Is there an example of a group $G$ that has the properties

1. the cohomological dimension of $G$ is infinite: $\operatorname{cd}(G) = \infty$,
2. $G$ is torsion-free,
3. $G$ is of type $F_\infty$,
4. $G$ does not contain Thompson's group $F$?

Those who don't know about Thompson's group can replace 4 by the stronger requirement

4'. $G$ does not contain an infinite-rank free abelian group.

Motivation: there are various known groups that satisfy 1-3 but all of them contain $F$. If one is bold one might wonder whether this because of what we know or because of $F$.

On a less speculative level, this is a variation of this question: two good reasons for 1 are for $G$ to have torsion or to contain infinite-rank free abelian groups. So how to assure it if the two are ruled out? The cited question did not require 3 although in comments IgorBelegradek and IanAgol speculate in that direction.

Is there an example of a group $G$ that has the properties

1. the cohomological dimension of $G$ is infinite: $\operatorname{cd}(G) = \infty$,
2. $G$ is torsion-free,
3. $G$ is of type $F_\infty$,
4. $G$ does not contain (an isomorphic copy of) Thompson's group $F$?

Those who don't know about Thompson's group can replace 4 by the stronger requirement

4'. $G$ does not contain an infinite-rank free abelian group.

Motivation: there are various known groups that satisfy 1-3 but all of them contain $F$. If one is bold one might wonder whether this because of what we know or because of $F$.

On a less speculative level, this is a variation of this question: two good reasons for 1 are for $G$ to have torsion or to contain infinite-rank free abelian groups. So how to assure it if the two are ruled out? The cited question did not require 3 although in comments IgorBelegradek and IanAgol speculate in that direction.