The unbounded dead-end depth property is not a group invariant
  The dead-end depth of an element g of a group with finite generating set A is the distance from g to the complement of the radius d(1,g) closed ball, in the word metric d associated to A. We exhibit a finitely presented group K with two finite generating sets A and B such that dead-end depth is unbounded on K with respect to A but is at most two with respect to B. 
Tim R. Riley and Andrew D. Warshall