We prove that for any closed surface of genus at least four, and any punctured surface of genus at least two, the space of ending laminations is connected. A theorem of E. Klarreich implies that this space is homeomorphic to the Gromov boundary of the complex of curves. It follows that the boundary of the complex of curves is connected in these cases, answering the conjecture of P. Storm. Other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate Kleinian groups.
We show that if S is a finite type orientable surface of negative Euler characteristic which is not the 3-holed sphere, 4-holed sphere or 1-holed torus, then the ending lamination space of S is connected, locally path connected and cyclic.
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian punctured surface groups without accidental parabolics. In this note we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. This extends earlier work done for closed surface groups.
We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence $$1\to H\to G \to Q \to 1 $$ of hyperbolic groups. These laminations arise in different contexts: existence of Cannon-Thurston maps; closed geodesics exiting ends of manifolds; dual to actions on $\R-$trees. We use the relationship between these laminations to prove quasiconvexity results for finitely generated infinite index subgroups of $H$, the normal subgroup in the exact sequence above.
Yair Minsky showed that punctured torus groups are classified by a pair of ending laminations (ν_-,ν_+). In this note, we show that there are ending laminations ν_+ such that for any choice of ν_-, the punctured torus group is transcendental as a subgroup of PSL_2 C.
We define an ending lamination for a Weil-Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil-Petersson metric, these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem 1.1) for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex. As an application, we establish fundamentals of the topological d…
We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichmüller geodesics does not hold for WP geodesics.
We present a new proof of the bi-Lipschitz model theorem, which occupies the main part of the Ending Lamination Conjecture proved by Minsky and Brock-Canary-Minsky. Our proof is done by using techniques of standard hyperbolic geometry as much as possible.
We show that if S is a finite type orientable surface of genus g and p punctures where 3g+p > 4, then EL(S) is (n-1)-connected and (n-1)-locally connected where dim(PML(S))=2n+1=6g+2p-7. Furthermore, if g=0, then EL(S) is homeomorphic to the p-4 dimensional Nobeling space.
In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one. This completes the proof of the conjectural picture of Cannon-Thurston maps for surface groups.
Thurston's Ending Lamination Conjecture states that a hyperbolic 3-manifold N with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when N has incompressible ends relative to its cusps follows readily. The main ingredient is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in math.GT/0302208, and a subsequent paper will establish the Ending Lamination Conjecture in general.
We characterize which cobounded quasigeodesics in the Teichmueller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path gamma in T, we show that gamma is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over gamma is a hyperbolic metric space. As an application, for complete hyperbolic 3-manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically inf…
In this paper we construct examples of Weil-Petersson geodesics with nonminimal ending laminations which have 1-dimensional limit sets in the Thurston compactification of Teichmüller space.
For geometrically finite Kleinian surface groups, Bonahon and Otal proved the existence part, and partly the uniqueness part of the bending lamination conjecture. In this paper, we generalise the existence part to general Kleinian surface groups including geometrically infinite ones. We furthermore prove the compactness of the set of Kleinian surface groups realising an arbitrarily fixed data of bending laminations and ending laminations. Our proof is independent of that of Bonahon and Otal.
We show that the strong asymptotic class of Weil-Petersson (WP) geodesics with narrow end invariant and bounded annular coefficients is determined by the forward ending lamination. This generalizes the Recurrent Ending Lamination Theorem of Brock-Masur-Minsky. As an application we provide a symbolic condition for divergence of WP geodesic rays in the moduli space.
In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit set a circle in the Thurston compactification.
We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations. Applications include an improvement to the bounded geometry versions of Thurston's ending lamination conje…
When 1 -> H -> G -> Q -> 1 is a short exact sequence of three infinite, word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an "ending lamination" on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a de…
We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.
We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author. We study finite covers between such surfaces and show that lifts of nonseparating curves to the nonseparating curve graph of the cover span quasiconvex subgraphs which are infinite diameter and not coarsely equal to the nonseparating curve graph of the cover. In the finite type case, we also reprove a theorem of Hamenstädt identifying the Gromov boundary wi…
We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have `bounded geometry' (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed using the ending invariants of the group and the complex of curves. These results are directly analogous to those obtained in the case of punctured-torus surface groups. In that setting the ending invari…
We study Weil-Petersson (WP) geodesics with narrow end invariant and develop techniques to control length-functions and twist parameters along them and prescribe their itinerary in the moduli space of Riemann surfaces. This class of geodesics is rich enough to provide for examples of closed WP geodesics in the thin part of the moduli space, as well as divergent WP geodesic rays with minimal filling ending lamination. Some ingredients of independent interest are the following: A strength version of Wolpert's Geodesic Limit Theorem proved in Sec.4. The stability of hierarchy resolution paths bet…
Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\H^3\cup Ω_G)/G$. Suppose that for each boundary component of $M$, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit $Γ$ of quasi-conformal deformations of $G$ such that there is a homeomorphism $h$ from $\mathrm{Int} M$ to $\H^3/Γ$ compatible with the natural isomorphism from $G$ to $Γ$, the given laminations are unrealisable in $\H^3/Γ$, and the …
Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus $g$ for which the space of measures has the maximal dimension $3g-3$, for example. We also s…
We give the first part of a proof of Thurston's Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a ``Lipschitz model'' for the thick part of the corresponding hyperbolic manifold. This enables us to describe the topological structure of the thick part, and to give a-priori geometric bounds.
