What is the difference between the $Spin(3,1)$ group and the $SO(3,1)$ group?

Question

What is the difference between the $Spin(3,1)$ group and the $SO(3,1)$ group?

Answer 1

1. The spin group $Spin^+(3,1;\mathbb{R})\cong SL(2,\mathbb{C})$ is the double cover of the restricted Lorentz group $SO^+(3,1;\mathbb{R})$, cf. e.g. this Phys.SE post and links therein.

2. The spin group $Spin(3,1;\mathbb{R})$ is the double cover of the special Lorentz group $SO(3,1;\mathbb{R})$.

3. The pin group $Pin(3,1;\mathbb{R})$ is the double cover of the Lorentz group $O(3,1;\mathbb{R})$.

References:

1. M. Rausch de Traubenberg, Clifford Algebras in Physics, arXiv:hep-th/0506011; p. 9.