TITLE: Can Suslin (or Aronszajn) lines ever be orderings of abelian groups?
QUESTION [15 upvotes]: I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way?
Let $\mathcal{C}$ be a class of (torsion-free) abelian groups, and $L$ be the first-order language of linear orderings. Let $LO(\mathcal{C})$ be the theory of ordered abelian groups belonging to the class $\mathcal{C}$. Recall an abelian group $A$ is ordered by the binary relation $<$ if $<$ is a linear order on $A$ such that whenever $g < h$ then $g+k < h+k$ for all $g, h, k \in A$. 
Let $\mathcal{K}_{LO(\mathcal{C})} = \lbrace A \mid L : A \models LO(\mathcal{C}) \rbrace$ be the class of all linear orderings which are orderings of members in $\mathcal{C}$. 
Suppose $\mathcal{C}$ is the class of all abelian groups. So $\mathcal{K}_{LO(\mathcal{C})}$ is a pseudo-elementary class. From A.C. Morel's work (\textit{Structure and order structure in Abelian groups}, Colloq. Math. 19 (1968), 199-209), the $\aleph_{1}$ countable members of $\mathcal{K}_{LO(\mathcal{C})}$ are known up to isomorphism.
What uncountable lines can turn up in $\mathcal{K}_{LO(\mathcal{C})}$?
Is it consistent that $\mathcal{K}_{LO(\mathcal{C})}$ contains a Suslin line (or some but not all Suslin lines, or all Suslin lines if these exist)? If one adds a Cohen real, does $\mathcal{K}_{LO(\mathcal{C})}$ contain a Suslin line?
Does $\mathcal{K}_{LO(\mathcal{C})}$ contain an Aronszajn line, i.e. a line obtained by squashing an Aronszajn tree?
Thank you for any help.

REPLY [6 votes]: This answer might be late, but I found the following papers related:

(A) Koppelberg, Sabine,
  ``Groups cannot be Souslin ordered'',  Arch. Math. (Basel) 29 (1977), no. 3, 315–317.

In this paper it is shown that there are no Suslin ordered groups. 
I may also mention that the problem of whether a Suslin order may be the underlying order of an ordered field or of some more general algebraic structure is also discussed in the following paper:

(B) Felgner, Ulrich,
  ``Das Problem von Souslin für geordnete algebraische Strukturen''. Set theory and hierarchy theory (Proc. Second Conf., Bierutowice, 1975), pp. 83–107. Lecture Notes in Math., Vol. 537, Springer, Berlin, 1976.