Metric conformal structures and hyperbolic dimension

Metric Conformal Structures and Hyperbolic Dimension

For any hyperbolic complex X and a ∈ X we construct a visual metric ď = ďa on ∂X that makes the Isom(X)-action on ∂X bi-Lipschitz, Möbius, symmetric and conformal. We define a stereographic projection of ďa and show that it is a metric conformally equivalent to ďa. We also introduce a notion of hyperbolic dimension for hyperbolic spaces with group actions. Problems related to hyperbolic dimensi...

Conformal dimension and canonical splittings of hyperbolic groups

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic groups and show an interesting relationship between conformal dimension and some canonical splittings of the group.

Conformal Dimension and the Quasisymmetric Geometry of Metric Spaces

Moduli of Flat Conformal Structures of Hyperbolic Type

To each flat conformal structure (FCS) of hyperbolic type in the sense of Kulkarni-Pinkall, we associate, for all θ ∈ [(n − 1)π/2, nπ/2[ and for all r > tan(θ/n) a unique immersed hypersurface Σr,θ = (M, ir,θ) in H of constant θ-special Lagrangian curvature equal to r. We show that these hypersurfaces smoothly approximate the boundary of the canonical hyperbolic end associated to the FCS by Kul...

The metric dimension and girth of graphs

A set $Wsubseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,vin V(G)$ there exists $win W$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $dim(G)$. In this paper, it is proved that in a connected graph $...