1.1 Historical development and motivation

Geometric Group Theory emerged in the late 20th century, blending group theory, topology, and geometry. It traces back to Max Dehn's early 1900s work on word problems, gaining momentum with Mikhail Gromov's 1980s research on hyperbolic groups and negatively curved spaces.

The field uses tools like quasi-isometries and CAT(0) geometry to study groups through their actions on geometric spaces. It's grown rapidly since the 1990s, connecting with low-dimensional topology, dynamics, and computer science, offering new insights into classical group theory problems.

Origins of Geometric Group Theory

Early Foundations and Synthesis

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• Geometric group theory emerged in late 20th century synthesized group theory, topology, and geometry
• Roots trace back to Max Dehn's works in early 1900s on word and conjugacy problems in group theory
• Field gained momentum in 1980s with Mikhail Gromov's work on hyperbolic groups and negatively curved spaces
• Development influenced by study of fundamental groups in topology and geometric approach to Kleinian groups (groups of isometries of hyperbolic space)

Key Concepts and Tools

• Introduction of quasi-isometries crucial for studying groups up to "large-scale" equivalence
• Quasi-isometries preserve geometric properties at large scales while allowing for local distortions
• Incorporation of CAT(0) geometry expanded scope of geometric group theory
• CAT(0) spaces generalize non-positive curvature and include Euclidean spaces, hyperbolic spaces, and trees

Growth and Diversification

• Field experienced rapid growth and diversification since 1990s
• Connections established with various areas of mathematics (low-dimensional topology, dynamics, computer science)
• Low-dimensional topology links include study of 3-manifolds and their fundamental groups
• Dynamical aspects involve group actions on geometric spaces and their ergodic properties
• Computer science applications include algorithmic problems in group theory and complexity theory

Key Contributors to Geometric Group Theory

Foundational Figures

• Max Dehn made foundational contributions through work on word and conjugacy problems
• Dehn studied surface groups, groups associated with closed surfaces of genus $g \geq 2$
• Mikhail Gromov revolutionized field by introducing hyperbolic groups and Gromov hyperbolicity
• Gromov hyperbolicity captures notion of negative curvature in metric spaces
• William Thurston's work on 3-manifolds and geometrization impacted geometric approach to group theory
• Thurston's geometrization conjecture classified 3-manifolds into eight geometric types

Modern Pioneers

• James Cannon developed techniques for studying hyperbolic groups
• Cannon contributed to theory of automatic groups, groups with finite-state automata for word problems
• Pierre de la Harpe and Alain Valette studied Kazhdan's property (T) and its geometric implications
• Property (T) relates to group representations and has applications in ergodic theory and expander graphs
• Martin Bridson and André Haefliger expanded geometric tools through work on CAT(0) spaces and cubical complexes
• CAT(0) spaces generalize non-positively curved spaces, while cubical complexes provide combinatorial models for geometric group actions

Motivations for Geometric Group Theory

Geometric Understanding of Groups

• Desire to understand groups through actions on geometric spaces (Kleinian groups, surface groups)
• Kleinian groups act on hyperbolic 3-space, while surface groups act on hyperbolic plane
• Need for new tools to study infinite groups (fundamental groups of manifolds, automorphism groups of geometric structures)
• Recognition that group-theoretic properties have natural geometric interpretations
• Example: word length in a group corresponds to distance in its Cayley graph

Unification and Problem-Solving

• Goal of unifying branches of mathematics (group theory, topology, geometry) under common framework
• Pursuit of geometric solutions to algorithmic questions in group theory (word problem)
• Word problem asks whether two words represent the same group element
• Ambition to develop geometric classification theory for finitely generated groups
• Classification aims to categorize groups based on their large-scale geometric properties
• Analogous to classification of finite simple groups, a major achievement in 20th-century mathematics

Fundamental Questions in Geometric Group Theory

Algebraic-Geometric Relationships

• Investigate relationship between algebraic properties of groups and geometric properties of spaces they act on
• Example: hyperbolic groups act properly and cocompactly on hyperbolic spaces
• Classify groups up to quasi-isometry and study quasi-isometry invariants
• Quasi-isometry invariants include growth rate, ends, and asymptotic dimension

Geometric Methods for Classical Problems

• Develop and apply geometric methods to solve classical group-theoretic problems
• Address word problem using geometric techniques (Dehn's algorithm for hyperbolic groups)
• Tackle isomorphism problem through geometric invariants
• Study group actions on various geometric and combinatorial structures (trees, CAT(0) spaces, buildings)
• Trees provide models for free groups and amalgamated products
• Buildings generalize symmetric spaces and appear in the study of algebraic groups

Asymptotic Properties and Connections

• Investigate asymptotic properties of groups (growth rates, amenability, property (T))
• Growth rates measure how quickly the size of ball in Cayley graph increases with radius
• Explore connections between geometric group theory and other areas (topology, dynamical systems, operator algebras)
• Develop geometric understanding of algorithmic and computational aspects of group theory
• Study theory of automatic groups and random walks on groups
• Automatic groups have finite-state machines recognizing their multiplication tables
• Random walks on groups relate to heat diffusion and provide insights into group structure