On Computing Geodesics in Baumslag-Solitar Groups

TL;DR

This paper introduces a new normal form for elements in Baumslag-Solitar groups that closely approximates geodesic paths, and provides polynomial-time algorithms for computing these forms when p divides q.

Contribution

It defines the peak normal form for Baumslag-Solitar groups and demonstrates polynomial-time computability when p divides q, advancing understanding of geodesic problems in these groups.

Findings

Peak normal form is close to length-lexicographical form and is geodesic.

Polynomial-time algorithm for computing peak normal form when p divides q.

Reduction of geodesic computation to Britton-reduced words with specific t-sequences.

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.