Lecture I:  Monday 10 January, 2005 at 15:30

DECISION PROBLEMS, CURVATURE, AND THE UNIVERSE OF FINITELY PRESENTED GROUPS

One may reasonably take the view that the most basic of finitely presented groups are the finite groups and that the next class worthy of mention is formed by the virtually cyclicgroups. What comes next?

One might start taking direct products and pass to consider ationof virtually abelian groups, or one might allow free products
and pass to consideration of virtually free groups; proceeding in th eformer  vein, one might enlarge the class of groups considered progressively to include virtually-nilpotent, solvable, then amenable groups --- ``the amenable side of the universe"; proceeding in the latter vein one maps out the hyperbolic/ nonpositively curved side of the universe.
 
In this talk I shall sketch a map of this "universe of finitely presented groups", letting it emerge from Dehn's original formulation of the decision problems at the heart of infinite group theory. Curvature enters the discussion through Gromov's theorem that the groups that admit "the most efficient" solution to the word problem are precisely the hyperbolic groups. The endpoint of the lecture is an explanation of the state of the art concerning our knowledge about the Word, Conjugacy, and Isomorphism problems for groups, and the prominent role that curvature plays in this understanding.

Lecture II:  Wednesday, 12 January, 2005  at 15:30

SUBDIRECT PRODUCTS OF HYPERBOLIC GROUPS, LOGIC, AND KAEHLER GEOMETRY

A free group might be thought a rather banal group, and likewise a direct product of free groups. But when one examines the subgroup sturcture of such a  direct product, a remarkable amount of complex structure emerges. Taking up this theme, we ask  what are the subgroups of direct products of hyperbolic groups?
 
For the most basic hyperbolic groups -- virtually free groups -- the classification of subdirect products  is non-trivial  but remarkably restricted in the presence of suitable finiteness hypotheses. I shall explain why this rigidity extends to subdirect products in the class of surface groups and  the next natural enlargement of the class of free
  groups --- the groups with the same elementary theory as the free groups as illuminated by Sela.
 

Applications of these results to the study of Kaehler manifolds arise in recent work of Delzant and Gromov.
 

 In contrast, we expose highly diverse behaviour among the finitely presented subgroups of direct products of more general hyperbolic groups, even in the 2-dimensional and Kleinian cases.

Lecture III:  Thursday, 13 January, 2005  at 15:30

BALANCED PRESENTATIONS OF GROUPS AND PROBLEMS OF ANDREWS-CURTIS AND GROTHENDIECK
 
A finite presentation of a group is said to be balanced if the number of generators is the same as the number of relations. I shall begin this talk by reviewing why this condition in important in the study of low-dimensional manifolds and complexes. I shall then explain some novel constructions of balanced presentations that have implications in topology, particularly to the construction of homology 4-spheres and to the conjectures of Andrews-Curtis and Zeeman. Finally, I shall explain how a certain class of balanced presentations played a key role in my solution with Grunewald of Grothendieck's problem concerning groups with isomorphic profinite completions.