Ara Basmajian (Hunter College)
From Collars on Riemann surfaces to Tubes in Complex Ball quotients
Abstract
The celebrated Keen collar lemma guarantees that a simple closed geodesic on a hyperbolic Riemann surface has a collar (tubular neighborhood) whose width only depends on its length. Viewing a Riemann surface as the quotient of the unit ball in the complex plane, a natural generalization is to ball quotients in higher dimensions where the $\text{Poincar}\acute{e}$ metric is replaced by the Bergman metric (also known as the complex hyperbolic metric). Such ball quotients are called complex hyperbolic manifolds. The focus of this talk will be on embedded complex geodesics in complex hyperbolic 2-manifolds; a complex geodesic has complex codimension one in the quotient complex 2-manifold. We prove a tubular neighborhood theorem for such a complex geodesic where the width of the tube depends only on the Euler characteristic of the embedded complex geodesic. We also derive an explicit estimate for this width. After giving a short history of the collar lemma generalizations and discussing the basics of complex hyperbolic geometry, we will discuss the ideas leading to the proof of this tubular neighborhood theorem. This is joint work with Youngju Kim.
Nikolay Bogachev (University of Toronto)
Random walks on hyperbolic lattices
Abstract
The question of the singularity at infinity of the hitting measure of random walks has a long history, originating from the work of Furstenberg in the 1960s. In 2011, Kaimanovich and Le Prince conjectured that the hitting measure of any finitely supported random walk on a discrete subgroup Г < SL(d,R) is singular at infinity. This conjecture now has many variations, and has been actively studied for lattices Г < G in semi-simple Lie groups, including G = SL(2,R), SO(n,1), SO(p,q), etc. Nonetheless, this question remains widely open even for hyperbolic lattices Г < SO(n,1), in the uniform case, and this is the main topic of this talk. In a joint work in progress with P. Kosenko and G. Tiozzo, we prove the singularity conjecture for the hitting measures of “many” random walks on “most” of cocompact Fuchsian (n=2) and Kleinian (n=3) groups. I will discuss this result and our proof, as well as the obstructions to generalizing our methods to all cocompact Kleinian groups or to higher-dimensional (n>3) uniform hyperbolic lattices.
Jacopo Chen (Scuola Normale Superiore)
Non-cobordant hyperbolic manifolds
Abstract
In all dimensions $n \ge 4$ not of the form $4m+3$, we show that there exists a closed hyperbolic $n$-manifold which is not the boundary of a compact $(n+1)$-manifold. The proof relies on the relationship between the cobordism class and the fixed point set of an involution on the manifold, together with a geodesic embedding of Kolpakov, Reid and Slavich. We also outline a possible approach to cover the dimensions $4m+3 \ne 2^k-1$.
Michelle Chu (University of Minnesota)
Salem numbers and commensurability classes of arithmetic hyperbolic manifolds
Abstract
Salem numbers are directly related to the lengths of geodesics in arithmetic hyperbolic manifolds. In this talk I will discuss recent results joint with Plinio Murillo on realizing Salem numbers in infinitely many commensurability classes of arithmetic hyperbolic manifolds of simplest type.
Sami Douba (IHES)
Conformally flat Gromov–Thurston polyhedra
Abstract
We discuss examples of Gromov-hyperbolic right-angled Coxeter groups with 3-sphere boundary that are not virtually Kleinian in dimension 4, but are nevertheless “quasi-Fuchsian” in one higher dimension.
Jean Lafont (Ohio State University)
High dimensional hyperbolic Coxeter groups that virtually fiber
Abstract
I’ll explain how to construct right-angled Coxeter groups that have arbitrarily high virtual cohomological dimension, are Gromov hyperbolic, and virtually algebraically fiber. This is joint work with Minemyer, Sorcar, Stover, and Wells.
Jason Manning (Cornell University)
Variations on negatively curved Dehn filling
Abstract
I’ll survey various constructions on hyperbolic manifolds yielding negatively (or nonpositively) curved spaces whose fundamental groups are quotients of the fundamental groups of the original spaces. These procedures can yield hyperbolic groups with interesting properties. I will mention joint projects with Koji Fujiwara, with Colby Kelln, and with Lorenzo Ruffoni.
Barry Minemyer (Commonwealth University – Bloomsburg)
Negatively curved metrics on branched covers and codimension two complements.
Abstract
Let M be a finite volume manifold whose universal cover is isometric to either hyperbolic space, a product of hyperbolic spaces, or complex hyperbolic space.
Let N be a compact, codimension two, totally geodesic submanifold of M. In this talk we will consider two manifolds constructed from the pair (M, N) and discuss whether or not they admit “nice” negatively (or non-positively) curved Riemannian metrics. The first manifold X is a finite cyclic branched cover of M about N, while the second manifold is S = M\N, the manifold obtained from M by “drilling out” N. In this talk we will survey all known results related to several types of “nice” metrics. These different results will depend greatly on how the submanifold N sits inside of M, of course. We will also discuss the general methods used to prove some of these results.
Julien Paupert (Arizona State University)
Complex hyperbolic deformations of cusped hyperbolic lattices
Abstract
After reviewing some classical results about rigidity and flexibility of lattices in semisimple Lie groups and known results in dimension 2, we will discuss the existence and potential discreteness/faithfulness of deformations of cusped lattices of SO(3,1) into SU(3,1). We will focus on the cases of small Bianchi groups (joint with M. Thistlethwaite) and the figure-eight knot group (joint with S. Ballas and P. Will), as well as the general behavior of bending deformations in all dimensions.
Plinio Murillo (Universidade Federal Fluminense)
On Salem numbers and arithmetic lattices
Abstract
The exponential length of closed geodesics of arithmetic hyperbolic n-orbifolds of the first type defined over \mathbb relates with Salem numbers of degree at most n+1. Is there a single arithmetic n-orbifold realizing all Salem numbers of degree d\leq n+1? The goal of this talk is to discuss recent results in joint work with Cayo Dória answering this and related questions for square-rootable Salem numbers of degree 4.
Zhenghao Rao (Rutgers University-New Brunswick)
Incompressible surfaces in closed hyperbolic manifolds
Abstract
In 2009, Kahn and Markovic proved the Surface Subgroup Theorem, and they constructed a ubiquitous collection of \pi_1-injective immersed surfaces in closed hyperbolic 3-manifolds. Hamenstadt later showed that any cocompact lattice of some simple rank 1 Lie group other than SO(2m,1) (m >= 1) has a surface subgroup. Recently, we constructed \pi_1-injective immersed surfaces in closed hyperbolic 2n-manifolds, which addresses the cases missing from Hamenstadt’s work. This is joint work with Jeremy Kahn.
Lorenzo Ruffoni (Binghamton University)
Convex cocompact groups in real hyperbolic spaces with limit set a Pontryagin sphere
Abstract
We construct two convex cocompact groups of isometries of real hyperbolic spaces with limit set a Pontryagin sphere. Both examples arise from right-angled Coxeter groups and admit special subgroups with limit set a Menger curve. One example is in H^4 and the other is 2-generated in H^6, which makes these examples optimal with respect to dimension and rank respectively. This is joint work with S. Douba, G.-S. Lee, and L. Marquis.
Junmo Ryang (Rice University)
Pseudo-Anosov Subgroups of Surface Bundles over Tori
Abstract
In 2002, Farb and Mosher introduced the notion of convex cocompactness in the mapping class group to capture coarse geometric information of the associated surface group extensions. Convex cocompact subgroups are necessarily finitely generated and purely pseudo-Anosov, but it is an open question whether the converse is true. Several partial results are known in certain settings, however. For example, work of Dowdall, Kent, Leininger, Russell, and Schleimer give a positive answer for subgroups of fibered 3-manifold groups (aka surface-by-cyclic extensions) naturally embedded in punctured mapping class groups via the Birman exact sequence. We present a generalization of this in the setting of surface-by-abelian extensions.
Connor Sell (UQAM)
Cusps of arithmetic hyperbolic manifolds
Abstract
The question of which flat manifolds can geometrically bound a hyperbolic manifold has led to the study of the cross-sections of the cusps of noncompact hyperbolic manifolds. For instance, McReynolds showed that every flat manifold occurs as such a cross-section in some arithmetic hyperbolic orbifold. In this talk, we will discuss conditions under which a given flat manifold B can or cannot appear. In particular, we will determine in which commensurability classes of noncompact arithmetic hyperbolic manifolds B occurs as a cusp cross-section, followed by some interesting examples if time permits. Joint work with Duncan McCoy.
Bena Tshishiku (Brown University)
Morse complexity of locally symmetric manifolds
Abstract
The Morse complexity of a homology class is the minimum number of critical points of a Morse function on a manifold representing the class, varying over all representatives. This invariant, introduced by Gromov, is analogous to the simplicial norm, with simplices replaced by handles. We will discuss what is (un)known about Morse complexity for hyperbolic manifolds and other locally symmetric manifolds. This is joint work with Fedya Manin and Shmuel Weinberger.
