Finding surface subgroups and virtual immersions

我们描述了三个项目的想法和方法。第一个是找到某些自然群类的曲面子群：负弯曲秩为1的局部对称空间的基本群，或者更一般地说，封闭的负弯曲流形的基本群; Gromov边界连通的双曲群;或者紧致曲面的映射类群。第二个项目是将一个几何上有限的开放3-流形的“虚拟浸入”计算到一个封闭的双曲3-或4-流形中，目的是将这些虚拟浸入组装到一个更复杂的3-流形的虚拟浸入中（并归纳地计算这种浸入的数量）。第三个项目是发展退化复杂结构的理论（类似于测量层），并使用这个理论来证明二次类映射的重整化的界限，并提供一个有限细分规则被共形实现的充分条件，着眼于证明卡农猜想。数学研究者的基本工作之一是对我们所定义的对象进行分类。流形是一个具有一定数量维度的空间，没有边界或边，但在范围上是有限的。二维流形的例子包括地球的表面和内管的表面;二维流形从世纪早期就被分类了。最近的结果，包括提议者的工作，使得有可能对三维流形进行分类;这种分类是通过在三维流形中找到二维物体来实现的，三维流形以这种方式划分它，以阐明它的结构。我们的最终目标是更深入地了解我们生活的三维空间。

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.