Related papers: An Interpolation from Sol to Hyperbolic Space

Motivated by collapsing of Riemannian manifolds and inhomogeneous scaling of left invariant Riemannian metrics on a real Lie group $G$ with a sub-group $H$, we introduce a family of interpolation equations on $G$ with a parameter…

We characterize the Lie groups with finitely many connected components that are $O(u)$-bilipschitz equivalent (almost quasiisometric in the sense that the sublinear function $u$ replaces the additive bounds of quasiisometry) to the real…

We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlev\'{e} transcendents. The main results…

We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with…

In this paper we deal with the class C of decomposable solvable Lie groups having dimension at most six. We determine those Lie groups in C and their subgroups which are the multiplication group Mult(L) and the inner mapping group Inn(L)…

We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space…

In this succinct note, it is showed that a partition function of equivalent classes of hyperbolic surfaces can be connected to an Ising model located on the boundary of the Poincare disc, as hinted by Poincare's Uniformization theorem and…

We study the isometry group of a globally hyperbolic spatially compact Lorentz surface. Such a group acts on the circle, and we show that when the isometry group acts non properly, the subgroups of $\mathrm{Diff}(\mathbb{S}^1)$ obtained are…

We investigate the finite-dimensional Lie groups whose points are separated by the continuous homomorphisms into groups of invertible elements of locally convex algebras with continuous inversion that satisfy an appropriate completeness…

We prove a real interpolation characterization for some non Euclidean H\"older spaces, built on the Lie structure induced by a class of ultra-parabolic Kolmogorov-type operators satisfying the H\"ormander condition. As a by-product we also…

We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric…

In this paper, we continue with the results in \cite{Pg} and compute the group of quasi-isometries for a subclass of split solvable unimodular Lie groups. Consequently, we show that any finitely generated group quasi-isometric to a member…

In this article we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in $\Bbb{P}^2_{\Bbb{C}}$. As a corollary we get that every discrete compact surface group in $\PO^+(2,1)$ admits a deformation…

We prove that ideal sub-Riemannian manifolds (i.e., admitting no non-trivial abnormal minimizers) support interpolation inequalities for optimal transport. A key role is played by sub-Riemannian Jacobi fields and distortion coefficients,…

Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily…

We show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an open subset of an arithmetic quotient of the real hyperbolic plane. Moreover, our main result says that the induced metric on this…

Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4…

In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric…

In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs $(P,\tau)$, where $P$ is a compact Riemann surface with a finite number of holes and punctures and $\tau:P\to P$ is an anti-holomorphic involution. We…

We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and…