Fans and generators of free abelian l-groups

Let $t_1,ldots,t_n$ be $ell$-group terms in the variables $X_1,ldots,X_m$. Let $hat t_1,ldots,hat t_n$ be their associated piecewise homogeneous linear functions. Let $G $ be the $ell$-group generated by $hat t_1, ldots,hat t_n$ in the free $m$-generator $ell$-group $mathcal A_m.$ We prove: (i) the problem whether $G$ is $ell$-isomorphic to $mathcal A_n$ is decidable; (ii) the problem whether $G$ is $ell$-isomorphic to $mathcal A_l$ ($l$ arbitrary) is undecidable; (iii) for $m=n$, the problem whether ${hat t_1,ldots,hat t_n}$ is a {it free} generating set is decidable. In view of the Baker-Beynon duality, these theorems yield recognizability and unrecognizability results for the rational polyhedron associated to the $ell$-group $G$. We make pervasive use of fans and their stellar subdivisions.