Kleinian Groups

Jørgensen number and arithmeticity

“…We begin by stating Poincaré's Polyhedron Theorem (see Section IV.H of [23] for notation and terminology), which we will use to construct fundamental polyhedra for arithmetic Jørgensen groups of parabolic type not treated in [17], [18], or [19].…”

Wild knots in higher dimensions as limit sets of Kleinian groups

“…We say that a point x is a limit point for the Kleinian group G, if there exists a point z ∈ S n and a sequence {g m } of distinct elements of G, with g m (z) → x. The set of limit points is Λ(G) (see [22], section II.D).…”

“…For this purpose a fundamental domain is very helpful. Roughly speaking, it contains one point from each equivalence class in Ω(G) (see [18], pages 78-79 and [22], pages 29-30).…”

“…Let Γ be the group generated by reflections I j through ∂B j , B j ∈ T . To guarantee that the group Γ is Kleinian we will use the Poincaré Polyhedron Theorem (see [8], [18], and [22]). This theorem establishes conditions for the group to be discrete.…”

“…In this case one can show that the sphere is necessarily fractal (possibly unknotted). Examples of wild knots in S 3 , which are limit sets of geometrically finite Kleinian groups, were obtained by Maskit [22], Kapovich [17], Hinojosa [13], and Gromov, Lawson and Thurston [9]. An example of a wild 2-sphere in S 4 which is the limit set of a geometrically finite Kleinian group was obtained by the second-named author [14] and, independently, by Belegradek [5] (see also [4] for a wild limit set S 2 → S 3 ).…”