/~hulpke/arithmetic.g

We report on experiments undertaken with the implementation. One major task is computing all congruence quotients of a finitely generated dense group $H\leq\mathrm{SL}(n,\mathbb{Z})$ from $\Pi(H)$, as explained in [4, Section 4.1].

5.1. Explicit order bounds

We will let $f(n)$ be a bound on the largest element order for the absolutely irreducible groups of degree $n$ in $\mathscr{C}_{6}\cup\mathscr{S}$ that are irreducible in their adjoint representation. The tables in [3, Section 8] furnish bounds for $n\leq 12$. We construct an example of each such group in $\mathscr{C}_{6}\cup\mathscr{S}$ using the Magma [2] implementation that accompanies [3]. Then we use GAP to calculate conjugacy class representatives and their orders.

For completeness, Table 1 gives maximal subgroup order, maximal element order, and the least common multiple of exponents. The column ‘Geometric’ lists the number $i$ of each Aschbacher class $\mathscr{C}_{i}$ that can occur.

We include, for degrees $n\in\{3,4,5,7,11\}$, the element order bounds from [4] for all groups in $\mathscr{C}_{6}\cup\mathscr{S}$. The rows with these bounds have $n\mathscr{S}$ in the Degree column. For $n=3,4,5$ the bounds agree, and so we have omitted the row beginning with $n$.

5.2. Implementation and experimental results

5.2.1. Triangle groups

Let $\Delta(3,3,4)$ be the triangle group $\langle a,b\mid a^{3}=b^{3}=(ab)^{4}=1\rangle$. In [15, Theorem 1.1], a four-dimensional real representation of $\Delta(3,3,4)$ is defined by

[TABLE]

[TABLE]

Let $H(k)=\langle\rho_{k}(a),\rho_{k}(b)\rangle$. If $k\in\mathbb{Z}$ then $H(k)\leq\mathrm{SL}(4,\mathbb{Z})$.

Let $F(k)$ be the image under $\rho_{k}$ of $\langle[a,b],[a,b^{-1}]\rangle$. Calculations by D. F. Holt (personal communication) using kbmag [12] establishes that the latter is a free subgroup of $\Delta(3,3,4)$. All groups $H(k)$ (resp. $F(k)$) are $2$-generated, and of the same structure; as $k$ varies we are just changing the size of matrix entries. Note that the entries of the generators of $F(k)$ have roughly twice the number of digits as those of $H(k)$. Our experiments justify that $H(k)$, $F(k)$ are dense (for $H(k)$ this follows independently from [15]), and non-arithmetic, i.e., thin. As $\mathscr{C}_{4}$ and $\mathscr{C}_{7}$ do not figure in degree $4$, the algorithm from Section 4.2 can be utilized here. This will illustrate the benefit of the improvements in Sections 4.1 and 4.2.

In Table 2, $M$ is the level of $\mathrm{cl}(H)$ and ‘Index’ is $|\mathrm{SL}(4,\mathbb{Z}):\mathrm{cl}(H)|$. We remark that computing $\Pi(H)$, $M$, and indices is not possible with our previous methods [4, 5]. Other columns give runtimes in seconds on a 3.7GHz Xeon E5 (2013 MacPro). Column $t_{A}$ gives the runtime of PrimesNonSurjectiveSL. Column $t_{I}$ gives the time of the Meataxe-based algorithm from Section 4.1. Due to the randomized nature of the Meataxe calculations, timings turned out to be variable. Consequently we give a timing of ten experiments and list minimum, maximum, and average runtime in the format min–max; average. Column $t_{B}$ gives runtimes of the algorithm in Section 4.2 (computing $\Pi(H)$ without $\mathrm{ad}(H)$), and the final column $t_{M}$ is runtime to compute $M$ and Index from $\Pi(H)$.

After computing $M$, we can find all congruence quotients of $H(k)$, and hence a set of finite quotients of $\Delta(3,3,4)$. We see from the results for $k=1$, $2$ that $\Delta(3,3,4)$ has quotients $\mbox{PSL}(4,p)$ for $p>2$. On the other hand, a calculation with the GAP operation GQuotients shows that $\Delta(3,3,4)$ has no quotient isomorphic to $\mbox{PSL}(4,2)$. Furthermore, since $\Delta(3,3,4)$ has quotients isomorphic to $\mathrm{Alt}(10)$, which cannot be a section of a matrix group of degree $4$ over a finite field, $H(k)$ is thin for all $k\in\mathbb{Z}$. The $F(k)$ are thin because they are free.

5.2.2. Other experiments

We used the following constructions of dense groups, including examples that permit tensor decomposition modulo some primes.

• (i)

Let $K(a,b,m)$ be the subgroup of $\mathrm{SL}(ab,\mathbb{Z})$ generated by $\mathrm{SL}(a,\mathbb{Z})\otimes\mathrm{SL}(b,\mathbb{Z})$ and the elementary matrix $mt_{1,a+1}$ (two generators per factor of the Kronecker product).

• (ii)

For distinct monic polynomials $p(x),q(x)\in\mathbb{Z}[x]$ of equal degree $n$, let $C(p,q)$ be the subgroup of $\mathrm{SL}(n,\mathbb{Z})$ generated by the companion matrices $C_{p}$ and $C_{q}$ for $p(x)$ and $q(x)$.

Regarding density of the $K(a,b,m)$, cf. [4, Lemma 3.15]. By [21, Theorem 1.5], $C(p,q)$ is dense if it is non-abelian, $C_{q}$ has infinite order, and $p(x)$ is irreducible with Galois group $\mathrm{Sym}(n)$.

The runtimes in Table 3 have the same interpretation as in Table 2. Some computations with the larger groups did not complete for several hours. In that event, the pertinent column entry is blank. Indices are not listed for space reasons.

5.2.3. Performance

The runtime to find $\Pi(H)$ is roughly proportional to the magnitudes of its elements. In fact, runtime is dominated by tests to ensure that no prime $p$ returned is a false positive, i.e., that the $p$-congruence image really is a proper subgroup of $\mathrm{SL}(n,p)$.

The timings show that the method of Section 4.2 is clearly superior to the default, with the Meataxe-based algorithm performing better unless matrix entries become very large. This pattern becomes more pronounced in larger degrees.

Acknowledgments

We thank Mathematisches Forschungsinstitut Oberwolfach for facilitation of our work through the programme ‘Research in Pairs’ in 2018. A. S. Detinko was supported by Marie Skłodowska-Curie Individual Fellowship grant H2020 MSCA-IF-2015, no. 704910 (EU Framework Programme for Research and Innovation). A. Hulpke was supported by Simons Foundation Collaboration Grant no. 524518 and National Science Foundation grant DMS-1720146.