It's perhaps worth mentioning a powerful construction of finitely presented residually finite groups due to Bridson--Grunewald (though the residual finiteness comes from work of Wise).

Wise produced a residually finite version of the Rips construction, which has the following statement.

For every finitely presented group $Q$ there is a residually finite hyperbolic group $\Gamma$ and a short exact sequence of groups

 

$1\to K\to\Gamma\to Q\to 1$

 

so that $K$ is finitely generated.

The usefulness of this statement lies in the fact that pathologies of $Q$ translate into pathologies of $K$. For instance, if $Q$ has unsolvable word problem then $K$ has unsolvable membership problem.

Unfortunately, the subgroup $K$ is almost never finitely presented, but one can partially rectify this using the "1--2--3 theorem" of Baumslag--Bridson--Miller--Short.

If

 

$1\to K\to\Gamma\stackrel{f}{\to} Q\to 1$

 

is a short exact sequence of groups with $K$ finitely generated and $Q$ of type $F_3$ (this is a higher-dimensional analogue of finite presentability, meaning that $Q$ has an Eilenberg--Mac Lane space with finite 3-skeleton) then the fibre product

 

$P=\{(\gamma_1,\gamma_2)\in\Gamma\times\Gamma\mid f(\gamma_1)=f(\gamma_2)\}$

 

is finitely presented.

Putting these together, we obtain a machine for translating pathologies of an arbitrary $F_3$ group $Q$ into pathologies of a finitely presented subgroup $P$ of the residually finite group $\Gamma\times\Gamma$. In fact, nowadays, one can do even better, since Haglund--Wise proved a linear version of the Rips construction. Here are some sample applications.

• Bridson--Grunewald found a proper finitely presented subgroup $Q$ of a finitely presented residually finite group $\Gamma\times\Gamma$ so that the inclusion map induces an isomorphism on the profinite completion.

• A related argument of Bridson--Miller, combined with the Haglund--Wise version of the Rips construction, shows that the isomorphism problem is unsolvable for finitely presented linear groups.

• Martino--Minasyan used this construction to give examples of conjugacy separable finitely presented groups with non-conjugacy separable subgroups of finite index.

• Bridson and I showed that the isomorphism problem is undecidable for profinite completions of finitely presented, residually finite groups.

It's perhaps worth mentioning a powerful construction of finitely presented residually finite groups due to Bridson--Grunewald (though the residual finiteness comes from work of Wise).

Wise produced a residually finite version of the Rips construction, which has the following statement.

For every finitely presented group $Q$ there is a residually finite hyperbolic group $\Gamma$ and a short exact sequence of groups

$1\to K\to\Gamma\to Q\to 1$

so that $K$ is finitely generated.

The usefulness of this statement lies in the fact that pathologies of $Q$ translate into pathologies of $K$. For instance, if $Q$ has unsolvable word problem then $K$ has unsolvable membership problem.

Unfortunately, the subgroup $K$ is almost never finitely presented, but one can partially rectify this using the "1--2--3 theorem" of Baumslag--Bridson--Miller--Short.

If

$1\to K\to\Gamma\stackrel{f}{\to} Q\to 1$

is a short exact sequence of groups with $K$ finitely generated and $Q$ of type $F_3$ (this is a higher-dimensional analogue of finite presentability, meaning that $Q$ has an Eilenberg--Mac Lane space with finite 3-skeleton) then the fibre product

$P=\{(\gamma_1,\gamma_2)\in\Gamma\times\Gamma\mid f(\gamma_1)=f(\gamma_2)\}$

is finitely presented.

Putting these together, we obtain a machine for translating pathologies of an arbitrary $F_3$ group $Q$ into pathologies of a finitely presented subgroup $P$ of the residually finite group $\Gamma\times\Gamma$. In fact, nowadays, one can do even better, since Haglund--Wise proved a linear version of the Rips construction. Here are some sample applications.

• Bridson--Grunewald found a proper finitely presented subgroup $Q$ of a finitely presented residually finite group $\Gamma\times\Gamma$ so that the inclusion map induces an isomorphism on the profinite completion.

• A related argument of Bridson--Miller, combined with the Haglund--Wise version of the Rips construction, shows that the isomorphism problem is unsolvable for finitely presented linear groups.

• Martino--Minasyan used this construction to give examples of conjugacy separable finitely presented groups with non-conjugacy separable subgroups of finite index.

• Bridson and I showed that the isomorphism problem is undecidable for profinite completions of finitely presented, residually finite groups.