Geometric Group Theory

Walter D. Neumann and Michael Shapiro

original PostScript file, 35 pages, 547 Kb
AtlasImage version, 40 pages - reformatted, with different section/page numbering.

Contents

Outline

0 Introduction

0.1 Dehn's problems: the word problem, the conjugacy problem, the isomorphism problem

0.2 Cayley graphs and hyperbolicity

0.3 The word problem and Cayley graphs

0.4 Dehn's solution to the word problem for hyperbolic surface groups

1 Cayley graphs and hyperbolicity

1.1 Hyperbolicity

1.2 Cayley graphs and group actions

1.3 Groups acting on hyperbolic spaces

2 Elementary properties of hyperbolic groups

2.1 Quasi-geodesics in a hyperbolic space

2.2 Hyperbolicity is a quasi-isometry invariant

2.3 Word and conjugacy problem for hyperbolic groups

2.4 Torsion in hyperbolic groups

2.5 The Rips Complex

2.6 The boundary of a hyperbolic group

3 Isoperimetric inequalities

3.1 Area of a word that represents the identity

3.2 Van Kampen diagrams

3.3 Dehn's function of a presentation

3.4 Isoperimetric functions and the word problem

3.5 Equivalence of isoperimetric functions

3.6 The word problem is geometric

4 The JSJ decomposition

4.1 Splittings of hyperbolic groups

5 Regular languages, automatic, bi-automatic and asynchronously automatic groups

5.1 Definitions

5.2 Fellow traveler property

5.3 Quadratic isoperimetric equality and quadratic time word problem

5.4 Closure properties

5.5 Famous classes of automatic groups

5.6 Cone types and falsification by fellow-traveler

5.7 Bi-automaticity

5.8 Asynchronous automaticity

6 Equivalence classes of automatic structures

6.1 The asynchronous fellow traveller property

6.2 Rationality

6.3 Bi-automaticity

7 Subgroups of automatic groups

7.1 Rational and quasiconvex subsets

7.2 Rational subgroups inherit automatic or hyperbolic structures

7.3 Subgroups of bi-automatic groups

8 Almost Convexity

8.1 Defintion

8.2 Building the Cayley graph

8.3 Almost convexity is not a group property

9 Growth functions and growth rates

9.1 Finite cone types implies rational growth function

9.2 Gromov's theorem: polynomial growth implie virtually nilpotent

Date received: September 19, 2000.