What type of modem do I need to connect my AirPort Extreme to BT infinity

The question is not really what type of modem that you need......it is rather whether one of the "approved" modems that BT provides will be compatible with the AirPort Extreme.

Unfortunately, the AirPort Extreme and other Apple routers were designed to work with standard DSL and cable type connections. Fibre is a new ballgame, and Apple has not yet learned all the rules for the new game.

Your best bet might be to post a list of BT "approved" modems here and then ask other users who might have the same product and an AirPort Extreme to comment on their experience.

You might also simply ask BT what product they would recommend to work with the AirPort Extreme. They may tell you that they will not support you if you use the Apple product, so that might be something to consider as well.