Factoring Euclidean isometries

Every isometry of a finite-dimensional Euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this paper we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimum length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved.

Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds

Let Γ1 and Γ2 be Bieberbach groups contained in the full isometry group G of ℝn. We prove that if the compact flat manifolds Γ1\ℝn and Γ2\ℝn are strongly isospectral, then the Bieberbach groups Γ1 and Γ2 are representation equivalent; that is, the right regular representations L2(Γ1\G) and L2(Γ2\G) are unitarily equivalent.

On isometric actions

To a cardinal k ≥ 2, we associate a simply-connected polyhedral surface Σk endowed with a bounded metric dk such that every group of cardinality k has an isometric, properly discontinuous action on (Σk, dk). If ℵ0 ≤ k ≤ 2ℵ0 and G is a group of cardinality k, then we extend (Σk, dk) to a simply-connected bounded metric space (MG, dG) such that the action of G extends to an isometric, properly discontinuous action on (MG, dG) and G is the full isometry-group of (MG, dG).