# Is there a simple group that is torsion-free, type $\textrm{F}_\infty$, and infinite dimensional?

Does there exist an example of a group that is:

1. Simple,
2. Torsion-free,
3. Of type $$\textrm{F}_\infty$$, and
4. 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.

• A silly and trivial comment: there's no need for $T$ and $V$ to get all but 2 -- any finite simple group will do the trick. Commented Nov 15, 2020 at 18:28
• Ha, good point Andy! Commented Nov 15, 2020 at 19:42
• But finite groups are have cohomological dimension zero over $\mathbf{Q}$, so $F$, $T$ and $V$ satisfy (4) in a stronger sense.
– YCor
Commented Nov 15, 2020 at 20:55
• True, and another natural strengthening of (4) that keeps all the examples relevant would be, "contains free abelian groups of arbitrarily large rank" (maybe this could be called "infinite algebraic dimension" or something). Commented Nov 15, 2020 at 23:19

Take the group of all piecewise-linear, orientation-preserving homeomorphisms $$f$$ of $$\mathbb{R}$$ that have breakpoints in $$\mathbb{Z}[1/6]$$ and slopes of the form $$2^i 3^j$$, commute with translation by 1, and also satisfy that if the slope of $$f$$ at $$x\in\mathbb{R}\setminus\mathbb{Z}[1/6]$$ is $$2^i 3^j$$ then the difference $$i-j$$ equals the (signed) number of integers strictly in between $$x$$ and $$x.f$$.
They prove that the commutator subgroup of this group is simple and of type $$\mathrm{F}_\infty$$, hence satisfies properties (1) and (3), and obviously satisfies (2) and (4) since it's left-orderable and contains Thompson's group $$F$$.