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About question 1. The answer is "no". Suppose that every finite group $G$ embeds into $SL_2(R_G)$ for a commutative $R_G$. Take the ultraproduct ${\mathcal G}$ of all $G$. It embeds into the ultraproduct of $SL_2(R_G)$ which is $SL_2(R)$ where $R$ is the ultraproduct of $R_G$. Since $R$ is a commutative ring, every finitely generated subgroups of $S_2(R)$ SL_2(R)$is residually finite by Malcev. But not all finitely generated subgroups of${\mathcal G}$are residually finite. See this paper for example. In case it is not clear how "approximably finite" groups relate to ultraproducts of finite groups, here is a corollary of Theorem 2 from Section 8, Chapter 4 of Malcev's "Algebraic systems": if every finite subset of a group$G$(with induced partial operation) is embedded into a finite group, then$G$is embedded into an ultraproduct of finite groups. (That is actually an easy statement.) Hence "approximably finite" groups are subgroups of ultraproducts of finite groups. The article I gave a link to contains examples of "approximably finite" but non-residually finite groups. 3 added 584 characters in body About question 1. The answer is "no". Suppose that every finite group$G$embeds into$SL_2(R_G)$for a commutative$R_G$. Take the ultraproduct${\mathcal G}$of all$G$. It embeds into the ultraproduct of$SL_2(R_G)$which is$SL_2(R)$where$R$is the ultraproduct of$R_G$. Since$R$is a commutative ring, every finitely generated subgroups of$S_2(R)$is residually finite by Malcev. But not all finitely generated subgroups of${\mathcal G}$are residually finite. See this paper for example. In case it is not clear how "approximably finite" groups relate to ultraproducts of finite groups, here is a corollary of Theorem 2 from Section 8, Chapter 4 of Malcev's "Algebraic systems": if every finite subset of a group$G$(with induced partial operation) is embedded into a finite group, then$G$is embedded into an ultraproduct of finite groups. (That is actually an easy statement.) Hence "approximably finite" groups are subgroups of ultraproducts of finite groups. The article I gave a link to contains examples of "approximably finite" but non-residually finite groups. 2 added 97 characters in body About question 1. The answer is "no". Suppose that every finite group$G$embeds into$SL_2(R_G)$for a commutative$R_G$. Take the ultraproduct${\mathcal G}$of all$G$. It embeds into the ultraproduct of$SL_2(R_G)$which is$SL_2(R)$where$R$is the ultraproduct of$R_G$. Since$R$is a commutative ring, every finitely generated subgroups of$S_2(R)$is residually finite by Malcev. But not all finitely generated subgroups of${\mathcal G}\$ are residually finite. See this paper for example.

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