# The Sol geometry

## What is Sol?

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.

Click on the button below to reveal a concrete model of Sol.

As a set of points, Sol is the usual 3-dimensional space $X = \mathbb R^3$ with coordinates $(x,y,z)$. The riemanian metric on Sol is given by $$ ds^2 = e^{-2z} dx^2 + e^{2z}dy^2 + dz^2.$$

Consider to points $p_1 = (x_1, y_1, z_1)$ and $p_2 = (x_2, y_2, z_2)$ in $X$. The group law in Sol is given by $$ p_1 \ast p_2 = (x_1 + e^{z_1}x_2, y_1 + e^{-z_1}y_2, z_1 + z_2) $$ The left action of Sol on itself is an action by isometries. Sol has finite index in its isometry group. More precisely ${\rm Isom}(X) = X \rtimes \mathbb D_8$, where $\mathbb D_8$ is the dihedral group of order eight.

## Some views of Sol

**Warning:**
Some of the real-time simulations below requires a powerful graphic card.
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Click on the button below to reveal the fly commands.

The default controls to fly in the scene are the following.
If you have a different keyboard, the keys should be the ones having the same **location** as the given ones on a QWERTY keyboard.

Command | QWERTY keyboard | AZERTY keyboard |
---|---|---|

Yaw left | a | q |

Yaw right | d | d |

Pitch up | w | z |

Pitch down | s | s |

Roll left | q | a |

Roll right | e | e |

Move forward | arrow up | arrow up |

Move backward | arrow down | arrow down |

Move to the left | arrow left | arrow left |

Move the the right | arrow right | arrow right |

Move upwards | ' | ù |

Move downwards | / | = |

HD pictures of Sol can be found in the gallery (in preparation)

## Features of Sol

Some features of Sol are described in the following Bridges paper.