Spring 2021 Yet Another Zoom

Abstract: What are the possible topological behaviors of an isometrically immersed hyperbolic plane (which we call geodesic plane) in a hyperbolic 3-manifold? When the 3-manifold has finite volume, every geodesic plane is closed or dense, due to Shah and Ratner independently. This remarkable rigidity can be generalized to certain 3-manifolds of infinite volume: when the 3-manifold is geometrically finite and acylindrical, every geodesic plane intersecting the interior of the convex core is closed or dense there, due to recent works of McMullen-Mohammadi-Oh and Benoist-Oh. A few questions naturally arise from their works; for example, is a geodesic plane closed in the interior of the convex core closed in the whole manifold? How do geodesic planes outside the convex core behave? And what about planes in other hyperbolic 3-manifolds? In this talk, we will focus on the first question and give a negative answer with an example. If time permits, we will also describe works in progress regarding the other two questions.

Abstract: The study of translation surfaces is a rich blend of dynamics and algebraic geometry. One of the most fruitful perspectives has been to look at moduli spaces of translation surfaces known as strata. While there has been spectacular progress in understanding their dynamical and algebro-geometric properties, the topology and geometric group theory of strata is an almost total mystery. In spite of this, there are strong indications that this will develop into an immensely rich topic. The current state of knowledge is such that I can give an essentially complete overview of what is currently known in an hour, with plenty of time left over to speculate about what might turn out to be true, and why geometric group theorists in particular should be dropping what they are doing to work on this stuff.

Abstract: We show how one can easily construct plenty of hyperbolic n-manifolds starting with a single right-angle polytope. We will focus on dimension 4 and show how these techniques can be used to build examples of hyperbolic 4-manifolds satisfying various types of topological conditions. If time permits, we discuss how to apply the Bestvina – Brady machinery to construct nice circle-valued maps that should be analogous to the fibrations in dimension 3. 

Abstract: TBA

Abstract: The idea for this expository talk is to introduce you to a topic, étale groupoids, that Tyrone Ghaswala and I met while trying to give a definition of a mapping class group of an orbifold that would satisfy not just low-dimensional topologists, but everyone who works with orbifolds. Orbifolds go back to the work of Satake and Thurston; the idea is a space locally modeled on the quotient of euclidean space by the action of a finite group. The problem is maps of orbifolds: a profusion of definitions exist, only some of which are composable. Étale groupoids are widely accepted as a solution to this problem, though as we’ll see, it’s a bit of a case of exchanging one problem for another. The goal of the talk is to define all the words in the title, giving examples along the way.

Abstract: Studying quasi-isometries between groups is a major theme in geometric group theory. Of particular interest are the situations where the existence of a quasi-isometry between two groups implies that the groups are equivalent in a stronger algebraic sense, such as being commensurable. I will survey some results of this type, and then talk about recent work with Daniel Woodhouse where we prove quasi-isometric rigidity for certain graphs of virtually free groups, which include “generic” cyclic HNN extensions of free groups.

Abstract: TBA

Abstract: We are motivated by looking for traces of hyperbolicity in a space or group which is not Gromov-hyperbolic. One previous approach in this direction is the notion of Morse quasigeodesics, which describes “negatively-curved” directions in the spaces; another previous approach is “higher rank hyperbolicity”. One example which illustrates the idea of higher rank hyperbolicity is that though the “thin triangles property” fails in products of two hyperbolic planes, a version of “thin tetrahedron property” holds true. We introduce the notion of Morse quasiflats, which unifies these two seemingly different approaches and applies to a wider range of objects. In the talk, I will provide motivations and examples for Morse quasiflats, as well as a discussion of related works by other people. We will also show that Morse quasiflats are asymptotically conical, and comment on potential applications. Based on joint work with B. Kleiner and S. Stadler.

Abstract: Automata have long been used to produced tractable groups with exotic properties. In this talk, we will discuss how to create lamplighter-like groups using bireversible automata, which are automata that produce particularly nice actions on a rooted tree.
This is a joint project with Benjamin Steinberg.

Abstract: A surface homeomorphism can be thought of as a dynamical system; its mapping torus is a 3-manifold that fibers of a circle; and a Riemannian metric on this 3-manifold determines a path metric on its universal cover. I will discuss how, in some cases, dynamical invariants of the surface homeomorphism, topological invariants of the mapping torus, and geometric invariants of the universal cover are all equivalent. For example, assuming the surface is closed and hyperbolic, then the following are equivalent:
1) the surface homeomorphism has finite order up to isotopy;
2) the mapping torus is finitely covered by a product of the surface and the circle; and
3) the universal cover “is” the product of the hyperbolic plane and the real line.
This is mostly an expository talk of “classical”/old results but, if time permits, I will end the talk with a discussion on the few results that have been translated into the setting of free group automorphisms.

Abstract: I will talk about joint work with Ivan Levcovitz, in which we provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups.  The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups, and can be chosen to be 2-dimensional.  We use techniques inspired by Stallings folds to establish the non-quasiconvexity of our subgroups.  This adds to the somewhat limited list of known methods of constructing non-quasiconvex subgroups of hyperbolic groups.  

Abstract:  A closed surface of genus>1 admits many hyperbolic structures. These structures can be reinterpreted as representations of \pi_1(S) into PSL(2,R). Could a representation from \pi_1(S) into PSL(n,R) for n>2 also enclose geometric information about S? Anosov representations provide a certain answer to this question. In this talk we will go over what Anosov representations are from the perspective of their action on symmetric spaces. Then I will talk about some of my current work, where I construct families of Anosov representations outside the Hitchin component.

Abstract: In this expository talk, I plan to give some basic background on the theory of random walks on groups. Depending on the geometry of the group, the behavior of random walks and the types of results one can show can vary wildly. We will consider groups from a few different settings (hyperbolic and otherwise) and explore how algebra, geometry, and boundary theory can come together to teach us about random walks.