UB Geometry and Topology Seminar 

Abstract: In this talk we answer questions regarding the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop α, the length of α along a balanced folding path is not larger than the maximum of its lengths at the end points. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counterexamples.

Abstract: We discuss the ways in which the geometry of a group $G$ constrains the possible behavior of symbolic dynamical systems over $G$. In particular, we explain our results with Chaim Goodman-Strauss and Yoav Rieck on SFTs over hyperbolic groups.

Abstract: The geodesicity condition for Gromov delta hyperbolic groups is a coarse local condition, in other words, in order to check whether a given edge path is a geodesic you need only to check that subsegments of length $m$, where $m$ depends on delta, are geodesics. The formal way of stating the above is to say that hyperbolic groups admit a regular language that reads all geodesics in the group. This seemingly simple observation has a lot of interesting consequences. For example, one can use it to show that hyperbolic groups have a rational growth function. Another consequence is the fact that the boundary of a hyperbolic group is a subshift of finite type (over $\mathbb{Z}$). Charney and Sultan introduced the notion of contracting boundaries for CAT(0) spaces, in short, these are the collection of all "hyperbolic directions" in a CAT(0) space. This talk will be about our theorem with Josh Eike proving that contracting geodesics in CAT(0) groups are realized by a regular language.

Abstract: Let $S_g$ be a closed oriented surface of genus $g \geq 2$ and $\mathcal{C}^1(S_g)$ be its curve complex—vertices are homotopy classes of essential simple closed curves with two vertices sharing an edge if they have disjoint representatives. It is known that $\mathcal{C}(S_g)$ is path connected , and the distance, $d(\alpha , \beta)$, between two vertices $\alpha , \beta \in \mathcal{C}^1(S)$ is just the minimal count of the number of edges in an edge-path between $\alpha$ and $\beta$. One can also consider, $ i(\alpha , \beta)$, the minimal intersection between curve representatives of $\alpha$ and $\beta$. This talk discusses how $i(\alpha , \beta)$ will grow as $d(\alpha, \beta)$ grows. This is joint work with Dan Margalit.

Abstract: Random right-angled Coxeter groups have been studied extensively using methods and results from random graph theory. Some of these methods can be extended to study random Coxeter groups in general. In this talk, I will introduce random Coxeter groups and give several results.

Abstract: Let $S_g$ be a closed oriented surface of genus $g \geq 2$ and $\mathcal{C}^1(S_g)$ be its curve complex—vertices are homotopy classes of essential simple closed curves with two vertices sharing an edge if they have disjoint representatives. It is known that $\mathcal{C}(S_g)$ is path connected , and the distance, $d(\alpha , \beta)$, between two vertices $\alpha , \beta \in \mathcal{C}^1(S)$ is just the minimal count of the number of edges in an edge-path between $\alpha$ and $\beta$. One can also consider, $ i(\alpha , \beta)$, the minimal intersection between curve representatives of $\alpha$ and $\beta$. This talk discusses how $i(\alpha , \beta)$ will grow as $d(\alpha, \beta)$ grows. This is joint work with Dan Margalit.

Abstract: Assuming that both temperature and pressure are continuous functions, we can conclude that there are always two antipodal points on Earth with exactly the same pressure and temperature. This is the two-dimensional version of the celebrated Borsuk-Ulam Theorem which states that for any continuous map from the n-dimensional sphere to n-dimensional real Euclidean space there is always a pair of antipodal points on the sphere that are identified by the map. Our quest to unravel topological mysteries in the Middle Earth of quantum spaces will begin with gentle preparations in the Shire of elementary topology. Then, after riding swiftly through the Rohan of $C^\ast$-algebras and Gelfand-Naimark Theorems, and carefully avoiding the Mordor of incomprehensible technicalities, we shall arrive in the Gondor of compact quantum groups acting freely on unital $C^\ast$-algebras. It is therein that the generalized Borsuk-Ulam-type statements dwell waiting to be proven or disproven. To end with, we will pay tribute to the ancient quantum group $SUq(2)$, and show how the proven non-trivializability of Pflaum's $SUq(2)$-principal instanton bundle is a special case of two different noncommutative Borsuk-Ulam-type conjectures. Time permitting, we shall also explain a general method to extend the non-trivializability result from the Pflaum quantum instanton bundle to an arbitrary finitely iterated equivariant join of $SUq(2)$ with itself. The latter is a quantum sphere with a free $SUq(2)$-action whose space of orbits defines a quantum quaternionic projective space. (Based on joint work with Paul F. Baum, Ludwik Dabrowski, Tomasz Maszczyk and Sergey Neshveyev.)

Abstract: In geometric group theory a finitely presented group is often identified with its Cayley graph. Unfortunately, the Cayley graph depends on the generating set of the group. When the generating set is changed the homeomorphism type of the graph can change quite a lot. However, there is a metric equivalence relation called a quasi-isometry (QI) and the Cayley graph *is* well defined up to quasi-isometries. Unfortunately quasi-isometries are a very weak geometric equivalence (in general there may be no continuous bijections between quasi- isometric spaces!). In this context a very natural question to ask is, if two groups are quasi-isometric how similar are they as groups? For some classes of groups the answer is quite surprising! For example, if a group $G$ is QI to $\pi_1(S)$ where $S$ is a surface group, then $G$ is (virtually) isomorphic to a surface group! A class of groups which exhibits this property is called quasi-isometrically rigid. Many classes of groups are quasi-isometrically rigid, free abelian groups, finite volume discrete subgroups of a non-compact Lie group, surface groups, etc. Often, some form of negative curvature plays an important role in proving rigidity style theorems. The Morse property is a natural generalization of the behavior of geodesics in negatively curved manifolds. The Morse boundary is a quasi-isometry invariant for finitely generated groups that serves a similar role that the boundary of hyperbolic n-space serves for hyperbolic manifold groups. I will introduce all of the objects in question and talk about some results for the morse boundary that make it a promising tool to study the quasi-isometric rigidity phenomena of some classes of groups, in particular some CAT(0) groups, Mapping Class groups, some artin/coxeter groups etc. This is joint work with Ruth Charney and Matt Cordes.