Related papers: Multiarc and curve graphs are hierarchically hyper…
After Fossas-Parlier, we consider two graphs $\mathcal{G}_{0}(S)$ and $\mathcal{G}_{\infty}(S)$, constructed from multicurves on connected, orientable surfaces of infinite-type. Our first result asserts that $\mathcal{G}_{\infty}(S)$ has…
In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…
We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type…
The strip map is a natural map from the arc complex of a bordered hyperbolic surface $S$ to the vector space of infinitesimal deformations of $S$. We prove that the image of the strip map is a convex hypersurface when $S$ is a surface of…
This article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality…
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…
We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse…
Hyperbolic geometry offers a natural focus + context for data visualization and has been shown to underlie real-world complex networks. However, current hyperbolic network visualization approaches are limited to special types of networks…
In this paper we consider a large family of graphs of hierarchically hyperbolic groups (HHG) and show that their fundamental groups admit HHG structures. To do that, we will investigate the notion of hierarchical quasi convexity and show…
This paper proves the existence of a chiral map with alternating automorphism group for every hyperbolic type. We present a set of constructions using permutations for when at least one parameter is even, and call on previously known…
We show that a bijection $f:\mathbb{H}^2\rightarrow\mathbb{H}^2$ of the hyperbolic plane that sends horocycles to horocycles (respectively hypercycles to hypercycles) is an isometry. This extends a previous result of J. Jeffers on geodesics…
We classify maximal quartic generalised projective special real curves up to equivalence. A maximal quartic generalised projective special real curve consists of connected components of the intersection of the hyperbolic points of a quartic…
We find Dirac-type sufficient conditions for a hypergraph $\mathcal H$ with few edges to be hamiltonian. We also show that these conditions provide that $\mathcal H$ is {\em super-pancyclic}, i.e., for each $A \subseteq V(\mathcal H)$ with…
A connected subgraph of a graph is isometric if it preserves distances. In this short note, we provide counterexamples to several variants of the following general question: When a graph $G$ is edge covered by connected isometric subgraphs…
There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…
It is well-known that every vertex-transitive graph admits a representation as a coset graph. In this paper, we extend this construction by introducing monodromy graphs defined through double cosets. Our main result establishes that every…
Simplicial surfaces describe the incidence relations between vertices, edges and faces of triangulated 2-dimensional manifolds in a purely combinatorial way. By considering only the incidences of edges and faces, simplicial surfaces are…
We give a sufficient condition under which the fundamental group of a reglued graph of surfaces is hyperbolic. A reglued graph of surfaces is constructed by cutting a fixed graph of surfaces along the edge surfaces, then regluing by…
We study a finite sequence of graphs, beginning with the curve graph and ending with a graph quasi-isometric to a tree. There is a Lipschitz map from one graph in the sequence to the next. This sequence was first introduced by Hamenst\"adt.…
We show that minimal length carrier graphs are not unique, but if M is in a large class of hyperbolic 3-manifolds, including the geometrically finite ones, then M has only finitely many minimal length carrier graphs and no two of them are…