Does there exist an example of a group that is:

- Simple,
- Torsion-free,
- Of type $\textrm{F}_\infty$, and
- Infinite dimensional (meaning of infinite cohomological dimension)?

Thompson's group $F$ has all these properties except not (1). Thompson's groups $T$ and $V$ have all these properties except not (2). The commutator subgroup $[F,F]$ has all these properties except not (3). My understanding is that things like Burger-Mozes groups have all these properties except not (4).

I'm curious whether there is an example with all four properties. Many "Thompson-like" constructions provide examples of groups with properties (3) and (4), but any of the examples I can think of only have one of property (1) or (2), not both. It's possible this question is just open, but outside the "Thompson world" I don't know much about infinite simple groups, so I may be missing something.