Related papers: Grafting, pruning, and the antipodal map on measur…

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third order differential equation determined by the conformal…

Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X(S, PSL(2,C)) and the affine cotangent…

In this paper we define a generalization of the pentagram map to a map on twisted polygons in the Grassmannian space Gr(n;mn). We define invariants of Grassmannian twisted polygons under the natural action of SL(nm), invariants that define…

The Smith embedding of a finite planar map with two marked vertices, possibly with conductances on the edges, is a way of representing the map as a tiling of a finite cylinder by rectangles. In this embedding, each edge of the planar map…

The study of very large graphs is a prominent theme in modern-day mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable…

Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map \sigma_f of the Teichmueller space T modelled on P_f to itself fixing the basepoint corresponding to the identity map…

Geometrization says `` any closed oriented three-manifold which is prime (not a connected sum) carries one of the eight Thurston geometries OR it has incompressible torus walls whose complementary components each carry one of four…

Let $\mathcal T$ be the Teichm\"{u}ller space of marked genus $g$, $n$ punctured Riemann surfaces with its bordification $\Tbar$ the {\em augmented Teichm\"{u}ller space} of marked Riemann surfaces with nodes, \cite{Abdegn, Bersdeg}.…

Let $\widetilde{X}$ be a locally finite complete Gromov hyperbolic metric graph with the geometric boundary consisting of infinitely many points. Suppose that there is a discrete subgroup of the isometry group $Iso(\widetilde{X})$ acting…

For closed hyperbolic $3$-manifolds $M$ with volume less than a constant $V$, we prove an inequality regarding the geometric $L^2$-norm and the topological Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock and…

In this paper we try to look at the compactification of Teichmuller spaces from a tropical viewpoint. We describe a general construction for the compactification of algebraic varieties, using their amoebas, and we describe the boundary via…

This paper is devoted to prove that if an Alexandrov space of curvature not less than $\kappa$ with a codimension one extremal subset which admits an isometric involution with respect to the induced length metric, then the metric space…

Given a finite Lie incidence geometry which is either a polar space of rank at least $3$ or a strong parapolar space of symplectic rank at least $4$ and diameter at most $4$, or the parapolar space arising from the line Grassmannian of a…

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this…

We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a…

We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of…

A Thurston map is a branched covering map $f\colon S^2\to S^2$ that is postcritically finite. Mating of polynomials, introduced by Douady and Hubbard, is a method to geometrically combine the Julia sets of two polynomials (and their…

We survey explicit coordinate descriptions for two (A and X) versions of Teichmuller and lamination spaces for open 2D surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the…

Let $V$ be a two-dimensional vector space over a field $\mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $\nu$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of…

A combinatorial map is a connected topological graph cellularly embedded in a surface. This monograph concentrates on the automorphism group of a map, which is related to the automorphism group of a Klein surface and a Smarandache manifold,…