@Article{	  dl11bs,
  author	= "Volker Diekert and J{\"u}rn Laun",
  title		= "On Computing Geodesics in {B}aumslag-{S}olitar Groups",
  journal	= "International Journal of Algebra and Computation",
  volume	= "21",
  pages		= "119--145",
  number	= "1-2",
  year		= "2011",
  abstract	= "We introduce the peak normal form of elements of the
		  Baumslag-Solitar groups BS(p,q). This normal form is very
		  close to the length-lexicographical normal form, but more
		  symmetric. Both normal forms are geodesic. This means the
		  normal form of an element $u^{-1}v$ yields the shortest
		  path between $u$ and $v$ in the Cayley graph. For
		  horocyclic elements the peak normal form and the
		  length-lexicographical normal form coincide. The main
		  result of this paper is that we can compute the peak normal
		  form in polynomial time if $p$ divides $q$. As consequence
		  we can compute geodesic lengths in this case. In
		  particular, this gives a partial answer to Question 1 in
		  Elder et al. 2009, arXiv.org:0907.3258. For arbitrary $p$
		  and $q$ it is possible to compute the peak normal form
		  (length-lexicolgraphical normal form resp.) also for
		  elements in the horocyclic subgroup and, more generally,
		  for elements which we call hills. This approach leads to a
		  linear time reduction of the problem of computing geodesics
		  to the problem of computing geodesics for Britton-reduced
		  words where the $t$-sequence starts with $t^{-1}$ and ends
		  with $t$. To solve the general case in polynomial time for
		  arbitrary $p$ and $q$ remains a challenging open problem."
}