The Geometry of the Word Problem for Finitely Generated Groups Book

Foreword     vii
Dehn Functions and Non-Positive Curvature   Noel Brady     1
Preface     3
The Isoperimetric Spectrum     5
First order Dehn functions and the isoperimetric spectrum     5
Definitions and history     5
Perron-Frobenius eigenvalues and snowflake groups     7
Topological background     9
Graphs of spaces and graphs of groups     10
The torus construction and vertex groups     11
Snowflake groups     15
Snowflake groups and the lower bounds     16
Upper bounds     22
Questions and further explorations     25
Dehn Functions of Subgroups of CAT(0) Groups     29
CAT(0) spaces and CAT(0) groups     31
Definitions and properties     31
M[kappa]-complexes, the link condition     33
Piecewise Euclidean cubical complexes     35
Morse theory I: recognizing free-by-cyclic groups     38
Morse functions and ascending/descending links     38
Morse function criterion for free-by-cyclic groups     42
Groups of type (F[subscript n] [Characters not reproducible] Z) x F[subscript 2]     45
LOG groups and LOT groups     45
Polynomially distorted subgroups     46
Examples: The double construction and the polynomial Dehn function     48
Morse theory II: topology of kernel subgroups     49
A non-finitely generated example: Ker(F[subscript 2 right arrow] Z)     51
A non-finitely presented example: Ker(F[subscript 2] x F[subscript 2 right arrow] Z)     53
A non-F[subscript 3] example: Ker(F[subscript 2] x F[subscript 2] x F[subscript 2 right arrow] Z)     56
Branched cover example     57
Right-angled Artin group examples     57
Right-angled Artin groups, cubical complexes and Morse theory     58
The polynomial Dehn function examples     60
A hyperbolic example     64
Branched covers of complexes     65
Branched covers and hyperbolicity in low dimensions     66
Branched covers in higher dimensions     70
The main theorem and the topological version     71
The main theorem: sketch     72
Bibliography     77
Filling Functions   Tim Riley     81
Notation     83
Introduction     85
Filling Functions     89
Van Kampen diagrams     89
Filling functions via van Kampen diagrams      91
Example: combable groups     94
Filling functions interpreted algebraically     99
Filling functions interpreted computationally     100
Filling functions for Riemannian manifolds     105
Quasi-isometry invariance     106
Relationships Between Filling Functions     109
The Double Exponential Theorem     110
Filling length and duality of spanning trees in planar graphs     115
Extrinsic diameter versus intrinsic diameter     119
Free filling length     119
Example: Nilpotent Groups     123
The Dehn and filling length functions     123
Open questions     126
Asymptotic Cones     129
The definition     129
Hyperbolic groups     132
Groups with simply connected asymptotic cones     137
Higher dimensions     141
Bibliography     145
Diagrams and Groups   Hamish Short     153
Introduction     155
Dehn's Problems and Cayley Graphs     157
Van Kampen Diagrams and Pictures     163
Small Cancellation Conditions     179
Isoperimetric Inequalities and Quasi-Isometries     187
Free Nilpotent Groups      197
Hyperbolic-by-free groups     201
Bibliography     205