Limit points of Kleinian groups and finite sided fundamental polyhedra

Abstract

Let G be a discrete subgroup of SL(2, C)/{• 1}. Then G operates as a discontinuous group of isometrics on hyperbolic 3-space, which we regard as the open unit ball B a in Euclidean 3-space E a. G operates on S 2, the boundary of B a, as a group of conformal homeomorphisms, but it need not be discontinuous there. The set of points of S 2 at which G does not act discontinuously is the limit set A(G).If we fix a point 0 in B a, then the orbit of 0 under G accumulates precisely at A(G).The approximation is, however, not uniform. We distinguish a class of limit points, called points o/aproximation, which are approximated very well by translates of 0. The set of points of approximation includes all loxodromic (including hyperbolic) fixed points, and includes no parabolic fixed points. In w 1 we give several equivalent definitions of point of approximation, and derive some properties. We remark that these points were first discussed by Hedlund [7].Starting with a suitable point 0 in B a, we can construct the Dirichlet fundamental polyhedron P0 for G. It was shown by Greenberg [5] that even if G is finitely generated, P0 need not have finitely many sides. Our next main result, given in w 2, is that if P0 is finite-sided, then every point of A(G) is either a point of approximation or a cusped parabolic fixed point (roughly speaking a parabolic fixed point is cusped if it represents the right number of punctures in (S2-A(G))/G).The above theorem has several applications: one of these is a new proof of the following theorem of Ahlfors [1].I/Po has finitely many sides, then the Euclidean measure o/ A(G) is either 0 or 4~.Our next main result, given in w 3, is that the above necessary condition for P0 to have finitely many sides is also sufficient. In fact, we prove that any convex fundamental polyhedron G has finitely many sides if and only if A(G) consists entirely of points of 1-742908 Acta mathematica 132. Imprim6 1r 18 Mars 1974

Subgroups of mapping class groups from the geometrical viewpoint

“…In our setting: a limit point λ in Λ G is said to be conical if every geodesic ray in T(S) with direction λ has a metric neighborhood containing infinitely many points in a G-orbit-we stress to the reader that the definition of conical is adapted from the geometric definition in Kleinian groups [6], rather than from the theory of convergence groups [15].…”

Geometry and topology of complex hyperbolic and Cauchy-Riemannian manifolds

“…The original definition of a geometrically finite manifold M (due to L.Ahlfors [ Ah]) came from an assumption that M may be decomposed into a cell by cutting along a finite number of its totally geodesic hypersurfaces. The notion of geometrical finiteness has been essentially used in the case of real hyperbolic manifolds (of constant sectional curvature), where geometric analysis and ideas of Thurston have provided powerful tools for understanding of their structure, see [BM,MA,Th,A1,A3]. Some of those ideas also work in spaces with pinched negative curvature, see [Bow].…”

Kleinian Groups in Higher Dimensions

“…The reason for this name comes from the shape of the r-neighborhood of the vertical geodesic α in the upper half-space model of H n+1 : It is a Euclidean cone with the axis α. Equivalently, one can describe the conical limit points of nonelementary groups as follows (see [14,39]): 4 ξ ∈ Λ(Γ) is a conical limit point if and only if for every η ∈ Λ(Γ) \ {ξ} there exists a point ψ and a sequence γ i ∈ Γ such that:…”

“…lim i γ i (ζ) = ξ for every ζ ∈ Λ(Γ) \ {ψ}. [14], B. Bowditch [39]) A Kleinian group Γ is geometrically finite if and only if each limit point ξ ∈ Λ(Γ) is either a conical limit point or a bounded parabolic point.…”

“…Definition 2.3. A limit point ξ ∈ Λ(Γ) is called a conical limit point if there exists a geodesic α ⊂ H n asymptotic to ξ, a point o ∈ H n , a number r < ∞, and a sequenceThe reason for this name comes from the shape of the r-neighborhood of the vertical geodesic α in the upper half-space model of H n+1 : It is a Euclidean cone with the axis α. Equivalently, one can describe the conical limit points of nonelementary groups as follows (see [14,39]): 4 ξ ∈ Λ(Γ) is a conical limit point if and only if for every η ∈ Λ(Γ) \ {ξ} there exists a point ψ and a sequence γ i ∈ Γ such that:i (η). The set of conical limit points of a Kleinian group Γ is denoted Λ c (Γ).…”