For what it's worth, I'm watching Resource Monitor and I don't see connections from AvastUI to anything.
For me AvastSvc.exe shows occasional connections to addresses that reverse-resolve to AVAST Software a.s., which seems reasonable given the software's online update facilities.
Note that I don't use anything but the 3 shields and the software updater.
Personally, if I saw my system maintaining a connection with Google I'd be concerned that Google has infected me with something. That's certainly their goal.
Something related to consider, which blocks trackers etc.:
http://winhelp2002.mvps.org/hosts.htmAlso if you see applications accessing servers by name that you'd rather they didn't access, nothing keeps you from adding additional lines to your hosts file.
-Noel