## 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.

### MSC:

 11E57 Classical groups 20G30 Linear algebraic groups over global fields and their integers

### Citations:

Zbl 0449.20051; Zbl 0239.20065
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### References:

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