A **left order** on a (discrete) group $G$ is a total order on $G$ satisfying $\forall g,h,k \in G: g < h \implies kg < kh$. A **right order** is defined symmetrically, and a **biorder** is an order that is both at once. An order is **scattered** if it has no densely ordered subset of cardinality at least two, where a set $S$ is **densely ordered** if $a, c \in S \wedge a \neq c \implies \exists b \in S: a < b < c$.

I'm interested in order types of left orders (and biorders, why not). It's easy to show that left orders on the integers $\mathbb{Z}$ all have order type $\mathbb{Z}$, and on $\mathbb{Z}^2$ there are two constructions of left orders, and as far as I can tell one gives order type $\mathbb{Z}^2$ in lex order (so scattered), and the other orders are dense (so not scattered).

Let $F_2$ be the free nonabelian group on two generators.

Does $F_2$ admit a scattered left order?

Does $F_2$ admit a scattered biorder?

I had an awesome application for this, but I broke it already. Now I'm just curious. I don't actually know what you get from the Magnus embedding $a \mapsto 1+a$, I admit I was too lazy (or scared?) to even give it real thought, and I did not notice a statement in the literature.

More generally, one may ask:

What are the order-types of left orders on $F_2$?

What are the order-types of biorders on $F_2$?

More generally, I'm interested in information on order types of orders on any torsion-free groups, there's plenty of literature on orders but I haven't seen much about order types.