The original Eilenberg-Ganea conjecture can be formulated as: any group of cohomological dimension 2 has geometric dimension 2. It remains unsettled.

On the other hand, various generalisations in which the ordinary cohomological (resp. geometric) dimensions are replaced with the cohomological (resp. geometric) dimensions with respect to a certain family of subgroups are known to be false. Examples of groups $G$ for which $\operatorname{cd}_\mathcal{F}(G)=2$ and $\operatorname{gd}_\mathcal{F}(G)=3$ have been found by Brady, Leary and Nucinkis when $\mathcal{F}$ is the family of finite subgroups, and by Fluch and Leary when $\mathcal{F}$ is the family of virtually cyclic subgroups.

This got me wondering about positive about positive Eilenberg-Ganea results.

Any class of groups which is closed under isomorphism and taking subgroups can be used to define a family of subgroups of any given group. One can then ask if the Eilenberg-Ganea conjecture holds for this particular family. For the trivial class this is unsettled; for the class of all groups the result is vacuously true because every group has cohomological and geometric dimension zero with respect to the family of all subgroups. Thus the question becomes:

Does there exist a (proper) class $\mathcal{X}$ of groups such that for $\mathcal{F}$ the family of subgroups of a given group consisting of groups in $\mathcal{X}$, every group $G$ such that $\operatorname{cd}_\mathcal{F}(G)=2$ has $\operatorname{gd}_\mathcal{F}(G)=2$?

The original Eilenberg-Ganea conjecture, which remains unsettled, can be formulated as: any (discrete) group $G$ of cohomological dimension $\operatorname{cd}(G)=2$ has geometric dimension $\operatorname{gd}(G)=2$. Recall that $\operatorname{cd}(G)$ is the projective dimension of the trivial module $\mathbb{Z}$ in the category of $G$-modules, while $\operatorname{gd}(G)$ is the smallest dimension of an $EG$ complex.

On the other hand, various analogues in which the ordinary cohomological (resp. geometric) dimensions are replaced with the cohomological (resp. geometric) dimensions with respect to a certain family of subgroups are known to be false.

Recall that a family of subgroups of $G$ is a collection $\mathcal{F}$ of subgroups closed under conjuation and taking subgroups. The most important examples are the trivial family $\{1\}$, the family $\mathcal{FIN}$ of finite subgroups, and the family $\mathcal{VCYC}$ of virtually cyclic subgroups.

A classifying space for $G$ with respect to $\mathcal{F}$ is a $G$-CW complex $E_\mathcal{F}(G)$ such that for all subgroups $H\le G$, the fixed point set $E_\mathcal{F}(G)^H$ is empty if $H\notin\mathcal{F}$ and contractible if $H\in \mathcal{F}$. The geometric dimension of $G$ with respect to $\mathcal{F}$, denoted $\operatorname{gd}_\mathcal{F}(G)$, is the smallest dimension of an $E_\mathcal{F}(G)$.

The orbit category of $G$ with respect to $\mathcal{F}$ is the category $\mathcal{O}_\mathcal{F}(G)$ with objects the $G$-sets $G/H$ for $H\in \mathcal{F}$ and morphisms the $G$-maps $G/H\to G/K$. The category $\mathcal{O}_\mathcal{F}(G)$-mod of modules over the orbit category (contravariant functors $\mathcal{O}_\mathcal{F}(G)\to \operatorname{Ab}$) is abelian and has enough projectives. The cohomological dimension $\operatorname{cd}_\mathcal{F}(G)$ is the projective dimension of the constant module $\underline{\mathbb{Z}}$ in the category $\mathcal{O}_\mathcal{F}(G)$-mod.

Examples of groups $G$ for which $\operatorname{cd}_\mathcal{F}(G)=2$ and $\operatorname{gd}_\mathcal{F}(G)=3$ have been found by Brady, Leary and Nucinkis when $\mathcal{F}=\mathcal{FIN}$, and by Fluch and Leary when $\mathcal{F}=\mathcal{VCYC}$.

This got me wondering about positive Eilenberg-Ganea results, i.e. families $\mathcal{F}$ for which $\operatorname{cd}_\mathcal{F}(G)=2$ implies $\operatorname{gd}_\mathcal{F}(G)=2$.

Any class of groups which is closed under taking subgroups can be used to define a family of subgroups of any given group. One can then ask if the Eilenberg-Ganea conjecture holds for this particular family. For the trivial class $(1)$ this is unsettled; for the classes $(\mathcal{FIN})$ and $(\mathcal{VCYC})$ of finite and virtually cylcic groups it is false; for the class of all groups it is vacuously true because every group has cohomological and geometric dimension zero with respect to the family of all subgroups. Thus the question becomes:

Does there exist a (proper) class $\mathcal{X}$ of groups such that $\operatorname{cd}_{\mathcal{F}(\mathcal{X})}(G)=2$ implies $\operatorname{gd}_{\mathcal{F}(\mathcal{X})}(G)=2$, where $\mathcal{F}(\mathcal{X})$ denotes the family of subgroups of $G$ in the class $\mathcal{X}$?