The mapping torus of an orientation preserving Anosov homeomorphism

$\varphi $ of the 2-torus (i.e., a compact 3-manifold fibering over the circle with fiber a torus

${T}^{2}$) admits a geometric structure modeled on the solvable geometry Sol. There is a natural section

${\Sigma}\subset M$ of the fibration

$M\to {S}^{1}$ (

$\varphi :{T}^{2}\to {T}^{2}$ lifts to a linear map of

${\mathbb{R}}^{2}$ and hence fixes the projection of the origin in

${\mathbb{R}}^{2}$). The main result of the paper states that there exists a family of hyperbolic cone structures on

$M$ with singular set

${\Sigma}$ parametrized by the cone angle

$\alpha \in (0,2\pi )$. When

$\alpha \to 2\pi $ this family collapses to a circle (the basis of the fibration

$M\to {S}^{1}$); also, the metrics can be rescaled in the direction of the fibers so that they converge to the Sol structure on

$M$ (when

$\alpha \to 0$ this family of cone manifolds converges to the complete hyperbolic structure on

$M\setminus {\Sigma}$). An explicit construction of the deformations of the Sol structure on

$M$ is given by using the Cartan splitting of the Lie algebra

$s{l}_{2}\left(\u2102\right)$ and associated Killing fields, and also a result about algebraic deformations of reducible representations proved in a previous paper by the authors [J. Reine Angew. Math. 530, 191–227 (2001;

Zbl 0964.57006)]. An example of such a manifold

$M$ is obtained by 0-surgery on the figure eight knot which has been considered by Jorgensen and Thurston; also, Hilden, Lozano and Montesinos constructed an explicit family of Dirichlet polyhedra for this manifold collapsing to a segment (whose ends are identified to give

${S}^{1}$), and the third named author of the present paper showed that this family of polyhedra can be rescaled to converge to a Sol structure.