Project Details



Award: DMS-0103208

Principal Investigator: Lee Mosher

The motivating theme of geometric group theory is that finitely

generated groups can be studied using topological and geometric

methods. Starting around 1980, Gromov proposed classifying

groups geometrically using the relation of quasi-isometry. He

demonstrated that the quasi-isometry class of a group G can often

be described explicitly in simple algebraic or geometric terms, a

process now referred to as ``quasi-isometric rigidity'' or

``QI-rigidity'' for the group G. The first part of this project

will be to investigate QI-rigidity problems for graphs of surface

groups and of abelian groups. An initial focus will be those

graphs of groups with the simplest algebraic structure, namely

semidirect products with free groups, determined by a

homomorphism from a free group into an appropriate automorphism

group such as the mapping class group of a surface. General

graphs of surface groups are determined by homomorphisms into the

commensurability mapping class group, a much more mysterious

object, and this point of view will be used to investigate

constructions of new and interesting examples. Also, recent work

on graphs of abelian groups has revealed a lot of rich structure,

suggesting a real possibility of obtaining QI-rigidity in many

new cases. In the second part of this project, the focus will be

to study geometric properties of free groups, motivated by

analogies between surface groups and free groups. In particular,

Thurston's ending lamination conjecture, an important goal in the

study of surface groups, has an analogue in the study of free

groups, in terms of classifying certain group actions up to

equivariant quasi-isometry. Pursuing this issue will require

generalizing many of the standard tools of surface groups, such

as geodesics in Teichmuller space, to the setting of free groups.

Geometric group theory is the study of infinitely symmetric

patterns. Popular examples called ``surface groups'' are

familiar from wallpaper symmetries and from the symmetries of

Escher's prints. Scientific examples occur in the symmetry

groups of crystalline arrays, and the symmetry groups of field

theories in particle physics. The development of topology

starting in the late 19th century, and the concomitant

development of combinatorial group theory, exhibited a direct

link between abstract groups and geometry. The need for deeper

understanding of this link has been demonstrated again and again

by different threads within 20th century mathematical

developments. Many of these threads were pulled together around

1980 by Gromov, whose proposed unification of geometric group

theory using the relation of ``quasi-isometry'' has been very

fruitful in the intervening twenty years. The focus of this

research project will be to investigate quasi-isometric

classification problems for several different types of symmetry

groups. In particular, by using a constructive technique known

as ``graphs of groups'', new symmetry groups can be constructed

out of familiar examples such as surface groups; these and

closely related constructions will be the subjects of this

research project.

Effective start/end date8/15/017/31/05


  • National Science Foundation: $104,961.00


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