CAREER: Surface bundles and logic in geometric group theory

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

This project revolves around finding solutions to equations over discrete groups. This is equivalent to understanding sets of homomorphisms between a pair of discrete groups (the domain representing the equations and the target the group of interest). The first half of this project is motivated by the theory of S-bundles, where S is a compact orientable surface. Surface bundles arise throughout pure mathematics. If B is a manifold (or other reasonable space), then there is a one-to-one correspondence between the set of isomorphism classes (oriented) of S-bundles over B and the set of (conjugacy classes of) homomorphisms from the fundamental group of B to the mapping class group of S. The PI intends to develop a general structure theory for this set when the fundamental group of B is finitely generated (which will always happen when B is compact, for example). In the second half of this project (joint work with H, Wilton), we study homomorphisms to free (and more generally torsion-free hyperbolic) groups. Broadly speaking, this is the class of `negatively-curved' groups. This study is motivated by the first-order logic of these groups. It is quite remarkable that the geometry of negative curvature has profound implications for first-order logic. We intend to study these first-order theories from an algorithmic point of view, and find general decision processes which determine if a logical sentence is true or false. Groups form a natural language for studying symmetries of mathematical objects. As such, they arise throughout mathematics, and their study is informed by many branches of mathematics. There are very few basic properties of a `symmetry', the most important of which is that it doesn't lose any information about the space, so it can be undone. In this project, we treat the groups as objects of intrinsic interest, although the questions that we ask take their motivation from topology, geometry, logic and computer science as well as from within group theory. Broadly speaking, we take a collection of equations over a group G, and try to understand the set of all solutions to these equations. In the first half of the project, the group G is the mapping class group of a surface, which captures much of the symmetry of a surface (a space which looks like the plane in small sets). Studying equations over the mapping class group is of fundamental interest in geometry and topology, through the study of surface bundles (a space which nearby any point decomposes into a pair of smaller sets, one of which is a surface). The study of equations over the mapping class group gives a general theory parametrizing surface bundles. The second half of this project studies equations over groups from the point of view of logic, and looks for general algorithms which decide if logical sentences are true of false. This project is jointly funded by the Topology Program and the Foundations Program.