A homomorphism between two complex instances of these objects must be a holomorphic map, and the regular Fréchet kind of these are   di eomorphism groups of compact manifolds.
The Sol geometry is the geometry of the semidirect product of one of these structures and R2. The Baker-Campbell-Hausdor  theorem connects these objects to algebras described by the same term, and Heisenberg and Lorenz name examples of these. A homomorphism between two complex instances of these objects must be a holomorphic map, and the regular Fréchet kind of these are   di eomorphism groups of compact manifolds. This groups are de ned by the requirement that the mapping (x,y) → x−1y is smooth from the product manifold into the group. For ten points, identify this class of groups which are also di erentiable manifolds.
The Sol geometry is the geometry of the semidirect product of one of these structures and R2.
Lie groups [do not accept  Lie algebras ]