Related papers: Bowditch representations in Gromov-hyperbolic spac…
We prove that the set of Bowditch representations (introduced by Bowditch in 1998, then generalized by Tan, Wong and Zhang in 2008) and the set of primitive-stable representations (introduced by Minsky in 2013) of the free group of rank two…
We prove the equivalence of two conditions on the primitive elements in an $SL(2,\mathbb C)$ representation of the free group $F_2$ on two generators, which may hold even when the image of $F_2$ is not discrete. One is Minsky's condition of…
The equivalence of two conditions on the primitive elements in an $SL(2,\mathbb C)$ representation of the free group $F_2 = <a,b>$, namely Minsky's condition of primitive stability and the $BQ$-conditions introduced by Bowditch and…
In this paper, we study representations from the four-punctured sphere group into isometry groups of Gromov-hyperbolic spaces. We prove that the set of simple-stable representations (in analogy with Minsky's notion of primitive-stability)…
The two largest known domains of discontinuity for the action of Out(F_2) on the PSL(2,C)-character variety of F_2 - defined by Minsky's primitive stability, and Bowditch's Q-conditions - were proven to be equal independently by Lee-Xu and…
For the action of the outer automorphism group of the rank two free group on the corresponding variety of PSL(2,C) characters, two domains of discontinuity have been known to exist that are strictly larger than the set of Schottky…
We study general representations of the free group on two generators into $SL(2,C)$, and the connection with generalized Markoff maps, following Bowditch. We show that Bowditch's Q-conditions for generalized Markoff maps are sufficient for…
Coarse median spaces simultaneously generalise the classes of hyperbolic spaces and median algebras, and arise naturally in the study of the mapping class groups and many other contexts. One issue with their definition as originally…
Based on the work of Farb, Bowditch, and Groves-Manning on discrete relatively hyperbolic groups, we introduce an approach to relative hyperbolicity for totally disconnected locally compact (TDLC) groups. For compactly generated TDLC…
Carrier graphs were first introduced for closed hyperbolic 3-manifolds by White. In this paper, we first generalize this definition to carrier graphs for representations of a rank two free group into the isometry group of hyperbolic three…
An irreducible representation of the free group on two generators X,Y into SL(2,C) is determined up to conjugation by the traces of X,Y and XY. We study the diagonal slice of representations for which X,Y and XY have equal trace. Using the…
In this note we use Yaman's dynamical characterization of relative hyperbolicity to prove a theorem of Bowditch about relatively hyperbolic pairs $(G,\mathcal{H})$ with $G$ hyperbolic. Our proof additionally gives a description of the…
In this paper we show that the existence of a non-parabolic local cut point in the Bowditch boundary $\partial(G,\mathbb{P})$ of a relatively hyperbolic group $(G,\mathbb{P})$ implies that $G$ splits over a $2$-ended subgroup. This theorem…
With the aid of a Gromov hyperbolic characterization of uniform domains, we first give an affirmative answer to an open question arisen by V\"ais\"al\"a under weaker assumption. Next, we show that the three-point condition introduced by…
We give a generalization to convex co-compact semigroups of a beautiful theorem of Patterson-Sullivan, telling that the critical exponent (that is the exponential growth rate) equals the Hausdorff dimension of the limit set (that is the…
This is the second article of a series of our recent works, addressing an open question of Bonk-Heinonen-Koskela [3], to study the relationship between (inner) uniformality and Gromov hyperbolicity in infinite dimensional spaces. Our main…
An algebraic extension of the rational numbers is said to have the $\textit{Bogomolov property}$ (B) if the absolute logarithmic Weil height of its non-torsion elements is uniformly bounded from below. Given a continuous representation…
Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of…
Let $\Gamma$ be a hyperbolic group and G be the isometry group of a Gromov-hyperbolic, properand geodesic metric space. We study the action of the outer automorphism group Out($\Gamma$) onthe set X($\Gamma$,G) of conjugacy classes of…
Our monograph presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Our work unifies and extends a long list of results by many authors. We make it a point to avoid any…