Related papers: Controlled Floyd Separation and Non Relatively Hyp…

We study conditions under which quasi-conformal homeomorphisms are quasi-isometries. We show that if two nilpotent geodesic Lie groups are quasi-conformally homeomorphic, then they are quasi-isometrically equivalent. We also give more…

We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the…

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichm\"uller space, Hitchin representations and geodesic currents. We add to…

Let $G = V, E$ be a simple connected undirected graph. A set $X \subseteq V$ is \emph{geodesically convex} if for any pair of vertices $x, y \in X$, all vertices on all shortest paths in $G$ from $x$ to $y$ are contained in $X$. A set $H…

This paper proves under certain conditions the existence of an algorithm, which detects relatively quasiconvex subgroups $H$ of relatively hyperbolic groups $(G,\mathbb{P})$. Additionally, this algorithm outputs an induced peripheral…

The first author and Oguni introduced a class of groups of non-positive curvature, called coarsely convex group. The recent success of the theory of groups which are hyperbolic relative to a collection of subgroups has motivated the study…

In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by…

We consider splittings of groups over finite and two-ended subgroups. We study the combinatorics of such splittings using generalisations of Whitehead graphs. In the case of hyperbolic groups, we relate this to the topology of the boundary.…

Let $G$ be a group hyperbolic relative to a finite collection of subgroups $\mathcal P$. Let $\mathcal F$ be the family of subgroups consisting of all the conjugates of subgroups in $\mathcal P$, all their subgroups, and all finite…

A result of Gersten states that if $G$ is a hyperbolic group with integral cohomological dimension $\mathsf{cd}_{\mathbb{Z}}(G)=2$ then every finitely presented subgroup is hyperbolic. We generalize this result for the rational case…

Let $G$ be a 1-connected, almost-simple Lie group over a local field and $\mathcal{S}$ a subsemigroup of $G$ with non-empty interior. The action of the regular hyperbolic elements in the interior of $\mathcal{S}$ on the flag manifold $G/P$…

In proper hyperbolic geodetic spaces we construct rooted $\mathbb R$-trees with the following properties. On the one hand, every ray starting at the root is quasi-geodetic; so these $\mathbb R$-trees represent the space itself well. At the…

For a given divison algebra of the quaternions we construct two types of units: Pell units and Gauss units. If K is a rational quadratic extension and G is a finite group, we classify R and G, s.t., the unit group U(RG) of augmentation one…

We study properties of "hyperbolic directions" in groups acting cocompactly on properly convex domains in real projective space, from three different perspectives simultaneously: the (coarse) metric geometry of the Hilbert metric, the…

We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of $Out(F_n)$ as subgroups such that an orbit map in the free…

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…

Consider the following classes of pairs consisting of a group and a finite collection of subgroups: $\mathcal{C}= \left\{ (G,\mathcal H) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right\}$ and $ \mathcal{D}= \left\{…

The first goal of this article is to investigate a refinement of previously-introduced strongly regular hyperbolic automorphisms of locally finite thick Euclidean buildings $\Delta$ of finite Coxeter system $(W,S)$. The new ones are defined…

We show that the set $SA(G)$ of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group $G$ is dense in the product of the sets $SA(P)$ over all maximal parabolic subgroups $P$. The set $BSA(G)$…

Let $G$ be a group hyperbolic relative to a collection of subgroups $\{H_\lambda ,\lambda \in \Lambda \} $. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{H_\lambda ,\lambda \in…