Finding surface subgroups and virtual immersions

点击翻译按钮获取中文摘要

We describe ideas and approaches to three projects. The first is to find surface subgroups for certain natural classes of groups: the fundamental group of a negatively curved rank 1 locally symmetric space, or more generally of a closed negatively curved manifold; a hyperbolic group whose Gromov boundary is connected; or the mapping class group of a compact surface. The second project is to count "virtual immersions" of a geometrically finite open 3-manifold into a closed hyperbolic 3- or 4-manifold, with an aim toward assembling these virtual immersions into a virtual immersion of a more complex 3-manifold (and inductively counting the number of such immersions). The third project is to develop the theory of degenerate complex structures (with something like measured laminations), and use this theory to prove bounds for renormalization of quadratic-like maps, and to provide a sufficient condition for a finite subdivision rule to be conformally realized, with an eye towards proving the Cannon conjecture. One of the fundamental jobs of the mathematical researcher is to classify the objects that we have defined. A manifold is a space with some number of dimensions, with no boundaries or edges but which is finite in extent. Examples of two-dimensional manifolds include the surface of the Earth and the surface of an inner tube; the two-dimensional manifolds have been classified since the early twentieth century. Recent results including the work of the proposer have made it possible to classify three-dimensional manifolds as well; this classification is made possible by finding two-dimensional objects within the three-dimensional manifold that divides it in such a way as to elucidate its structure. Our ultimate goal is to arrive at a much deeper understanding of the three-dimensional space that we live in.