Can you add friends through Gamecenter?

Yes. 

- Click on the GameCentre icon on your ipad/iphone while signed in to Gamecentre

- Click on "Friends" from the menu that runs horizontally along the bottom of the screen

- In the new window that opens up, on the top-left corner of the screen there will be a blue "+" icon next to the word "Friends". Click on the "+" icon

- This will open up a new box with "Friend Request" as the heading. Enter the email address or GameCentre nickname/ID of the person you want to add and then click "send". 

For a list of players who are currently looking for friends/neighbours then see this thread.

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Do you mean is there a way to get people who are already on your friends list but don't have the game to go and get the game?

Add me: Brian Birkett

Unfortunately this feature doesn't seem to exist anymore. Do you have any idea of how to add friends with iOS 10? There is no "+" sign anywhere even though I'm logged in with Game Center