Woojin Jeon

Let be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space so that there exists a continuous -equivariant map <!-- MATH $i:\partial G\to Z$ --> , which we call a Cannon-Thurston map. We obtain two characterizations (a dynamical one and a geometric one) of conical limit points in in terms of their pre-images under the Cannon-Thurston map . As an application we prove, under the extra assumption that the action of on has no accidental parabolics, that if the map is not injective, then there exists a non-conical limit point with <!-- MATH $|i^{-1}(z)|=1$ --> . This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if is a non-elementary torsion-free word-hyperbolic group, then there exists <!-- MATH $x\in \partial G$ --> such that is not a ``controlled concentration point'' for the action of on <!-- MATH $\partial G$ --> .

Let $G$ be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space $Z$ so that there exists a continuous $G$-equivariant map $i:\partial G\to Z$, which we call a \emph{Cannon-Thurston map}. We obtain two characterzations (a dynamical one and a geometric one) of conical limit points in $Z$ in terms of their pre-images under the Cannon-Thurston map $i$. As an application we prove, under the extra assumption that the action of $G$ on $Z$ has no accidental parabolics, that if the map $i$ is not injective then there exists a non-conical limit point $z\in Z$ with $|i^{-1}(z)|=1$. This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if $G$ is a non-elementary torsion-free word-hyperbolic group then there exists $x\in \partial G$ such that $x$ is not a "controlled concentration point" for the action of $G$ on $\partial G$.

Let $G,H$ be two Kleinian groups with homeomorphic quotients $\mathbb{H}^{3}/G$ and $\mathbb{H}^{3}/H$ . We assume that $G$ is of divergence type, and consider the Patterson–Sullivan measures of $G$ and $H$ . The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map $\widehat{k}$ from the limit set $\unicode[STIX]{x1D6EC}_{G}$ of $G$ to that of $H$ is either the restriction of a Möbius transformation or totally singular. In this paper, we shall show that such $\widehat{k}$ always exists. In fact, we shall construct $\widehat{k}$ concretely from the Cannon–Thurston maps of $G$ and $H$ .

We consider a finitely generated torsion free Kleinian group $H$ and a random walk on $H$ with respect to a symmetric nondegenerate probability measure $\mu$ with finite support. When $H$ is geometrically infinite without parabolics or when $H$ is Gromov hyperbolic with parabolics, we prove that the Patterson-Sullivan measure is singular with respect to the harmonic measure coming from $\mu$.

In this paper, we give a complete criterion for a discrete faithful representation $\rho:F_n \ra \pslc$ to be primitive stable. This will answer Minsky's conjectures about geometric conditions on $\H^3/\rho(F_n)$ regarding the primitive stability of $\rho$.