Conjugacy languages in groups
  We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin. 
Laura Ciobanu, Susan Hermiller, Derek Holt and Sarah Rees
math.GR