[Paper Review] On strong approximation for algebraic groups

Andrei S. Rapinchuk|arXiv (Cornell University)|Jul 18, 2012

Advanced Topology and Set Theory6 citations

TL;DR

This paper provides a comprehensive survey of strong approximation in algebraic groups, focusing on classical $S$-arithmetic groups and modern results for Zariski-dense subgroups. It establishes that $\mathrm{SL}_2(\mathbb{Z})$ satisfies strong approximation via elementary matrix generation, while $\mathrm{GL}_2(\mathbb{Z})$ does not, and generalizes these results using Weisfeiler's theorem on adelic closures of Zariski-dense subgroups.

ABSTRACT

This is an expanded version of my talk given at the workshop "Hot Topics: Thin Groups and Super-strong Approximation" (MSRI, Berkeley, February 6-10, 2012).

Motivation & Objective

• To survey known results on strong approximation in algebraic groups, particularly for $S$-arithmetic and Zariski-dense subgroups.
• To clarify the distinction between $\mathrm{SL}_2(\mathbb{Z})$ and $\mathrm{GL}_2(\mathbb{Z})$ in terms of strong approximation, despite similar defining equations.
• To present Weisfeiler's theorem on the adelic closure of Zariski-dense subgroups and its implications for the structure of such subgroups.
• To highlight applications in group theory, especially Lubotzky's alternative and subgroup growth in linear groups.
• To address technical challenges in positive characteristic, particularly in characteristics 2 and 3, via Pink's theory of minimal triples.

Proposed method

• Uses the inverse limit topology on $X(\hat{\mathbb{Z}}) = \varprojlim X(\mathbb{Z}/m\mathbb{Z})$ to define strong approximation via density of $X(\mathbb{Z})$ in $X(\hat{\mathbb{Z}})$.
• Applies the Chinese Remainder Theorem to identify $\hat{\mathbb{Z}} \simeq \prod_p \mathbb{Z}_p$, allowing the use of $p$-adic topologies on $X(\mathbb{Z}_p)$.
• Employs elementary matrix generation in $\mathrm{SL}_2(\mathbb{Z}/m\mathbb{Z})$ to prove surjectivity of reduction maps, establishing strong approximation for $\mathrm{SL}_2$.
• Utilizes the trace ring $A = \mathrm{tr}(\mathrm{Ad} \, \gamma)$ for $\gamma \in \Gamma$ to define a ring of definition for Zariski-dense subgroups.
• Applies Weisfeiler's theorem to show that for a Zariski-dense finitely generated subgroup $\Gamma \subset G(k)$, the adelic closure $\widehat{\Gamma}$ is open in $G(\widehat{A_b})$.
• Uses profinite completion $\widehat{A_b}$ and localization $A_b$ to construct a group scheme over $A_b$ such that $\Gamma'$ embeds densely in $G_{A_b}(\widehat{A_b})$.

Experimental results

Research questions

• RQ1 Under what conditions does the reduction map $\rho_m: \mathrm{SL}_2(\mathbb{Z}) \to \mathrm{SL}_2(\mathbb{Z}/m\mathbb{Z})$ remain surjective for all $m$?
• RQ2 Why does $\mathrm{GL}_2(\mathbb{Z})$ fail to satisfy strong approximation despite a similar defining equation to $\mathrm{SL}_2(\mathbb{Z})$?
• RQ3 How can Weisfeiler's theorem be used to describe the adelic closure of a Zariski-dense subgroup in a simply connected almost simple algebraic group?
• RQ4 What role do trace rings and their localizations play in the structure theory of Zariski-dense subgroups over algebraically closed fields?
• RQ5 How does strong approximation lead to structural dichotomies in linear groups, such as Lubotzky's alternative?

Key findings

• The reduction map $\rho_m: \mathrm{SL}_2(\mathbb{Z}) \to \mathrm{SL}_2(\mathbb{Z}/m\mathbb{Z})$ is surjective for all $m \geq 1$, establishing strong approximation for $\mathrm{SL}_2(\mathbb{Z})$.
• $\mathrm{GL}_2(\mathbb{Z})$ does not satisfy strong approximation because its image in $\mathrm{GL}_2(\mathbb{Z}/m\mathbb{Z})$ is not dense under the profinite topology.
• Weisfeiler's theorem guarantees that for a Zariski-dense finitely generated subgroup $\Gamma \subset G(k)$, there exists a ring $A_b$ such that $\Gamma'$ embeds densely in $G_{A_b}(\widehat{A_b})$.
• For $\Gamma = \langle g_1, \dots, g_d \rangle \subset \mathrm{SL}_n(\mathbb{Z})$, there exists $N = N(d,m,n)$ such that $\rho_p(\Gamma) = \mathrm{SL}_n(\mathbb{F}_p)$ for all primes $p > N$.
• Lubotzky's alternative states that a finitely generated linear group over a field of characteristic zero is either virtually solvable or admits a profinite completion mapping onto $G(\widehat{\mathbb{Z}_\Pi})$ for some finite set of primes $\Pi$.
• In positive characteristic 2 and 3, the trace ring may not be the correct ring of definition; Pink's theory of minimal triples provides the correct framework for such cases.