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.
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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 $S_2(R)$ 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. |
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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 $S_2(R)$ 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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