Say that a triple of real numbers $(a,b,c)$ is a realizable triple if there are matrices $A,B\in SL\_2(\mathbb{R})$ such that $tr (A)=a$, $tr (B)=b$, and $tr (AB)=c$. Question: what is the shape of the non-realizable set?


This is surely known, but I couldn't find an answer by myself, nor a reference on the www. It's easy to see that the set in question is non-empty (it contains the origin, for instance) and it lives in the 3-dimensional square $(-2,2)\times(-2,2)\times(-2,2)$. I'm guessing that the Fricke polynomial $a^2+b^2+c^2-abc$ should enter the picture as well.


Note that realizability is, well, a real problem but not a complex one.



tracial triples
Let $x, y, z$ be traces of $A, B, AB$ respectively. Define
$$
k(x,y,z)= x^2+y^2 + z^2 -xyz -2.
$$
Then a triple of real traces $(x, y, z)$ is realizable in $SL(2,R)$, unless it is realizable in $SU(2)$, the latter happens if and only if $x, y, z\in [-2,2]$ and $k(x,y,z)\le 2$. See Goldman's paper "Topological components of spaces of representations", Inventiones, 1988. 


