# Geometry and Topology Seminar

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 on the sixth floor of Wachman Hall.

• Monday May 6, 2019 at 16:00, Wachman 617
Lower bounds for separability of nilpotent and solvable groups

Mark Pengitore, The Ohio State University

In this talk, we introduce quantitative approaches to the study of separability in nilpotent and solvable groups. In particular, we will describe effective residual finiteness, effective subgroup separability, and effective conjugacy separability and discuss various results for asymptotic lower bounds of these properties for these classes of groups. Moreover, we introduce the algebraic, number theoretic, and geometric methods used in the construction of these lower bounds.

• Monday May 6, 2019 at 14:00, Wachman 617
Separability properties of finitely generated groups

Mark Pengitore, The Ohio State University

This talk will be an introduction to separability of finitely generated groups. The premise is that we can detect membership of interesting subsets of finitely generated groups such as the identity subgroup, finitely generated subgroups, and conjugacy classes via surjective group morphisms to finite groups. This idea can be interpreted in many distinct ways such as lifting of closed loops of manifolds to finite covers, topological properties of a totally disconnected compact topological group, and well approximation of elements in a metric space. One of the many applications of these ideas is a quantitative solution to the word problem, conjugacy problem, and other decision problems. In a more topological direction, another application is the constructing lifts of an immersed submanifold to an embedded submanifold in a finite cover. This talk will be expository and will explain connections between all of above ideas and motivate interest in separability.

• Wednesday May 1, 2019 at 14:30, Wachman 617
Exotic real projective Dehn surgery space

Jeff Danciger, University of Texas at Austin

We study properly convex real projective structures on closed 3-manifolds. A hyperbolic structure is one special example, and in some cases the hyperbolic structure may be deformed non-trivially as a convex projective structure. However, such deformations seem to be exceedingly rare. By contrast, we show that many closed hyperbolic manifolds admit a second convex projective structure not obtained through deformation. We find these examples through a theory of properly convex projective Dehn filling, generalizing Thurston’s picture of hyperbolic Dehn surgery space. Joint work with Sam Ballas, Gye-Seon Lee, and Ludovic Marquis.

• Tuesday April 23, 2019 at 14:30, Wachman 617
Distance formulae and quasi-cube complexes

Mark Hagen, University of Bristol

Abstract: Masur and Minsky's work on the geometry of mapping class groups, combined with more recent results about the geometry of CAT(0) cube complexes, motivated the introduction of the class of hierarchically hyperbolic spaces. A metric space $X$ is hierarchically hyperbolic if there is a set of (uniformly) Gromov-hyperbolic spaces $U$, each equipped with a projection from $X$ to $U$, satisfying various axioms that amount to saying that the geometry of $X$ is recoverable, up to quasi-isometry, from this projection data. Working in this context often allows one to promote facts about hyperbolic spaces to conclusions about highly non-hyperbolic spaces: mapping class groups, Teichmuller space, "most" 3-manifold groups, etc. In particular, many CAT(0) cube complexes -- including those associated to right-angled Artin and Coxeter groups -- are hierarchically hyperbolic.

The relationship between CAT(0) cube complexes and hierarchically hyperbolic spaces is intriguing. Just as, in a hyperbolic space, a collection of n points has quasiconvex hull quasi-isometric to a finite tree (i.e. 1-dimensional CAT(0) cube complex), in a hierarchically hyperbolic space, there is a natural notion of the quasiconvex hull of a set of n points, and it is quasi-isometric to a CAT(0) cube complex, by a result of Behrstock-Hagen-Sisto. The quasi-isometry constants depend on n in general. However, when each hyperbolic space U is quasi-isometric to a tree, it turns out that this dependence disappears. From this one deduces that, if $X$ is a metric space that is hierarchically hyperbolic with respect to quasi-trees, then $X$ is quasi-isometric to a CAT(0) cube complex. I will discuss this theorem and some of its group-theoretic consequences. This is joint work with Harry Petyt.

• Wednesday April 17, 2019 at 14:30, Wachman 617
Spectral Rigidity of q-differential Metrics

Marissa Loving, University of Illinois at Urbana Champaign

When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask which curves’ lengths do we really need to know? It is a classical result of Fricke that a hyperbolic metric on a surface is determined by its marked simple length spectrum. More recently, Duchin–Leininger–Rafi proved that a flat metric induced by a unit-norm quadratic differential is also determined by its marked simple length spectrum. In this talk, I will describe a generalization of the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential metric.

• Wednesday April 10, 2019 at 14:30, Wachman 617
Effective Special Covers of Alternating Links

Edgar Bering, Temple University

In 1982 Thurston stated the "virtual conjectures" for 3-manifolds: that every hyperbolic 3-manifold has finite covers that are Haken and fibered, with large Betti numbers. These conjectures were resolved by Agol and Wise in 2012, using the machinery of special cube complexes. Even before the work of Agol and Wise, but especially after, mathematicians have been interested in understanding the degree of these covers in terms of a manifold's invariants.

In joint work, David Futer and I give the first steps of a quantitative answer to this question in the setting of alternating link complements. Given an alternating link with n crossings we construct a special cover of degree less than n!. As a corollary, we bound the degree of a cover with Betti number at least k.

• Thursday April 4, 2019 at 17:45, University of Pennsylvania, room DRL 4C8
Filling metric spaces

Alex Nabutovsky, University of Toronto and IAS

PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

Abstract: The Uryson $k$-width of a metric space $X$ measures how close $X$ is to being $k$-dimensional. Several years ago Larry Guth proved that if $M$ is a closed $n$-dimensional manifold, and the volume of each ball of radius 1 in $M$ does not exceed a certain small constant $e(n)$, then the Uryson $(n-1)$-width of $M$ is less than 1. This result is a significant generalization of the famous Gromov inequality relating the volume and the filling radius that plays a central role in systolic geometry.

Guth asked if a much stronger and more general result holds true: Is there a constant $e(m)>0$ such that each compact metric space with $m$-dimensional Hausdorff content less than $e(m)$ always has $(m-1)$-dimensional Uryson width less than 1? Note that here the dimension of the metric space is not assumed to be $m$, and is allowed to be arbitrary.

Such a result immediately leads to interesting new inequalities even for closed Riemannian manifolds. In my talk I am are going to discuss a joint project with Yevgeny Liokumovich, Boris Lishak and Regina Rotman towards the positive resolution of Guth's problem.

• Thursday April 4, 2019 at 16:30, University of Pennsylvania, room DRL 4C8
Periodic Geodesics and Geodesic Nets on Riemannian Manifolds

Regina Rotman, University of Toronto and IAS

PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

Abstract: I will talk about periodic geodesics, geodesic loops, and geodesic nets on Riemannian manifolds. More specifically, I will discuss some curvature-free upper bounds for compact manifolds and the existence results for non-compact manifolds. In particular, geodesic nets turn out to be useful for proving results about geodesic loops and periodic geodesics.

• Thursday April 4, 2019 at 14:30, Wachman 617
Gonality and the character variety

Kate Petersen, Florida State University

• Wednesday April 3, 2019 at 14:30, Wachman 617
The Shape of Phylogenetic Treespace

Katherine St. John

City University of New York & American Museum of Natural History

Trees are a canonical structure for representing evolutionary histories. Many popular criteria used to infer optimal trees are computationally hard, and the number of possible tree shapes grows super-exponentially in the number of taxa. The underlying structure of the spaces of trees yields rich insights that can improve the search for optimal trees, both in accuracy and running time, and the analysis and visualization of results. We review the past work on analyzing and comparing trees by their shape as well as recent work that incorporates trees with weighted branch lengths. This talk will highlight some of the elegant questions that arise from improving search and visualizing the results in this highly structured space. All are welcome.

• Friday March 22, 2019 at 15:30, Wachman 617
Augmentations and immersed Lagrangian fillings

Dan Rutherford, Ball State University PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

Abstract: This is joint work with Y. Pan that applies previous joint work with M. Sullivan. Let $\Lambda \subset \mathbb{R}^{3}$ be a Legendrian knot with respect to the standard contact structure. The Legendrian contact homology (LCH) DG-algebra, $\mathcal{A}(\Lambda)$, of $\Lambda$ is functorial for exact Lagrangian cobordisms in the symplectization of $\mathbb{R}^3$, i.e. a cobordism $L \subset \mathit{Symp}(\mathbb{R}^3)$ from $\Lambda_-$ to $\Lambda_+$ induces a DG-algebra map, $f_L:\mathcal{A}(\Lambda_+) \rightarrow \mathcal{A}(\Lambda_-).$ In particular, if $L$ is an exact Lagrangian filling ($\Lambda_-= \emptyset$) the induced map is an augmentation $\epsilon_L: \mathcal{A}(\Lambda_+) \rightarrow \mathbb{Z}/2.$

In this talk, I will discuss an extension of this construction to the case of immersed, exact Lagrangian cobordisms based on considering the Legendrian lift $\Sigma$ of $L$. When $L$ is an immersed, exact Lagrangian filling a choice of augmentation $\alpha$ for $\Sigma$ produces an induced augmentation $\epsilon_{(L, \alpha)}$ for $\Lambda_+$. Using the cellular formulation of LCH, we are able to show that any augmentation of $\Lambda$ may be induced by such a filling.

In the morning background talk, at 11:00am, I will cover augmentations and immersed Lagrangian fillings.

• Friday March 22, 2019 at 14:00, Wachman 617
Commensurability classes of fully augmented pretzel links

Christian Millichap, Furman University PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

Abstract: Fully augmented links (FALs) are a large class of links whose complements admit hyperbolic structures that can be explicitly described in terms of combinatorial information coming from their respective link diagrams. In this talk, we will examine an infinite subclass of FALs that are constructed by fully augmenting pretzel links and describe how to build their hyperbolic structures. We will then discuss how we can use the geometries of these link complements to analyze arithmetic properties and commensurability classes of these links. This is joint work with Jeff Meyer (CSSB) and Rollie Trapp (CSSB).

The morning background talk, at 9:30am, will be an exploration of hyperbolic structures on link complements.

• Thursday March 21, 2019 at 14:30, Wachman 617
Exploring algebraic rigidity in mapping class groups

Nicholas Vlamis, CUNY Queen's College

A classical theorem of Powell (with roots in the work of Mumford and Birman) states that the pure mapping class group of a connected, orientable, finite-type surface of genus at least 3 is perfect, that is, it has trivial abelianization. We will discuss how this fails for infinite-genus surfaces and give a complete characterization of all homomorphisms from pure mapping class groups of infinite-genus surfaces to the integers. This characterization yields a direct connection between algebraic invariants of pure mapping class groups and topological invariants of the underlying surface. This is joint work with Javier Aramayona and Priyam Patel.

• Wednesday March 13, 2019 at 16:00, Wachman 527
Local to global morse properties, convexity and hierarchically hyperbolic spaces.

Davide Spiriano, ETH Zurich

In a Gromov hyperbolic space, geodesics satisfies the so-called Morse property. This means that if a geodesic and a quasi-geodesic share endpoints, then their Hausdorff distance is uniformly bounded. Remarkably, this is an equivalent characterization of hyperbolic spaces, meaning that all consequences of hyperbolicity can be ascribed to this property. Using this observation to understand hyperbolic-like behaviour in spaces which are not Gromov hyperbolic has been a very successful idea, which led to the definition of important geometric objects such as the Morse boundary and stable subgroups. Another strong consequence of hyperbolicity is the fact that local quasi-geodesics are global quasi-geodesics. This allows detecting global properties on a local scale, which has far-reaching consequences. The goal of this talk is twofold. Firstly, we will prove results that are known for hyperbolic groups in a class of spaces satisfying generalizations of the above properties. Secondly, we show that the set of such spaces is large and contains several examples of interest, i.e. CAT(0) spaces and hierarchically hyperbolic spaces.

• Wednesday March 13, 2019 at 14:30, Wachman 617
Free products and random walks in acylindrically hyperbolic groups

Carolyn Abbott, University of California Berkley Imagine you are standing at the point 0 on a number line, and you take a step forward or a step backwards, each with probability 1/2. If you take a large number of steps, is it likely that you will end up back where you started? What if you are standing at a vertex of an 4-valent tree, and you take a step in each of the 4 possible directions with probability 1/4? This process is special case of what is called a random walk on a space. If the space you choose is the Cayley graph of a group (as these examples are), then a random walk allows you to choose a "random" or "generic" element of the group by taking a large number of steps and considering the label of the vertex where you end up. One can ask what properties a generic element of the group is likely to have: for example, is it likely that the element you land on has infinite order? In this talk, I will focus on the class of the class of so-called acylindrically hyperbolic groups, which contains many interesting groups, such as mapping class groups, outer automorphism groups of free groups, and right-angled Artin and Coxeter groups, among many others. I will discuss the algebraic and geometric properties of subgroups generated by a random element and a fixed subgroup.

• Wednesday February 20, 2019 at 14:30, Wachman 617
Coherence and lattices

Matthew Stover, Temple University

I will survey (in)coherence of lattices in semisimple Lie groups, with a view toward open problems and connections with the geometry of locally symmetric spaces. Particular focus will be placed on rank one lattices, where I will discuss connections with reflection groups, "algebraic" fibrations of lattices, and analogies with classical low-dimensional topology.

• Friday February 15, 2019 at 16:00, Haverford College, room TBA
TBA

Francesco Lin, Princeton University PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract TBA

• Friday February 15, 2019 at 14:30, Haverford College, room TBA
TBA

Oleg Lazarev, Columbia University

PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract TBA

• Wednesday February 13, 2019 at 14:30, Wachman 617
Simplicial complexes, configuration spaces, and ‘chromatic’ invariants

Andrew Cooper, NC State

Given a space $X$, the configuration space $F(X,n)$ is the space of possible ways to place $n$ points on $X$, so that no two occupy the same position. But what if we allow some of the points to coincide?

The natural way to encode the allowed coincidences is as a simplicial complex $S$. I will describe how the configuration space $M(S,X)$ obtained in this way gives rise to polynomial and homological invariants of $S$, how those invariants are related to the cohomology ring $H^*(X)$, and what this has to do with the topology of spaces of maps into $X$.

I will also mention some potential applications of this structure to problems arising from international relations and economics.

This is joint work with Vin de Silva, Radmila Sazdanovic, and Robert J Carroll.

• Wednesday February 6, 2019 at 14:30, Wachman 617
CAT(0) cubical groups with uniform exponential growth

Thomas Ng, Temple University

Abstract: A group is said to have uniform exponential growth if the number of elements that can be spelled with words of bounded length is bounded below by a single exponential function over all generating sets. In 1981, Gromov asked whether all groups with exponential growing group in fact have uniform exponential growth. While this was shown not to be the case in general, it has been answered affirmatively for many natural classes of groups such as hyperbolic groups, linear groups, and the mapping class groups of a surface. In 2018, Kar-Sageev show that groups acting properly on 2-dimensional CAT(0) cube complexes by loxodromic isometries either have uniform exponential growth or are virtually abelian by explicitly exhibiting free semigroups whose generators have uniformly bounded word length whenever they exist. These free semigroups witness the uniform exponential growth of the group. I will explain how certain arrangements of hyperplane orbits can be used to build loxodromic isometries generating free semigroups and then describe how to use the convex hull of their axes and the Bowditch boundary to extend Kar and Sageev's result to CAT(0) cube complexes with isolated flats. This is joint work with Radhika Gupta and Kasia Jankiewicz.

• Wednesday January 30, 2019 at 14:30, Wachman 617
Circle packings and Delaunay circle patterns for complex projective structures

Andrew Yarmola, Princeton University

Abstract: At the interface of discrete conformal geometry and the study of Riemann surfaces lies the Koebe-Andreev-Thurston theorem. Given a triangulation of a surface $S$, this theorem produces a unique hyperbolic structure on $S$ and a geometric circle packing whose dual is the given triangulation. In this talk, we explore an extension of this theorem to the space of complex projective structures - the family of maximal $CP^1$-atlases on $S$ up to Möbius equivalence. Our goal is to understand the space of all circle packings on complex projective structures with a fixed dual triangulation. As it turns out, this space is no longer a unique point and evidence suggests that it is homeomorphic to Teichmüller space via uniformization - a conjecture by Kojima, Mizushima, and Tan. In joint work with Jean-Marc Schlenker, we show that this projection is proper, giving partial support for the conjectured result. Our proof relies on geometric arguments in hyperbolic ends and allows us to work with the more general notion of Delaunay circle patterns, which may be of separate interest. I will give an introductory overview of the definitions and results and demonstrate some software used to motivate the conjecture. If time permits, I will discuss additional ongoing work with Wayne Lam.

• Wednesday January 23, 2019 at 14:30, Wachman 617
Surfaces almost transverse to circular pseudo-Anosov flows

Michael Landry, Yale University

Let $M$ be a closed hyperbolic 3-manifold which fibers over $S^1$, and let $F$ be a fibered face of the unit ball of the Thurston norm on $H^1(M;R)$. By results of Fried, there is a nice flow on $M$ naturally associated to $F$. We study surfaces which are almost transverse to $F$ and give a new characterization of the set of homology directions of $F$ using Agol’s veering triangulation of an auxiliary cusped 3-manifold.