On Computing Geodesics in Baumslag-Solitar Groups

We introduce the peak normal form for elements of the BaumslagSolitar groups BS(p, q). This normal form is very close to the lengthlexicographical normal form, but more symmetric. Both normal forms are geodesic. This means the normal form of an element uv yields the shortest path between u and v in the Cayley graph. 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 [4]. For arbitrary p and q it is possible to compute the peak normal form 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 and ends with t. To solve the general case in polynomial time for arbitrary p and q remains a challenging open problem.