Related papers: Large-scale geometry of graphs interpolating betwe…
A family of interpolating graphs $\calC (S, \xi)$ of complexity $\xi$ is constructed for a surface $S$ and $-2 \leq \xi \leq \xi (S)$. For $\xi = -2, -1, \xi (S) -1$ these specialise to graphs quasi-isometric to the marking graph, the pants…
The fine curve graph of a surface is a graph whose vertices are essential simple closed curves and whose edges connect disjoint curves. Following a rich history of hyperbolicity of various graphs associated to surfaces, the fine curve graph…
The paper discusses quantum graphs with a vertex coupling which interpolates between the common one of the $\delta$ type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the…
The pants graph has proved to be influential in understanding 3-manifolds concretely. This stems from a quasi-isometry between the pants graph and the Teichm\"uller space with the Weil-Petersson metric. Currently, all estimates on the…
We study arc graphs and curve graphs for surfaces of infinite topological type. First, we define an arc graph relative to a finite number of (isolated) punctures and prove that it is a connected, uniformly hyperbolic graph of infinite…
We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex…
This note is about the geometry of the pants graph P(S), a natural simplicial graph associated to a finite type topological surface S where vertices represents pants decompositions. The main result in this note ascserts that for a…
We consider large random graphs with prescribed degrees, such as those generated by the configuration model. In the regime where the empirical degree distribution approaches a limit $\mu$ with finite mean, we establish the systematic…
We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.
We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense.…
We introduce a spatial graph and hypergraph model that smoothly interpolates between a graph with purely pairwise edges and a graph where all connections are in large hyperedges. The key component is a spatial clustering resolution…
We introduce a coarse perspective on relations of the $SU(2)$-Witten-Reshetikhin-Turaev TQFT, the Weil-Petersson geometry of the Teichm\"uller space, and volumes of hyperbolic 3-manifolds. Using data from the asymptotic expansions of the…
Initial steps in the study of inner expansion properties of infinite Cayley graphs and other infinite graphs, such as hyperbolic ones, are taken, in a flavor similar to the well-known Lipton-Tarjan square root separation result for planar…
Hypergraphs are a generalization of graphs in which edges can connect any number of vertices. They allow the modeling of complex networks with higher-order interactions, and their spectral theory studies the qualitative properties that can…
We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs…
This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs,…
The L-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel shapes in the plane. A subfamily of these graphs are {L, |, --}-contact graphs which are the contact graphs of axis parallel L, |,…
A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in…
We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author.…
This paper investigates quasi-isometries between graphs with variable edge lengths. A quasi-isometry is a mapping between metric spaces that approximately preserves distances, allowing for a bounded amount of additive and multiplicative…