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In a discrete group generated by hyperplane reflections in the$n$-dimensional hyperbolic space, the reflection length of an element is theminimal number of hyperplane reflections in the group that suffices to factorthe element. For a Coxeter group that arises in this way and does not splitinto a direct product of spherical and affine reflection groups, the reflectionlength is unbounded. The action of the Coxeter group induces a tessellation ofthe hyperbolic space. After fixing a fundamental domain, there exists abijection between the tiles and the group elements. We describe certain pointsin the visual boundary of the $n$-dimensional hyperbolic space for which everyneighbourhood contains tiles of every reflection length. To prove this, we showthat two disjoint hyperplanes in the $n$-dimensional hyperbolic space withoutcommon boundary points have a unique common perpendicular.

Reflection length at infinity in hyperbolic reflection groups