This workshop is hosted at Simis, during December 8-12, with Wednesday a break with no class. 4 speakers, each speaker gives 4 lectures.
Alessandra Iozzi (ETH Zürich):
Title: Actions on the circle via bounded cohomology
Abstract: We review some of the (more or less) classical results on the topic, from the one obtained with Gromov original definition to those obtained using the functiorial approach developed by Burger and Monod.
Chris Leininger (Rice University):
Title: Coarse Geometry of Surface Bundles
Abstract: William Thurston revolutionized three-manifold topology in the mid-to-late 1970’s with his Geometrization Program. Among other things, he proved that a surface bundle over the circle admits a hyperbolic metric if and only if it is atoroidal (an obvious necessary condition). This led to many questions about more general surface bundles in the early 1990’s. A motivating question is: Does an atoroidal four-manifold that fibers over a surface admit a hyperbolic metric? In fact, it quickly became clear that a more fundamental question is: Do atoroidal four-manifolds that fiber over a surface exist? This question remained open for roughly 30 years, and a part of these talks will focus on my work with Autumn Kent from last year, proving that the answer to this second question is `yes’.
The fibered hyperbolic four-manifold question above is likely too naive to have a positive answer, but a more robust question was formalized at the end of the last century, and asks whether an atoroidal surface bundle over any compact space has Gromov hyperbolic fundamental group (again, a necessary condition). A surface bundle is completely determined by its monodromy homomorphism to the mapping class group Mod(S) of the fiber surface S. Farb and Mosher defined a notion of convex cocompactness for subgroups of Mod(S) which they showed precisely captured the Gromov hyperbolicity of the associated surface bundle when the base of the bundle is a graph (Hamenstädt extended this to all surface bundles). The later talks in this series will describe convex cocompatness and explain our conjecture that the four-manifolds Autumn and I constructed should have Gromov hyperbolic fundamental group. If there is time, I will also explain work with Dowdall, Durham, Russell, and Sisto on a variety of generalizations of convex cocompactness and explain how, at least in examples, this is related to other coarse geometric properties of the fundamental group of the bundle.
Mahan Mj (Tata Institute of Fundamental Research (TIFR), Mumbai):
Title: Models of 3-manifolds and applications
Abstract: We shall describe a number of models of degenerate ends of hyperbolic 3-manifolds that were developed based on the work of Minsky and his collaborators. In particular, we shall dwell on two of the simplest models: bounded geometry models suited for 3-manifolds with a lower bound on injectivity radius; and i-bounded geometry models suited for 3-manifolds with an upper bound on the boundaries of Margulis tubes. We shall discuss recent applications of this model to cubulability of certain hyperbolic surface-by-free groups (joint with Manning and Sageev) and Hausdorff dimension of non-conical limit sets (joint with Yang).
Alan Reid (Rice University):
Title: The geometry and topology of hyperbolic manifolds.
Abstract: In this series of lectures we introduce the class of arithmetic hyperbolic manifolds and orbifolds, and study how the geometry and topology of these manifolds and orbifolds relate to their number theoretic construction. Particular attention will be paid to those examples in dimensions 2 and 3.
组织者：
- Xiaolong Han (Shanghai Institute for Mathematics and Interdisciplinary Sciences）
- Jiming Ma (Fudan University)
- Xiaolei Wu (Fudan University)
- Shi Wang (ShanghaiTech University)
简体中文 
English