Some deformations of codimension two discrete groups (Q1088029)

An isometry of an hyperbolic n-space \(H^ n\) which fixes exactly one point at infinity is called parabolic. For an hyperbolic n-space with \(n\leq 3\) viewed in the half-space model with the fixed point at infinity, a parabolic element is simply a Euclidean translation which preserves the hyperbolic space. Such a parabolic isometry is called pure. For \(n\geq 4\), there exist impure parabolic elements. Viewed from the same perspective, impure parabolics necessarily have the form \(x\to j(x)=r(x)+b\), r orthogonal, r not the identity, b not in the range of id-r. John Morgan has asked whether there exists a cofinite volume discrete hyperbolic group on \(H^ n\) containing pure parabolic elements which can be deformed to a discrete hyperbolic group on \(H^{n+k}\) with impure parabolics. The author shows by example that the answer is 'yes' for \(k>1\). He notes and proves that the answer is 'no' for \(k=1\).