Finer Coarse Geometry

点击翻译按钮获取中文摘要

In geometric group theory and geometric topology, one usually studies groups and spaces through their coarse geometry, focusing only on properties that are preserved by QIs (quasi-isometries, or maps with bounded additive and multiplicative distortion of distances). This viewpoint is forced on us if we want to study infinite finitely-generated groups without specifying a generating set, since the graphs that record each group's geometry are only mutually related by QIs. In this project, the PI proposes to study group statistics that are large-scale but not invariant under quasi-isometry, and so may depend nontrivially on a choice of generating set. This point of view facilitates the study of randomness and asymptotic density in groups. The PI has studied free abelian groups and the discrete Heisenberg group in this way, opening up several avenues for current and future research. Another object of particular interest is Teichmueller space, the parameter space for many kinds of geometric structures on surfaces. The proposal contains descriptions of ongoing and planned work developing ideas for measuring "typical'' geometric properties in Teichmueller space, the mapping class group, and the complex of curves using data that is destroyed by quasi-isometry. For instance, one can show in this way that even though Teichmueller space is not a hyperbolic space, certain characteristics of hyperbolicity hold on average, or up to measure. The techniques developed to study this space, which has many features of negative curvature but does not satisfy any of the usual curvature conditions, are promising for applicability in other settings, such as relatively hyperbolic groups. A third main component of the proposal returns to the world of quasi-isometry invariants, developing new families of filling functions that appear to make finer geometric distinctions than some of their predecessors in the literature.This project belongs to a world of ideas that has a growing number of practical applications. For instance, it is useful in a wide range of applications (like communications networks) to have sparse but well-connected graphs. In mathematics these are called expanders; the first families of examples came from Cayley graphs of groups, and later it was realized that randomization over regular graphs also produces good expanders with high probability. Expansion constants, like a host of other geometric statistics, are "fine" and not QI invariant. Generally, the question of understanding the nature of an object by randomly sampling its points appears across many areas of applied mathematics, and the topics in this proposal are clustered around this theme.