The mapping class group and its finitely generated subgroups

Ohio State University, The, Columbus OH

Investigators

Abstract

The Principal Investigator proposes to study the mapping class group and its various normal finitely generated subgroups. The project comprises three main parts. There is a well-known dictionary for translating results for the special linear group over the integers into conjectures for the mapping class group. In the first part of the project the Principal Investigator aims to further this connection by studying homological questions surrounding an important object associated to the mapping class group called the curve complex which is analogous to the Tits building for the special linear group. An important subgroup of the mapping class group is the Torelli group which one may view as the difference between the mapping class group and arithmetic groups such as the special linear group. The second part of the project aims to reprove Johnson's finite generation of the Torelli group from a geometric perspective. The Principal Investigator sees this as a step towards a possible finite presentation for the Torelli group. The third part of the project looks at the geometry of various finitely generated normal subgroups of the Torelli group. Two-dimensional surfaces are fundamental mathematical objects that appear throughout mathematics, physics, and other sciences. In mathematics surface symmetries are a key ingredient in the construction and understanding of three-dimensional objects. In string theory, a branch of theoretical physics, surfaces arise as paths of strings in space-time called "worldsheets." Surface symmetries have even arisen in the work of Jack Cowan (University of Chicago) on understanding how images are processed by the visual cortex in the human brain. Symmetries of surfaces have been studied mathematically since at least the time of Euclid's Elements, but their modern study has developed rapidly over the past century. The Principle Investigator proposes a study of symmetries of surfaces from three points of view. Firstly, the Principal Investigator intends to further a well-known analogy between symmetries of surfaces and symmetries of multidimensional grids (the latter having a well-developed theory of their own). In the second and third parts of the project the Principal Investigator intends to investigate a particularly mysterious collection of surface symmetries called the Torelli group which one may view as the difference between symmetries of surfaces and symmetries of multidimensional grids.