**THE MALLAT FAMILY FUND FOR RESEARCH IN MATHEMATICS **

**invites**** you to a**

**SPECIAL LECTURE SERIES**** **

**to**** be presented by**

The lectures will be held in

Room 232

Technion - Israel Institute of Technology

Lecture
I:

**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:

**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:

**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.