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This paper shows that the answer to 2) is false in the category of finitely presented residually finite groups. As Greg points out, this is different from the category of finitely presented linear groups though.

Addendum: In a paper of Bridson and Miller (which I found from Igor's link to Miller's survey), they show that the isomorphism problem for subgroups of $\Gamma\times\Gamma\times F$ is undecidable, where $\Gamma$ is a particular hyperbolic group (which is free-by-finitely generated) and $F$ is free. As mentioned in the paper, Mosher constructed free-by-surface hyperbolic groups, which therefore could work as $\Gamma$. These groups embed in the mapping class group of the once-punctured surface, so if mapping class groups of once-punctured surfaces are linear, this would answer 2). However, the only mapping class groups known to be linear are the punctured sphere/braid groups and the genus 2 mapping class group.

This paper shows that the answer to 2) is false in the category of finitely presented residually finite groups. As Greg points out, this is different from the category of finitely presented linear groups though.

Addendum: In a paper of Bridson and Miller (which I found from Igor's link to Miller's survey), they show that the isomorphism problem for subgroups of $\Gamma\times\Gamma\times F$ is undecidable, where $\Gamma$ is a particular hyperbolic group (which is free-by-finitely generated) and $F$ is free. As mentioned in the paper, Mosher constructed surface-by-free hyperbolic groups, which therefore could work as $\Gamma$. These groups embed in the mapping class group of the once-punctured surface, so if mapping class groups of once-punctured surfaces are linear, this would answer 2). However, the only mapping class groups known to be linear are the punctured sphere/braid groups and the genus 2 mapping class group.

This paper shows that the answer to 2) is false in the category of finitely presented residually finite groups. As Greg points out, this is different from the category of finitely presented linear groups though.

Addendum: In a paper of Bridson and Miller (which I found from Igor's link to Miller's survey), they show that the isomorphism problem for subgroups of $\Gamma\times\Gamma\times F$ is undecidable, where $\Gamma$ is a particular hyperbolic group (which is free-by-finitely generated) and $F$ is free. As mentioned in the paper, Mosher constructed free-by-surface hyperbolic groups, which therefore could work as $\Gamma$. These groups embed in the mapping class group of the once-punctured surface, so if mapping class groups of once-punctured surfaces are linear, this would answer 2). However, the only mapping class groups known to be linear are the punctured sphere/braid groups and the genus 2 mapping class group.

This paper shows that the answer to 2) is false in the category of finitely presented residually finite groups. As Greg points out, this is different from the category of finitely presented linear groups though.

Addendum: In a paper of Bridson and Miller (which I found from Igor's link to Miller's survey), they show that the isomorphism problem for subgroups of $\Gamma\times\Gamma\times F$ is undecidable, where $\Gamma$ is a particular hyperbolic group (which is free-by-finitely generated) and $F$ is free. As mentioned in the paper, Mosher constructed free-by-surface hyperbolic groups, which therefore could work as $\Gamma$. These groups embed in the mapping class group of the once-punctured surface, so if mapping class groups of once-punctured surfaces are linear, this would answer 2). However, the only mapping class groups known to be linear are the punctured sphere/braid groups and the genus 2 mapping class group.

This paper shows that the answer to 2) is false in the category of finitely presented residually finite groups. As Greg points out, this is different from the category of finitely presented linear groups though.

This paper shows that the answer to 2) is false in the category of finitely presented residually finite groups. As Greg points out, this is different from the category of finitely presented linear groups though.

Addendum: In a paper of Bridson and Miller (which I found from Igor's link to Miller's survey), they show that the isomorphism problem for subgroups of $\Gamma\times\Gamma\times F$ is undecidable, where $\Gamma$ is a particular hyperbolic group (which is free-by-finitely generated) and $F$ is free. As mentioned in the paper, Mosher constructed free-by-surface hyperbolic groups, which therefore could work as $\Gamma$. These groups embed in the mapping class group of the once-punctured surface, so if mapping class groups of once-punctured surfaces are linear, this would answer 2). However, the only mapping class groups known to be linear are the punctured sphere/braid groups and the genus 2 mapping class group.