Conical limit points and the Cannon-Thurston map

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 $\partial G$.Comment: various minor updates; final version; to appear in Conformal Geometry and Dynamic

Theoretical and Experimental Studies of the Dechlorination Mechanism of Carbon Tetrachloride on a Vivianite Ferrous Phosphate Surface

Chlorinated organics are the principal and most frequently found contaminants in soil and groundwater, generating significant environmental problems. Over the past several decades, Fe-containing minerals naturally occurring in aquatic and terrestrial environments have been used as natural electron donors, which can effectively dechlorinate a variety of chlorinated organics. However, a full understanding of the reaction mechanism of the dechlorination pathway cannot be obtained by experimental investigations alone, due to the immeasurability of chemical species formed over a short reaction time. In this report, we describe experiments and density functional theory (DFT) calculations carried out to investigate the complex reduction pathway of carbon tetrachloride (CT) on a vivianite (Fe^(II)_3(PO_4)_2·8H_2O) surface. Our results indicate that chloroform (HCCl_3) and formate are the primary transformation products. The experimental results reveal that the reduction kinetics of CCl_4 can be dramatically accelerated as the pH is increased from 3 to 11. On the basis of the DFT calculations, we found that HCCl_3 can be formed by ^•CCl_3 and :CCl_3^(–*) on a deprotonated vivianite surface (an adsorbate on vivianite is denoted using an asterisk). In addition, :CCl_3^(–*) can be nonreductively dechlorinated to form :CCl_2^* followed by sequential nucleophilic attack by OH^(–*), resulting in the formation of :CCl(OH)^* and :C(OH)_2^*, which are responsible for production of CO and formate, respectively. The results obtained from this study can facilitate the modeling of systems of other halogenated species and minerals, which will provide fundamental insight into their corresponding reaction mechanisms