Torus suspension

Given a matrix $A \in {\rm SL}(2, \mathbb R)$, one can produce a 3-dimensional manifold $M$ by suspending the 2-torus $T$ according to $A$. More precisely $M$ is the quotient of $T \times [0,1]$ by the equivalence relation identifying $(x,0)$ with $(Ax,1)$. The universal cover $X$ of $M$ can be identified with one of the eight Thurston geometries

• if $A$ has finite order, then $X = \mathbb E^3$.
• if $A$ is a Dehn twist, then $X = {\rm Nil}$.
• if $A$ is Anosov, then $X = {\rm Sol}$.

The Sol geometry

Sol is a 3-dimensional solvable Lie group. It can also be seen as the universal cover of the suspension of a 2-torus by an Anosov matrix.