Studying Apollonian Circle Packings using Group Theory

This video is in response to a colleague who asked for short videos about how we use group theory in our research. One of my interests is Apollonian circle packings.
Some links:
Wikipedia: en.wikipedia.o...
An article in New Scientist by Dana Mackenzie: www.americansc...
A good math paper to start: arxiv.org/abs/...
The main result I mention at the end, in the direction of the local-global conjecture, is due to Bourgain and Kontorovich:
link.springer....