Affine Deligne Lusztig varieties (ADLVs) are certain algebraic varieties associated to semisimple algebraic groups which have a Bruhat-Tits-building. We will explain how the geometry and combinatorics of the fundamental apartment of the building can be used to study nonemptiness and dimensions of ADLVs. Eventually all can be reduced to show existence and study the behaviour of certain positively folded galleries in affine Coxeter complexes.

Finally we will explain how one can obtain from nonemptiness and dimensions of ADLVs new insight on reflection length of elements of affine Coxeter groups.

This is joint work with Liz Milicevic and Anne Thomas.