Bounding Reflection Length in an Affine Coxeter Group

In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on R is bounded above by 2n and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups are the only Coxeter groups with a uniform bound on reflection length. Every Coxeter group W has two natural generating sets: the set S used in its standard presentation and the set R of reflections formed by collecting all conjugates of the elements in S. The first generating set leads to the standard length function `S : W → N and the second is used to define the reflection length function `R : W → N. When W is finite, both length functions are uniformly bounded for trivial reasons and are fairly well understood. On the other hand, when S is infinite, i.e., when W has infinite rank, it is easy to show that both length functions are unbounded by considering products of elements in S. Thus, we restrict our attention to infinite Coxeter groups of finite rank. For these Coxeter groups the function `S is always unbounded because there are only finitely many group elements of a given length as a consequence of the fact that S is finite. Our main result is that `R remains bounded for affine Coxeter groups and we provide an explicit optimal upper bound. Theorem A (Explicit affine upper bounds). If W is an affine Coxeter group that naturally acts faithfully and cocompactly on R then every Date: May 19, 2011. Work of McCammond partially supported by an NSF grant. Work of Petersen partially supported by an NSA Young Investigator grant. 1For finite W these generating sets and length functions exhibit an interesting duality: the maximum value of `S is |R| and the maximum value of `R is |S|. See [1] for further details and for additional illustrations of this phenomenon.