2007 Tahoe will not start

Cables. Sometimes the 0g battery cables can sever between terminal clamp and copper wire, unseen. And/Or ground wire to block, frame or chassis must be clean and tight

Try a set of jumper cables, from Black -- Neg battery post to a good ground on chassis or block. Use both cables, see what happens

It will jump off but will not start again if cut off

Same thing happens? Clicking? I am not sure if the solenoid comes with starter on that car, Tahoe.... With new battery, and starts with jump I still say cables. But I am not a GM guy, throwing guesses because nobody else has chimed in that will know. But if I may, when you jump it, and it does start.... do you hook up ground to battery (on your Tahoe) to battery Neg-- or to frame? And just to be sure we are on the same page, when jumped it starts and will run until until you shut it off... then just clicks when you try to re-start? If so, jump start it again and let it idle for at least 15 minutes (or more) then shut off and try again... see what happens. OR... do you hear the starter motor 'whirring'.... spinning, but not catching flywheel to crank over?

Does the starter look like this? (click for full screen)

Better yet....