Thurston Map Research Articles

Thurston maps and asymptotic upper curvature

A Thurston map is a branched covering map from \(\mathbb {S}^2\) to \(\mathbb {S}^2\) with a finite postcritical set. We associate a natural Gromov hyperbolic graph \(\mathcal {G}=\mathcal {G}(f,{\mathcal {C}})\) with an expanding Thurston map \(f\) and a Jordan curve \({\mathcal {C}}\) on \(\mathbb {S}^2\) containing \({{\mathrm{post}}}(f)\). The boundary at infinity of \(\mathcal {G}\) with associated visual metrics can be identified with \(\mathbb {S}^2\) equipped with the visual metric induced by the expanding Thurston map \(f\). We define asymptotic upper curvature of an expanding Thurston map \(f\) to be the asymptotic upper curvature of the associated Gromov hyperbolic graph, and establish a connection between the asymptotic upper curvature of \(f\) and the entropy of \(f\).

On Cannon–Thurston maps for relatively hyperbolic groups

Abstract.Baker and Riley gave an inclusion of a free group of rank 3 in a hyperbolic group for which the Cannon–Thurston map is not well-defined. By using their result, we show that every non-elementary hyperbolic group can be included in some hyperbolic group in such a way that the Cannon–Thurston map is not well-defined. In fact we generalize their result to every non-elementary relatively hyperbolic group.

Boundary values of the Thurston pullback map

For any Thurston map with exactly four postcritical points, we present an algorithm to compute the Weil-Petersson boundary values of the corresponding Thurston pullback map. This procedure is carried out for the Thurston map f ( z ) = 3 z 2 2 z 3 + 1 f(z)=\frac {3z^2}{2z^3+1} originally studied by Buff, et al. The dynamics of this boundary map are investigated and used to solve the analogue of Hubbard’s Twisted Rabbit problem for f f .

Topological characterization of canonical Thurston obstructions

Let $f$ be an obstructed Thurston map with canonical obstruction$\Gamma_f$. We prove the following generalization of Pilgrim'sconjecture: if the first-return map $F$ of a periodic component $C$ ofthe topological surface obtained from the sphere by pinching thecurves of $\Gamma_f$ is a Thurston map then the canonical obstructionof $F$ is empty. Using this result, we give a complete topologicalcharacterization of canonical Thurston obstructions.

The universal Cannon–Thurston map and the boundary of the curve complex

In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent–Leininger–Schleimer and Mitra, we construct a universal Cannon–Thurston map from a subset of the circle at infinity for the closed surface group onto the boundary of the curve complex of the once-punctured surface. Using the techniques we have developed, we also show that the boundary of this curve complex is locally path-connected.

On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

To each once-punctured-torus bundle, T φ , over the circle with pseudo-Anosov monodromy φ, there are associated two tessellations of the complex plane: one, Δ ( φ ) , is (the projection from ∞ of) the triangulation of a horosphere at ∞ induced by the canonical decomposition into ideal tetrahedra, and the other, CW ( φ ) , is a fractal tessellation given by the Cannon–Thurston map of the fiber group switching back and forth between gray and white each time it passes through ∞. In this paper, we fully describe the relation between Δ ( φ ) and CW ( φ ) .

Cannon–Thurston maps for pared manifolds of bounded geometry

Let N^h be a hyperbolic 3-manifold of bounded geometry corresponding to a hyperbolic structure on a pared manifold (M,P). Further, suppose that (\partial{M} - P) is incompressible, i.e. the boundary of M is incompressible away from cusps. Further, suppose that M_{gf} is a geometrically finite hyperbolic structure on (M,P). Then there is a Cannon- Thurston map from the limit set of M_{gf} to that of N^h. Further, the limit set of N^h is locally connected. This answers in part a question attributed to Thurston.

The Cannon–Thurston map for punctured-surface groups

Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.

On the Hausdorff dimension of the Gieseking fractal

Let φ:R→C be the Cannon–Thurston map associated to the Gieseking manifold; thus φ is also the Cannon–Thurston map associated to the complement of the figure-eight knot. There are sets ER and EC such that φ decomposes into two maps ER→EC and R−ER→C−EC, where the former is (at-least-two)-to-one, while the latter is a homeomorphism. It is known that ER has Hausdorff dimension zero. Let d denote the Hausdorff dimension of EC. We show that EC is a countable union of Jordan curves of Hausdorff dimension d, and that d=1.35±0.2. In particular, the two-dimensional Lebesgue measure of EC is zero, and the Jordan curves are not rectifiable. We also show that d is the critical exponent of a Poincaré series of a discrete semigroup of isometries of hyperbolic three-space, and describe a computer experiment that suggests that d is near 1.2971.

Canonical Thurston Obstructions

We refine Douady and Hubbard's proof of Thurston's topological characterization of rational functions by proving the following theorem. Let f:S2→S2 be a branched covering with finite postcritical set Pf and hyperbolic orbifold. Let Γc denote the set of all homotopy classes γ of nonperipheral, simple closed curves in S2−Pf such that the length of the unique geodesic homotopic to γ tends to zero under iteration of the Thurston map induced by f on Teichmüller space. Then either Γc is empty, and f is equivalent to a rational function, or else Γc is a Thurston obstruction.