Ends of Diabolical Groups

In 1982, Conway introduced the angel-devil game, which is played on an infinite chess board.  For fixed k, the angel moves at most distance k from its current position on its turn.  The devil then blocks a square permanently.  The devil wins if the angel eventually has no legal moves left.  Berlekamp showed the devil wins against the 1-angel.  Conway asked whether there exists k such that the k-angel has a winning strategy against the devil.  This was resolved independently by Kloster, Máthé, and Bowditch in 2006.  Bowditch proposed playing the game on Cayley graphs of finitely generated groups.  A group for which the devil beats the k-angel for every k is called diabolical.  We will explore the ends of these diabolical groups.