According to a report from The Information, Google held a series of talks with Verizon Wireless in January covering a range of issues. A source familiar with the talks says one item on the agenda was Google's interest in becoming a mobile virtual network operator (MVNO) in markets where it offers Google Fiber. This isn't the first time a Google phone carrier has been rumored, but this might be the first time Android is strong enough to withstand the inevitable blowback from established carriers.
To run an MVNO, Google would license spectrum from carriers like T-Mobile and Verizon – it's the same thing companies like Ting and Straight Talk do in the US market. However, there is no guarantee Verizon would be interested in offering Google access to it's expansive LTE network. T-Mobile might be more receptive to a deal, but its footprint is quite a bit smaller. If Verizon was interested, Google would most likely have to implement some form of VoLTE (voice over LTE) to avoid getting stuck with CDMA voice.
The tie in with Google Fiber markets would give the company the opportunity to get a handle on being a carrier without upsetting the balance too much right away. Google could also leverage the fiber it has laid down to offload mobile traffic to WiFi, thus reducing its usage of cellular bandwidth. The Play Store device section could also be used to sell phones compatible with the service. That's one reason T-Mobile's GSM/LTE network would make more sense for a Google MVNO in the short term.
Expanding any future mobile service nationwide would be risky for Google – carriers might not take kindly to such competition from Google, and as an MVNO, it would rely on the carriers for its spectrum. However, Android is dominant in mobile, which might give Google enough clout to carve out a sliver of the market for itself. Maybe ruffling feathers is on the agenda anyway – CEO Larry Page hasn't been shy about expressing his distain for traditional carriers. Maybe Google Fiber is only the beginning.