What spectral feature was used to confirm that S0-2 is "single and available"? What is Brγ?
Brγ is a line in the emission spectrum of Hydrogen, or single-electron (and therefore) hydrogen-like Rydberg atoms in general. In particular, it is the emission due to an electron moving from the 7th to the 4th energy level.

Br is short for Brackett, after Frederik Brackett, and gamma because 


$\alpha: \ \ 5 \rightarrow 4,$ 
$\beta:  \ \ 6 \rightarrow 4,$ 
$\gamma: \ \ 7 \rightarrow 4,$ 


...and so on.

Brγ has a wavelength of 2166nm, as you would get from the Rydberg formula, which is well in the infra-red part of the spectrum. It is probably chosen as it has low absorption by the dust towards the centre of the galaxy.

Screenshot from here.