The bid by Google wasn't for publicity.
Google wants in to this space. That is the 'local' relationship space. Google has been horrible on building relationships. One could consider their lack of acumen to be due to arrogance or bad manners.
Google failed against Netflix which AFAIK El Reg reported on this as well as the regular press like CNN (Money).
Groupon gives them 2 things. A patent which is a barrier to entry by competitors, and a built in customer base.
IMHO Groupon should have taken the 6 Billion deal from Google. They are not worth that much and Google would have lost a lot of face. Of course being the 'Chocolate Factory'... they can always print up whatever money they want. And one has to ask if Google could have swung the deal through the FTC and EU.