ABSTRACTS OF BRIDSON'S LECTURES

NON-POSITIVE CURVATURE IN GROUP THEORY

Proposed Abstracts of Lectures

The following outline is what is planned. It is to be expected that further developments between now and August 2004 will effect the content particularly of the lectures on the fifth day.

DAY 1

Introduction: core issues in group theory and the natural emergence of non-positive curvature

MORNING: What is geometric group theory?

What are the mutual benefits of its interactions with other areas of mathematics; what are some of the main issues, and why; why is unconstrained group theory arbitrarily hard (in a quantifiable sense) and what are reasonable constraints that lead to a tractable, rich and effective theory; which classes of groups demand special attention, and why?
A brief review of why finite presentations of groups are the natural objects of study. The heart of combinatorial group theory: the word, conjugacy and isomorphism problems and their geometric/topological origins (Dehn 1912). Topological group theory: topological models of groups -- the standard 2-complex, its strengths and weaknesses, ambiguity, strategies for improvement; K(Γ, 1), finiteness issues; manifold models, geometric properties (e.g. symplectic, complex, Kälhler), aspherical models, uniqueness issues (Borel conjecture etc.); subgroups and towers.

AFTERNOON: Naturally arising classes of groups in the above context

1-relator groups, 3-manifold groups, Kähler groups, Thompson's groups. Non-Positive Curvature lurking among almost all of these basic issues: CAT(0) and hyperbolicity; Cartan-Hadamard Theorem giving K(Γ, 1); non-positive curvature in Dehn's original context; non-positive curvature for 3-manifolds; small-cancellation and hyperbolicity.

DAY 2

The two strands of the theory: actions on spaces and intrinsic geometry

MORNING: Geometric Group Theory

Strand 1: illuminating groups and spaces by studying (and constructing) group actions (preferably by isometries). Complexes of groups and developability. Examples of core algebraic problems solved by the Strand 1 approach, e.g. conjugacy problem for Coxeter groups.
Strand 2: finitely generated groups as geometric objects (àla Gromov). Quasi-isometries and intrinsic geometry. Limits and ultralimits. Quasi-isometric rigidity. Asymptotic cones of free groups (whence hyperbolic groups). Special classes emerging: Nilpotent groups, hyperbolic groups. QI-invariant formulation of non-positive curvature?

AFTERNOON:Connecting the core of combinatorial group theory to geometry

The word problem as measured by Dehn functions; connection to isoperimetric properties of manifolds via the Filling Theorem; the isoperimetric spectrum; the clear demarkation of hyperbolic groups and non-positively curved groups (again).
The isolation of the class of hyperbolic groups: hyperbolic versus atoroidal, cf. 3-manifold theory. What other ideas enter from 3-manifold theory? JSJ decomposition, automorphisms of (hyperbolic) groups; the special role of mapping class groups and automorphism groups of free groups.

DAY 3

Basic Theory 1

MORNING:From negative to non-positive curvature

We investigate how complex the universe of groups becomes as one moves beyond hyperbolic groups in the direction of non-positive curvature. Moving in this direction, we focus on the following fundamental issues: Expanding on the role of non-positive curvature in Strand 1 of geometric group theory: the basic theory and highlights of CAT(0) geometry; curvature and devlopability of complexes of groups; key issues and open questions.

AFTERNOON: Expanding on Strand 2 of geometric group theory

A brief review of the theory of hyperbolic groups; focus on the difficulties of casting non-positive curvature in terms of the intrinsic geometry of groups. Semihyperbolic groups, combable groups, automatic groups, quadratic isoperimetric inequalities---recent results discriminating amongst the main classes, geometric and topological consequences.

DAY 4

Basic Theory 2

MORNING:Dehn's problems

Confronting Dehn's fundamental problems (which force us to consider negative/non-positive curvature in the first place): the Markov/automatic properties intrinsic in the negatively curved case and the (controlled) manner in which they break down in the non-positively curved case; a description of the major open issues. How can we determine when groups or spaces are isomorphic? How can curvature help? The conjugacy and isomorphism problems for classes of negatively curved groups and spaces, particularly the subtle differences that arise on passing to different manifestations of non-positive curvature, and the horrors that can lurk among the subgroups of these seemingly benign groups?

AFTERNOON: General issues of decidability

An account of the extent to which undecidability is to be found among non-positively curved groups --- this begins with a brief review of r.e. sets of integers encoded into groups, the Higman embedding theorem, proof (given the Higman theorem) of the existence of finitely presented groups with unsolvable word problem and promotion to the undecidability of isomorphism problems for groups and spaces. Positive isomorphism results in the presence of negative curvature, and for 3-manifolds, including the theorems of Farrell-Jones and Sela. The contrasting theorems of Bridson in the combable case. The delicacy of deciding which subgroups of semihyperbolic groups are themselves semihyperbolic. Positive results in low dimensions, tameness and wildness results in higher dimensions.

DAY 5

Further Topics

MORNING: Aspects of Rigidity

Mostow rigidity, automorphisms of hyperbolic groups, the theorems of Rips-Sela and Paulin. The special role of mapping class groups and (outer) automorphism groups of free groups. The extent to which these special groups are non-positively curved. Brief survey of q.i-rigidity and Super-rigidity, Zimmer programme, Helly-type theorems, property (T), etc.

AFTERNOON: Hyperbolic versus Atoroidal, and the Tits Alternative

A summary of the many contexts in which the hyperbolic/non-hyperbolic dichotomy holds; likewise the Tits alternative. Theorems establishing the equivalence of hyperbolic and atoroidal properties in various contexts (3-manifolds, analytic manifolds, new results in 2003--2004). Counterexamples and open questions. A similar analysis of the Tits alternative. (New results expected 2003--2004.)
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