Dead ends and rationality of complete growth series

We are interested in the NG-rationality and NG-algebraicity of the complete growth series of finitely generated groups. It is shown that dead ends of arbitrarily large depths form an obstruction to NG-rationality. In the case of the 3-dimensional Heisenberg group H3(Z), we prove that the complete series is not NG-algebraic for any generating set. Dead ends are also used to show that complete growth series of higher Heisenberg groups are not NG-rational for specific generating sets. Using a more general version of this obstruction, we prove that complete growth series of some lamplighter groups are not NG-rational either. This work provides the first examples of groups exhibiting a difference between rationality of standard growth series, and rationality of complete growth series.