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**The congruence topology, Grothendieck duality and thin groups.**
*(English)*
Zbl 1448.11080

Let \(\Gamma\) be a finitely generated subgroup of \(\mathrm{GL}_n(\mathbb{Z})\). For a representation \(\rho: \Gamma\to \mathrm{GL}(V)\) on a lattice \(V\), let \(\hat{\rho}:\hat{V}\to \mathrm{GL}(\hat{V})\) be the natural continuous extension to the profinite completions. Let \(\mathrm{Cl}_{\rho}(\Gamma)=\{g\in\hat{\Gamma}\,:\,\hat{\rho}(g)(V)\subseteq V\}\). In [Am. J. Math. 102, 663–689 (1980; Zbl 0449.20051)] the first author proved that \(\mathrm{Cl}(\Gamma):=\bigcap_{\rho}\mathrm{Cl}_{\rho}(\Gamma)\) coincides with the group \(\mathrm{Cl}_{\mathbb{Z}}(\Gamma)\), which is essentially defined in [A. Grothendieck, Manuscr. Math. 2, 375–396 (1970; Zbl 0239.20065)] and called the Grothendieck closure of \(\Gamma\).

The main theorem of the paper under review asserts that the equality \(\Gamma=\mathrm{Cl}(\Gamma)\) holds if and only if \(\Gamma\) has the so-called congruence subgroup property. As a corollary, Conjecture A of [Lubotzky, loc. cit.] is confirmed. That is, if \(\Gamma\) is a congruence subgroup of \(G(\mathbb{Q})\), where \(G\) is a simply connected semisimple algebraic group over \(\mathbb{Q}\), then \(\mathrm{Cl}(\Gamma)=\Gamma\) happens if and only if \(\Gamma\) has the strict congruence subgroup property. In particular, if \(\Gamma=\mathrm{SL}_2(\mathbb{Z})\) or \(\Gamma\) is a finitely generated free group on at least two generators, then \(\mathrm{Cl}(\Gamma)\neq\Gamma\).

A classical theorem of Chevalley says that a homomorphism of complex algebraic groups sends a closed subgroup to a closed subgroup. In the course of proving their main theorem, the authors obtained an analogue of Chevalley’s theorem for congruence subgroups.

The techniques in this paper also have a very interesting application to thin subgroups of \(\mathrm{GL}_n(\mathbb{Z})\). Recall that \(\Gamma\) is called a thin subgroup of \(\mathrm{GL}_n(\mathbb{Z})\) if it is of infinite index in \(G\cap\mathrm{GL}_n(\mathbb{Z})\), where \(G\) is the Zariski closure of \(\Gamma\) in \(\mathrm{GL}_n\). Theorem 0.10 of the present paper provides a purely group theoretic criterion for the thinness: \(\Gamma\) is thin if and only if \(\Gamma\) does not have a finite index subgroup with an infinite abelianization or \(\mathrm{Cl}(\Gamma)/\Gamma\) is not compact.

The main theorem of the paper under review asserts that the equality \(\Gamma=\mathrm{Cl}(\Gamma)\) holds if and only if \(\Gamma\) has the so-called congruence subgroup property. As a corollary, Conjecture A of [Lubotzky, loc. cit.] is confirmed. That is, if \(\Gamma\) is a congruence subgroup of \(G(\mathbb{Q})\), where \(G\) is a simply connected semisimple algebraic group over \(\mathbb{Q}\), then \(\mathrm{Cl}(\Gamma)=\Gamma\) happens if and only if \(\Gamma\) has the strict congruence subgroup property. In particular, if \(\Gamma=\mathrm{SL}_2(\mathbb{Z})\) or \(\Gamma\) is a finitely generated free group on at least two generators, then \(\mathrm{Cl}(\Gamma)\neq\Gamma\).

A classical theorem of Chevalley says that a homomorphism of complex algebraic groups sends a closed subgroup to a closed subgroup. In the course of proving their main theorem, the authors obtained an analogue of Chevalley’s theorem for congruence subgroups.

The techniques in this paper also have a very interesting application to thin subgroups of \(\mathrm{GL}_n(\mathbb{Z})\). Recall that \(\Gamma\) is called a thin subgroup of \(\mathrm{GL}_n(\mathbb{Z})\) if it is of infinite index in \(G\cap\mathrm{GL}_n(\mathbb{Z})\), where \(G\) is the Zariski closure of \(\Gamma\) in \(\mathrm{GL}_n\). Theorem 0.10 of the present paper provides a purely group theoretic criterion for the thinness: \(\Gamma\) is thin if and only if \(\Gamma\) does not have a finite index subgroup with an infinite abelianization or \(\mathrm{Cl}(\Gamma)/\Gamma\) is not compact.

Reviewer: Yong Hu (Guangdong)

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\textit{A. Lubotzky} and \textit{T. N. Venkataramana}, Algebra Number Theory 13, No. 6, 1281--1298 (2019; Zbl 1448.11080)

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