9 papers
Gromov boundary of the Grand Arc graph
Carolyn Abbott, Assaf Bar-Natan, Arya Vadnere
We describe a dense subset of the Gromov boundary of the grand arc graph of an infinite-type surface as a space of geodesic laminations, analogous to Klarreich's description of the…
The non-peripheral curve graph and divergence in big mapping class groups
Assaf Bar-Natan, Yulan Qing, Kasra Rafi
We introduce a numerical invariant measuring the end-complexity of and use it to organize coarse-geometric features of Map(). Our main tool is the \emph{non-periphera…
Convex structures of the unit tangent spheres in Teichm{ü}ller space
Assaf Bar-Natan, Ken'Ichi Ohshika, Athanase Papadopoulos
We analyse the convex structure of the Finsler infinitesimal balls of the Thurston metric on Teichm{ü}ller space. We analyse the convex structure of the Finsler unit ball in the ta…
Geodesic Envelopes in the Thurston Metric on Teichmuller space
Assaf Bar-Natan
The Thurston metric on Teichmuller space, first introduced by W. P. Thurston is an asymmetric metric on Teichmuller space defined by $d_{Th}(X,Y) = \frac12 log\sup_α \frac{l_α(Y)}{…
Big Flip Graphs and Their Automorphism Groups
Assaf Bar-Natan, Advay Goel, Brendan Halstead +3
In this paper, we study the relationship between the mapping class group of an infinite-type surface and the simultaneous flip graph, a variant of the flip graph for infinite-type…
The grand arc graph
Assaf Bar-Natan, Yvon Verberne
In this article, we construct a new simplicial complex for infinite-type surfaces, which we call the grand arc graph. We show that if the end space of a surface has at least three…
The Gerrymandering Jumble: Map Projections Permute Districts' Compactness Scores
Assaf Bar-Natan, Elle Najt, Zachary Schutzman
In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well-established, yet competing, notions of geographic compactne…
Conjugation curvature for Cayley graphs
Assaf Bar-Natan, Moon Duchin, Robert Kropholler
We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as "medium-scale" because it is neither infinitesimal nor asymptotic, but based on a chosen finite…
Arcs on Punctured Disks Intersecting at Most Twice with Endpoints on the Boundary
Assaf Bar-Natan
Let be the -punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most…