Fall 2021

9/13: Organizational meeting

Title: The Markov ordering of the rationals

Abstract: A rational number p/q determines a simple closed curve on a once-punctured torus, which then has a well-defined length when the torus is endowed with a complete hyperbolic metric. When the metric is chosen so that the torus is “modular” (that is, when its holonomy group is conjugate into PSL(2,Z)), the lengths of the curves have special arithmetic significance, with connections to Diophantine approximation and number theory. Taking inspiration from McShane’s elegant proof of Aigner’s conjectures, concerning the (partial) ordering of the rationals induced by hyperbolic length on the modular torus, I will describe how hyperbolic geometry can be used to characterize monotonicity of the order so obtained along lines of varying slope in the (q,p)-plane.

9/27: Chandrika Sadanand (Bowdoin)

Title: Hyperbolic surfaces with cone points and polygonal billiards

Abstract: Consider a polygon-shaped billiard table in the hyperbolic plane on which a ball can roll along geodesics and reflect off of edges infinitely. In joint work with Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of the polygon and the set of possible sequences of edges visited consecutively by billiard balls rolling and reflecting around the polygon. In order to do this, we made an arguably more interesting characterization: when a hyperbolic metric with cone points on a surface is determined by the geodesics that do not pass through cone points . In this talk, we will explore these characterizations and the tools used to prove them.

9/29: Michael Freedman (Microsoft)

Title: Controlled Mather Thurston Theorems

Abstract: The “c-principle” is a cousin of Gromov’s h-principle in which cobordism rather than homotopy is required to (canonically) solve a problem. We show that in certain well-known c-principle contexts only the mildest cobordisms, semi-s-cobordisms, are required. In physical applications, the extra topology (a perfect fundamental group) these cobordisms introduce could easily be hidden in the UV. This leads to a proposal to recast gauge theories such as EM and the standard model in terms of flat connections rather than curvature. See arXiv:2006.00374

10/4: Andrew Zimmer (UW-Madison)

Title: Anosov representations of geometrically finite Fuchsian groups

Abstract: Anosov representations are representations of word hyperbolic groups into semisimple Lie groups with good geometric properties. In this talk I will describe how the theory extends very nicely to geometrically finite Fuchsian groups. I will also focus on the case of representations into PGL(d,R) and avoid using an Lie theory. This is joint work with Richard Canary and Tengren Zhang.

10/6: Daniel Gulbrandsen (UW-Milwaukee)

Title: Collapsing CAT(0) Cube Complexes

Abstract: While it is known that all finite CAT(0) cube complexes are collapsible, the question of whether infinite CAT(0) cube complexes are collapsible remains largely unaddressed. In this talk, we will provide suitable definitions that extend the notion of collapsibility beyond finite cube complexes. Then, we will show that all (locally finite) CAT(0) cube complexes are collapsible. Our proof applies to the finite case as well and therefore provides a novel proof that all finite CAT(0) cube complexes are collapsible.

10/11: Zachary Munro (McGill)

Title: Cubical Dimension of Random Groups

Abstract: There are a number of interesting "phase transition" results in Gromov's density model of random groups. We will focus on those results concerning random group actions on CAT(0) cube complexes. Ollivier and Wise showed that random groups with density d<1/5 act non-trivially (i.e. without global fixed point) on CAT(0) cube complexes w.o.p. This result was strengthened by MacKay and Przytycki to d<5/24. It is of note that these cube complexes can be taken to be finite-dimensional and the actions cocompact. On the other hand, Zuk proved that for d>1/3 a random group has w.o.p. Kazhdan's property (T) and every action on a cube complex has a fixed point. This talk will center around the cubical dimension of random group actions. In particular, we will show how one can show random groups cannot act nontrivially on CAT(0) square complexes.

10/18: Eric Samperton (UIUC)

Title: Hyperbolic geometry does not help topological quantum computation

Abstract: I’ll show that many TQFT-style invariants of knots that are hard to compute remain hard to compute even if we only consider *hyperbolic* knots. “Hard” here can mean various things—it depends on the invariant, and it also depends on what “compute” is supposed to mean. I’ll spend most of the talk developing the background necessary to make these things precise. I’ll finish with a brief sketch of the proof, which exploits results of Johnson-Moriah that compute bridge distances of so-called “highly twisted plat knots.”

10/20: Justin Lanier (University of Chicago)

Title: Constraining mapping class group homomorphisms using finite subgroups

Abstract: There are many natural homomorphisms from mapping class groups of surfaces to other groups, and there are also some surprising ones. One tool that is often used to constrain homomorphisms of mapping class groups is analyzing elements of finite order in the domain and target. I will discuss an extension of this approach that analyzes finite non-cyclic subgroups. This yields a new short proof of a theorem of Aramayona–Souto that constrains homomorphisms between mapping class groups of different closed surfaces. This approach also yields results on constraining homomorphisms from mapping class groups to homeomorphism groups of low-dimensional spheres, extending work of Franks–Handel. This is joint work with Lei Chen.

10/25, 11/1: Craig Guilbault (UWM)

Title: Group boundaries for semidirect products with Z

Abstract: Bestvina introduced the notion of a Z-structure on a group G to provide an axiomatic treatment of group boundaries that simultaneously generalizes visual boundaries of CAT(0) groups and Gromov boundaries of hyperbolic groups. The definition requires G to act geometrically on a "nice" space X and for that space to admit a "nice" compactification $\bar{X}$ (a Z-compactification). In addition, one requires that translates of compact subsets of X get small in $\bar{X}$. An equivariant version of this definition -- an EZ-structure -- is also of interest. Many non-CAT(0) and non-hyperbolic groups have been shown to admit (E)Z-structures, but the general question of which groups admit these structures remains open.
In these talks I will provide a broad introduction to the above topic, then discuss recent joint work with Burns Healy and Brian Pietsch in which we explore Z- and EZ-structures for semidirect products $G \rtimes Z$.

10/28: Bena Tshishiku (Brown)

Title: Convex cocompact subgroups of the Goeritz group

Abstract: This talk is about hyperbolicity of surface group extensions and a question of Farb-Mosher about whether purely pseudo-Anosov subgroups of mapping class groups are convex cocompact. I will explain this problem and give an answer for subgroups of the genus-2 Goeritz group, which is the group of mapping classes of a genus-2 surface that extend to the genus-2 Heegaard splitting of the 3-sphere.

11/3, 11/8: Prayagdeep Parija (UWM)

Title: How to take (non-trivial) random quotients of Hyperbolic groups?

Abstract: According to Gromov for "density" <1/2 most random quotients of a free group via reduced words of length l are infinite-hyperbolic. Ollivier generalized this to random quotients of Hyperbolic groups under various density-models (plain words/reduced words/geodesic words of length l).
We will present and discuss the set of axioms given by Ollivier to be satisfied by various density-models of random quotients to get infinite-hyperbolic quotients.

11/10, 11/17: Arka Bannerjee (UWM)

Title: On computation of coarse cohomology

Abstract: Roe introduced the notion of coarse cohomology of a metric space that roughly measures the way in which uniformly large bounded sets in the space fit together. In general, coarse cohomology is hard to compute. However, if the space is proper and uniformly contractible then Roe proved that its coarse cohomology coincides with the compactly supported cohomology of the space. In my talk, I will give a brief introduction to coarse cohomology theory and then discuss a recent work that generalizes Roe's theorem that allows computation of coarse cohomology for more general spaces.

11/15: Grant Lakeland (Eastern Illinois)

Title: Systoles of hyperbolic punctured spheres

Abstract: The systole of a surface is the shortest essential, non-peripheral loop on the surface. The length of the systole of a hyperbolic surface may be bounded in terms of the complexity of the surface. In the case of punctured spheres, we use some results in planar graph theory to prove new bounds when the surface is arithmetic, and to find some surface types where arithmetic surfaces fail to maximize the systole length. This is joint work with Clayton Young.

11/22, 11/29: William Braubach (UWM)

Title: Coarse Homotopy Extension Property and its Applications

Abstract: For a space X and a subset A, if every homotopy on A can be extended to a homotopy on X, then the pair (X,A) has the Homotopy Extension Property. This is a very useful tool in many fields of study and an important question to ask is if this can be extended to the Coarse setting. In this talk, I will give a detailed look at a Coarse Version of the Homotopy Extension Property and many useful applications of it.

12/1: Jeffrey Rolland (UWM)

Title: A Necessary and Sufficient Condition for a Self-Diffeomorphism of a Smooth Manifold to be the Time-1 Map of the Flow of a Differential Equation

Abstract: In topological dynamics, one considers a topological space $X$ and a self-map $f: X \to X$ of $X$ and studies the self-map's properties. In global analysis, one considers a smooth manifold $M^n$ and a differential equation $\xi: M \to TM$ on $M$ and studies the flow $\Phi_t: M \times \BR \to M$ of the differential equation. We consider a necessary and sufficient condition for a self-diffeomorphism $f$ of a manifold $M$ to be the time-1 map $\Phi_1$ of the flow of a differential equation on $M$.

12/6, 12/8: Ric Ancel (UWM)

Title: TBD

Abstract: TBD