The avastUI.exe has nothing to do with your internet connection, so shouldn’t stop you from connecting.
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You are right. When you shut it down, the Avast icon disappears and you no longer have access to to the Avast interface.

So what does this mean? Does it mean that the whole Avast program is not stored on the home computer, and you have to allow this connection to Google in order to see the Avast interface?