Fall 2022

Abstract: Gromov introduced some “hyperbolization” procedures, i.e. some procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis then developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. Their procedure has been used to construct examples of manifolds and groups that exhibit various pathologies, despite having negative curvature. We construct actions of the resulting groups on CAT(0) cube complexes. As an application, we obtain that they are virtually special, hence linear over the integers and residually finite. This is joint work with J. Lafont.

Abstract: We will start by introducing the Teichmüller space of a surface, which parametrizes the possible conformal structures it supports. By defining this space analytically, we can equip it with the L^p metrics, of which the Teichmüller and Weil-Petersson metrics are special cases. We will discuss the incompleteness of the L^p metrics on Teichmüller space and what we know about their completions.

Abstract: Sela introduced limit groups in his work on the Tarski problem, and showed that each limit group has a cyclic hierarchy. We prove that a class of relatively hyperbolic groups, equipped with a hierarchy similar to the one for limit groups, is shown to be closed under quasi-isometry. Additionally, these groups share some of the algebraic properties of limit groups. In this talk I plan to present motivation for and introduce the class of groups studied, as well as present some of the results for this class.

Abstract: We study the asymptotic behavior of certain foliations of ends of quasi-Fuchsian manifolds. We introduce a class of such foliations, which we call Asymptotically Poincaré families as they are asymptotic to a family of surfaces determined naturally by the Poincaré metric on the Riemann surface at infinity. We prove the limiting behavior of any Asymptotically Poincaré family is completely determined by the geometry of the quasi-Fuchsian manifold and the conformal structure of the surface at infinity. We then apply our main results to foliations by constant curvature surfaces.

Abstract: Suppose a group G has a finite K(G,1) space X, and suppose we have a sequence of deeper and deeper regular finite sheeted covers of X, so that the corresponding sequence of normal subgroups intersect at {1}.

What can we say about homology of these covers? Rationally, the answer is given by the celebrated Lück Approximation Theorem: the normalized Betti numbers of the covers limit to the L^2-Betti numbers of G.

I will discuss this and the corresponding notions for the torsion part of homology. I will also explain recent computations, joint with Avramidi and Schreve, for right-angled Artin groups, and some consequences.

Abstract: Suppose you have a compact orientable surface with one boundary component. It is known that the Dehn twist about a curve isotopic to the boundary component is not quite like all the other Dehn twists. For one thing, it is central in the mapping class group of the surface. I will prove that such Dehn twists are co-final in every left-ordering of the mapping class group, making them even cooler than originally thought! I will then discuss what this tells us about mapping class group actions on the real line and the fractional Dehn twist coefficient.

This is joint with Adam Clay.

Abstract: Let I = [0,1]. Possibly wild links K and L in a 3-manifold M are (non-ambiently) isotopic if there is a level-preserving embedding e : K × I → M × I such that e(K × {0}) = K × {0} and e(K × {1}) = L × {1}.

Motivating Conjecture: Every (wild) knot in S³ is isotopic to an unknot.

It is not known whether the Bing sling – a wild knot in S³ that pierces no disk – is isotopic to an unknot.

Proposition: Every knot in S³ that pierces a disk – even a wild disk – is isotopic to an unknot.

Let K, L be links in a 3-manifold M. K is semi-isotopic to L if there is an annulus A ⊂ M³ × I such that A ∩ (M³ × ∂I) = ∂A = (K × {0}) ∪ (L × {1}) and there is a homeomorphism e : K × [0,1) → A – (L × {1}) such that e(K × {t}) ⊂ M³ × {t} for every t ∈ [0,1). A thickening of K in M is a compact 3-manifold T ⊂ M such that K ⊂ int(T), the inclusion of K in T is a homotopy equivalence and T is a disk bundle over a link. A link J is a core of a thickening T if J is the 0-section of a disk bundle structure on T. K is thickenable if it has a thickening.

We generalize a technique of C. Giffen called shift-spinning to prove:

Theorem 1. If a link K in a 3-manifold M has a thickening T with core J, then K is semi-isotopic to J.

Theorem 2. Every knot is S³ has a thickening which is an unknotted solid torus.

Corollary 1. Every knot in S³ is semi-isotopic to an unknot.

Using a result of S. Melikhov, we obtain:

Corollary 2. There is a non-thickenable 2-component link in S³.

We also recover the following unpublished result of C. Giffen:

Corollary 3. F-isotopic links are I-equivalent.

Abstract: What does a random quotient of a group look like? Gromov introduced the density model of quotients of free groups. The density parameter d measures the rate of exponential growth of the number of relators compared to the size of the Cayley ball. Using this model, he proved that for d<1/2 a random quotient of a free group is non-elementary hyperbolic. Ollivier extended Gromov's result to show that for d<1/2 a random quotient of even a non-elementary hyperbolic group is non-elementary hyperbolic.

Żuk/Kotowski-Kotowski proved that for d>1/3 a typical quotient of a free group has Property (T). We show that for 1/3<d<1/2 (in a closely related density model) a random quotient of a non-elementary hyperbolic group is non-elementary hyperbolic and has Property-(T).

This provides an answer to a question of Gromov (and Ollivier).

Abstract: Consider a Euclidean obtuse triangle. Can you subdivide it into triangles that are all strictly acute? How many triangles did you use? How many would you really need? This question dates back to the late Martin Gardner and was first published in a magazine of recreational mathematics for a general audience. But the issue of finding ``nice'' subdivisions of various objects is a fundamental one which arises naturally in many fields: computational geometry, physics and computer graphics all benefit from having well-behaved meshes at their disposition (typically in the form of angle bounds). Perhaps surprisingly, the very existence of acute triangulations offers a very substantial mathematical challenge. While they are known to exist for any polygon in the plane, it is also known that they cannot exist in dimensions higher than 5 - and close to nothing is known about their existence even in 3 dimensions! Moreover, virtually all previous efforts were focused solely on the Euclidean case, where the two dimensional case has been comprehensively studied and solved. In this talk, I will provide extensive background on this fascinating topic and discuss a new constructive method to show the existence of acute triangulations for the general class of constant curvature polygonal complexes. This result lays a stepstone towards the most general Riemannian setting.

Abstract: Mazur and Poénaru constructed the first compact, contractible manifolds distinct from disks. More recently, Sparks modified Mazur's construction and defined Jester manifolds. Sparks used Jester manifolds to produce compact, contractible 4-manifolds distinct from the 4-disk that split as the union of two 4-disks meeting in a 4-disk. We discuss the problem of distinguishing these 4-manifolds from one another and from the 4-disk. We also discuss relevant results on knots in S^1xS^2, a conjecture on two such knots, and a conjecture on hyperbolic triangle groups.

Abstract: Thurston proved that a non-Lattés branched cover of the sphere to itself is either equivalent to a rational map (that is: conjugate via a mapping class), or has a topological obstruction. The Nielsen–Thurston classification of mapping classes is an analogous theorem in low-dimensional topology. We unify these two theorems with a single proof, further connecting techniques from surface topology and complex dynamics. Moreover, our proof gives a new framework for classifying self-covering spaces of the torus and Lattés maps. This is joint work with Jim Belk and Dan Margalit.

Abstract: In this talk we will define what it means for a cube complex to be collapsible. In particular, our definition will apply to the case that the complex is not finite. Then, we will show that all locally-finite CAT(0) cube complexes are collapsible. The process will yield an inverse sequence of finite convex subcomplexes whose inverse limit provides a Z-compactification of the complex in which the boundary (which we call the cubical boundary) incorporates properties of both the visual and Roller boundaries.

Abstract: The Rips exact sequence is a useful tool for producing examples of groups satisfying combinations of properties that are not obviously compatible. It works by taking as an input an arbitrary finitely presented group Q, and producing as an output a hyperbolic group G that maps onto Q with finitely generated kernel. The ``output group" G is crafted by adding generators and relations to a presentation of Q, in such a way that these relations create enough ``noise" in the presentation to ensure hyperbolicity. One can then lift pathological properties of Q to (some subgroup of) G. Among other things, Rips used his construction to produce the first examples of incoherent hyperbolic groups, and of hyperbolic groups with unsolvable generalised word problem.

In this talk, I will explain Rips’ result, describe a variation of it that produces cubulated hyperbolic groups of any desired cohomological dimension, and survey some tools and concepts related to these constructions, including classical small cancellation theory, cubulated groups, and –time permitting— the cubical version of small cancellation theory.

Abstract: For a finite metric graph, the signed measure that maximizes a potential defined via pairwise effective resistance, which is the canonical, or Arakelov measure, have been studied extensively and had many applications to combinatorics and tropical geometry. We studied the (non negative) measure that maximizes such potential as well as the signed measure with given support that maximizes such potential, and found many applications of them on combinatorics as well. This is a joint work with Farbod Shokrieh and Harry Richman.

Abstract: In Algebraic Topology it is well known that both the 0^th homotopy set and 0^th homology group keep track of path connected components of a space. We find a similar relationship occurring in the coarse setting but only for spaces that have a specific property. In this talk we will first discuss the property needed for a space to have a connection between low level coarse homotopy and coarse homology and then define the relationship.

Abstract: How many simple curves can you fit on a surface so that every pair intersects at most once? (To make this answer finite, you should count homotopy classes of curves.) I will survey what’s known about answers to this question. This talk (and the following one) will be expository and chronological: from Malestein-Rivin-Theran, to Piotr Przytycki, to Aougab-Biringer-Gaster, and culminating with Josh Greene’s currently best bound.

Abstract: Groves-Manning and Osin studied group theoretic Dehn fillings of Relatively hyperbolic groups. This tool has been useful in the proof of various problems, most notably that of the virtual Haken conjecture. Given a relatively hyperbolic group pair, one is interested to know what properties persist under Dehn fillings. Groves and Manning showed that connectedness of Bowditch boundary persists provided the peripheral subgroups are virtually polycyclic. We will present an outline of a proof that does not require this hypothesis on the peripheral subgroups. In particular we show that if the Bowditch boundary of a relatively hyperbolic group pair (G, P), is connected and without cut points then for sufficiently long M-finite Dehn fillings, the resulting Bowditch boundary is also connected and without cut points.

Abstract: Hyperbolic Coxeter groups with Sierpinski carpet boundary was investigated by Swiatkowski. And hyperbolic right-angled Coxeter group with Gromov boundary as Menger curve was studied by Daniel Danielski. Also, Haulmark, Hruska, and Sathaye produced the first known examples of non-hyperbolic CAT(0) groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on n vertices for n greater than or equal to 5. Recently, Danielski and Swiatkowski gave complete characterizations (in terms of nerves) of the word hyperbolic Coxeter groups whose Gromov boundary is homeomorphic to the Sierpi´nski curve and to the Menger curve, respectively. In our presentation, we find new examples with both hyperbolic 4 and nonhyperbolic groups which state: the punctured torus admits a triangulation that is a nerve of right-angled Coxeter group with Menger curve boundary. The construction in Haulmark, Hruska, and Sathaye’s paper depended on a slight extension of Sierpinski’s theorem on embedding 1–dimensional planar compacta into the Sierpinski carpet. However, our methods depend on a perturbing trick for paths and special techniques for nullity condition; also, we exploit good properties of Pontryagin surface.

Abstract: There is a deep analogy between Kleinaian groups and subgroups of the mapping class group. Inspired by this, Farb and Mosher defined convex cocompact subgroups of the mapping class group in analogy with convex cocompact Kleinian groups. These subgroups have since seen immense study, producing surprising applications to the geometry of surface group extension and surface bundles. In particular, Hamenstadt plus Farb and Mosher proved that a subgroup of the mapping class groups is convex cocompact if and only if the corresponding surface group extension is Gromov hyperbolic.

Among Kleinian groups, convex cocompact groups are a special case of the geometrically finite groups. Despite the progress on convex cocompactness, no robust notion of geometric finiteness in the mapping class group has emerged. Durham, Dowdall, Leininger, and Sisto recently proposed that geometric finiteness in the mapping class group might be characterized by the corresponding surface group extension being hierarchically hyperbolic instead of Gromov hyperbolic. We provide evidence in favor of this hypothesis by proving that the surface group extension of the stabilizer of a multicurve is hierarchically hyperbolic.