Project Details
Description
This project will focus on the geometry of two important classes of
groups: mapping class groups of surfaces; and outer automorphism groups
of free groups. Mapping class groups will be studied using the tools of
large scale geometry, with the aim of learning about quasi-isometric
properties of mapping class groups and proving quasi-isometric rigidity.
Along the way we expect to develop new tools for understanding
quasi-isometric properties of relatively hyperbolic groups. Outer
automorphism groups are not as well understood as mapping class groups,
and will be studied from a more foundational point of view. The focus
will be to develop analogies with the theory of Teichmuller spaces which
is a very well developed tool for studying mapping class groups, with the
particular aim of developing analogies with the ending lamination
conjecture.
Geometric group theory is the study of symmetry groups associated to
highly symmetric geometries of infinite extent, such as a crystal lattice
in 3-dimensional space, whose symmetry group is the set of all motions of
space which move each point of the lattice to another point of the
lattice. The theory of groups was developed in the 19th century for
studying symmetry, and it quickly developed into a deep and rich
mathematical tool. Groups and geometries go hand-in-hand: every group has
a corresponding geometry, and one can use tools of geometry to learn about
the group. This project will focus on two classes of groups which have
assumed importance in the field of topology. Given a surface, such as the
surface of a doughnut with one or more holes, the 'mapping class group' of
that surface describes all the ways in which that surface can be mapped
back onto itself. Similarly, the 'outer automorphism group of a free
group' is a group that describes all of the ways in which a finite
1-dimensional network or graph can be mapped back onto itself. In both
cases, we will study geometries associated to these groups, with the aim
of learning more about the groups themselves. In the case of mapping
class groups our aim is to prove one of the central geometric conjectures,
known as quasi-isometric rigidity. In the case of outer automorphism
groups of free groups, where the geometry is less well understood, we will
attempt to build new tools and make a deeper study of existing tools, in
order to reach a better understanding of the geometry.
Status | Finished |
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Effective start/end date | 7/1/04 → 6/30/08 |
Funding
- National Science Foundation: $174,507.00