arXiv subjects
Publications and source records attributed to Assaf Bar-Natan.
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 Gromov boundary of the curve complex. After showing that the grand arc graph satisfies a bounded geodesic image theorem, we also prove that the boundary is not compact.
We introduce a numerical invariant $\zeta(\Sigma)$ measuring the end-complexity of $\Sigma$ and use it to organize coarse-geometric features of Map($\Sigma$). Our main tool is the \emph{non-peripheral curve graph} $C_{\rm np}(\Sigma)$, whose vertices are those essential simple closed curves that cannot be pushed out of every compact subsurface, with edges given by disjointness. Assuming Map($\Sigma$) is CB-generated and $\zeta(\Sigma)\ge 5$, we prove that $C_{\rm np}(\Sigma)$ is connected, has infinite diameter, is Gromov hyperbolic, and that the Map($\Sigma$)-action has unbounded orbits. As applications, we show that if $\zeta(\Sigma)\ge 4$ then Map($\Sigma$) has infinite coarse rank, and if $\zeta(\Sigma)\ge 5$ then Map($\Sigma$) has at most quadratic divergence, hence is one-ended.
We analyse the convex structure of the Finsler infinitesimal balls of the Thurston metric on Teichm{\"u}ller space. We analyse the convex structure of the Finsler unit ball in the tangent space at each point of Teichm\''uller space of a closed surface of genus $\geq 2$ equipped with Thurston's metric. We obtain a characterisation of faces, exposed faces and extreme points of such a unit sphere. In particular, we prove that every face has a unique naturally associated chain-recurrent geodesic lamination such that this face consists of the unit tangent vectors which are linear combinations of stretch vectors along maximal chain-recurrent geodesic laminations containing the given one. We show that a face is exposed if and only if its associated chain-recurrent geodesic lamination is the support of a measured lamination. Furthermore, we show that a point on a tangent unit sphere is an extreme point if and only if it is a stretch vector along some maximal chain-recurrent geodesic lamination. The last result gives an affirmative answer to a conjecture whose answer was known positively in the case where the surface is either the once-punctured torus or the 4-punctured sphere. Our main results also provide an alternative approach to the topological part of the infinitesimal rigidity result concerning Thurston's metric and the equivariance property of stretch vectors.
The Thurston metric on Teichmuller space, first introduced by W. P. Thurston is an asymmetric metric on Teichmuller space defined by $d_{Th}(X,Y) = \frac12 log\sup_α \frac{l_α(Y)}{l_α(X)}$. This metric is geodesic, but geodesics are far from unique. In this thesis, we show that in the once-punctured torus, and in the four-times punctured sphere, geodesics stay a uniformly-bounded distance from each other. In other words, we show that the \textbf{width} of the geodesic envelope, E(X,Y) between any pair of points $X,Y \in T(S)$ (where $S = S_{1,1}$ or $S = S_{0,4}$) is bounded uniformly. To do this, we first identify extremal geodesics in Env(X,Y), and show that these correspond to stretch vectors, proving a conjecture of Huang, Ohshika and Papadopoulos. We then compute Fenchel-Nielsen twisting along these paths, and use these computations, along with estimates on earthquake path lengths, to prove the main theorem.
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 different self-similar equivalence classes of maximal ends, then the grand arc graph is infinite-diameter and $δ$-hyperbolic. In this case, we also show that the mapping class group acts on the grand arc graph by isometries and acts on the visible boundary continuously. When the surface has stable maximal ends, we also show that this action has finitely many orbits.
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 surfaces defined by Fossas and Parlier. We show that the extended mapping class group is isomorphic to a proper subgroup of the automorphism group of the flip graph, unlike in the finite-type case. This shows that Ivanov's metaconjecture, which states that any "sufficiently rich" object associated to a finite-type surface has the extended mapping class group as its automorphism group, does not extend to simultaneous flip graphs of infinite-type surfaces.
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 radius parameter. We argue that it gives the foundation for a definition of Ricci curvature well adapted to geometric group theory, beginning by observing that the sign can easily be characterized in terms of conjugation in the group. With this conjugation curvature $κ$, abelian groups are identically flat, and in the other direction we show that $κ\equiv 0$ implies the group is virtually abelian. Beyond that, $κ$ captures known curvature phenomena in right-angled Artin groups (including free groups) and nilpotent groups, and has a strong relationship to other group-theoretic notions like growth rate and dead ends. We study dependence on generators and behavior under embeddings, and close with directions for further development and study.
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 compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper score, the convex hull score, and the Reock score, and these scores are used to compare two or more districts or plans. In this paper, we prove mathematically that any map projection from the sphere to the plane reverses the ordering of the scores of some pair of regions for all three of these scores. Empirically, we demonstrate that the effect of using the Cartesian latitude-longitude projection on the order of Reock scores is quite dramatic.
Let $D_n$ be the $n$-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 $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs intersecting at most twice must have a corner or a spur.