Linking and discreteness in hyperbolic 4-space

A pair of isometries of the 4-dimensional hyperbolic space is called linked if they can be expressed as compositions of two involutions, one of which is common to both isometries. While every pair of isometries of hyperbolic space in dimensions 2 and 3 is linked, not all pairs of isometries of hyperbolic 4-space are linked. One type of such an involution is called half-turn which is an orientation preserving elliptic isometry with a 2-dimensional fixed point set. We provide some geometric conditions for such a pair to be linked by half-turns. Here we develop a theory of pencils, twisting planes and half-turn banks that gives results about each of the pair-types of isometries and their simultaneous factorization. In order to provide conditions under which a given pair is linked via a half-turn, sets of hyperplanes in hyperbolic 4-space are defined for each orientation preserving isometry that enables one to locate the half-turns for which linking is possible. Once a pair is linked, known conditions about discreteness of the group, generated by a pair of isometries, in lower dimensional hyperbolic spaces can be generalized to some linked pairs in dimension 4. If a pair has a common invariant hyperplane or plane, the known conditions such as compact-core-geodesic-intersection and non-separating-disjoint-circles apply.Ph. D.Includes bibliographical referencesIncludes vitaby Andrew Ebitner Silveri