Geodesic
Bounding reflection length in an affine Coxeter group.
Journal of Algebraic Combinatorics, Tome 34 (2011) no. 4, pp. 711-719.

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Summary: 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 $\Bbb R ^{ n }$ is bounded above by $2 n$, 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.
Keywords: keywords Coxeter group, reflection length 
@article{JAC_2011__34_4_a1,
     author = {McCammond, Jon and Petersen, T.Kyle},
     title = {Bounding reflection length in an affine {Coxeter} group.},
     journal = {Journal of Algebraic Combinatorics},
     pages = {711--719},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2011__34_4_a1/}
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McCammond, Jon; Petersen, T.Kyle. Bounding reflection length in an affine Coxeter group.. Journal of Algebraic Combinatorics, Tome 34 (2011) no. 4, pp. 711-719. http://geodesic.mathdoc.fr/item/JAC_2011__34_4_a1/