⭕Groups and Geometries Unit 13 – Geometric Group Theory

Key Concepts and Definitions

• Geometric group theory studies finitely generated groups as geometric objects by considering their Cayley graphs and the actions on metric spaces
• A group is a set equipped with a binary operation that satisfies the group axioms: closure, associativity, identity, and inverses
• Finitely generated groups can be generated by a finite set of elements, and every element in the group can be expressed as a finite product of these generators and their inverses
• The Cayley graph of a group $G$ with respect to a generating set $S$ is a directed graph where vertices correspond to elements of $G$ and edges connect elements that differ by multiplication by a generator from $S$
• The word metric on a group $G$ with respect to a generating set $S$ is defined as $d(g_1, g_2) = \min\{n \mid g_1^{-1}g_2 = s_1 \cdots s_n, s_i \in S \cup S^{-1}\}$
  • It measures the shortest path distance between elements in the Cayley graph
• A group action of a group $G$ on a set $X$ is a function $G \times X \to X$ that satisfies the compatibility conditions $e \cdot x = x$ and $(gh) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and $x \in X$
• A metric space is a set equipped with a distance function that satisfies the metric axioms: non-negativity, symmetry, and triangle inequality

Fundamental Groups and Spaces

• The fundamental group $\pi_1(X, x_0)$ of a topological space $X$ with basepoint $x_0$ is the group of homotopy classes of loops based at $x_0$
  • Homotopy is an equivalence relation between continuous functions that can be continuously deformed into each other while keeping the endpoints fixed
• The fundamental group captures the essential one-dimensional holes or loops in a space (circles in a torus or a figure-eight)
• Spaces with isomorphic fundamental groups are homotopy equivalent, meaning they have the same topological shape up to continuous deformations
• The fundamental group is a topological invariant and can be used to distinguish non-homeomorphic spaces (a circle and a disk)
• The universal cover of a connected, locally path-connected, and semi-locally simply connected space $X$ is a simply connected space $\tilde{X}$ that covers $X$ with a projection map $p: \tilde{X} \to X$
  • The fundamental group $\pi_1(X)$ acts freely and properly discontinuously on the universal cover $\tilde{X}$ by deck transformations
• The quotient space of the universal cover by the action of the fundamental group is homeomorphic to the original space $X \cong \tilde{X} / \pi_1(X)$

Word Problems and Presentations

• The word problem for a finitely presented group $G = \langle S \mid R \rangle$ asks whether two words in the generators represent the same element in the group
  • It is a fundamental decision problem in group theory and is undecidable in general
• A group presentation is a description of a group in terms of generators and relations, written as $G = \langle S \mid R \rangle$, where $S$ is a set of generators and $R$ is a set of relations (words in the generators that equal the identity)
• The free group $F(S)$ on a set $S$ is the group of reduced words in the alphabet $S \cup S^{-1}$ with the operation of concatenation and reduction
• A group $G$ is a quotient of a free group $F(S)$ by the normal subgroup $N$ generated by the relations $R$, denoted as $G \cong F(S) / N$
• The word problem for a finitely presented group is solvable if and only if the group is recursive (has a recursive presentation)
• The Dehn function of a finitely presented group measures the complexity of the word problem by quantifying the area of minimal disk diagrams needed to show that a word represents the identity
  • It is a quasi-isometry invariant and can be used to classify groups up to quasi-isometry

Cayley Graphs and Group Actions

• The Cayley graph of a group encodes the algebraic structure of the group as a geometric object
  • It is a regular graph where the degree of each vertex equals the size of the generating set
• The action of a group on its Cayley graph by left multiplication is free (no non-identity element fixes a vertex) and transitive (any vertex can be mapped to any other vertex)
• The Cayley graph is a geometric realization of the group and can be used to study geometric properties of the group (growth rate, ends, hyperbolicity)
• A group action is proper if the stabilizers of points are finite and cocompact if the quotient space is compact
• The Švarc–Milnor lemma states that if a group acts properly and cocompactly on a proper geodesic metric space, then the group is finitely generated and quasi-isometric to the space
  • It provides a connection between the geometry of a space and the algebraic properties of a group acting on it
• The Bass–Serre theory studies groups acting on trees and provides a way to decompose groups as amalgamated free products or HNN extensions
  • It is a powerful tool for understanding the structure of groups and constructing new examples

Quasi-Isometries and Geometric Properties

• A quasi-isometry between metric spaces $(X, d_X)$ and $(Y, d_Y)$ is a map $f: X \to Y$ such that there exist constants $A \geq 1$ and $B \geq 0$ satisfying:
  • $\frac{1}{A} d_X(x_1, x_2) - B \leq d_Y(f(x_1), f(x_2)) \leq A d_X(x_1, x_2) + B$ for all $x_1, x_2 \in X$
  • Every point in $Y$ is within distance $B$ of the image of $f$
• Quasi-isometries capture the large-scale geometry of metric spaces and provide a notion of equivalence between spaces that may differ in their small-scale details
• The growth function of a finitely generated group measures the size of balls in the Cayley graph as a function of the radius
  • It is a quasi-isometry invariant and can be used to classify groups as polynomial growth, exponential growth, or intermediate growth (Grigorchuk group)
• The number of ends of a finitely generated group is a quasi-isometry invariant and measures the number of connected components at infinity of the Cayley graph
  • Groups can have 0, 1, 2, or infinitely many ends (integers, free groups, infinite dihedral group)
• The Gromov boundary of a hyperbolic group is a compact metric space that encodes the asymptotic geometry of the group and is a quasi-isometry invariant
  • It can be used to study the dynamics of group actions on hyperbolic spaces (limit sets, Patterson–Sullivan measures)

Hyperbolic Groups

• A finitely generated group is hyperbolic if its Cayley graph is a hyperbolic metric space, meaning it satisfies the thin triangles condition: there exists a constant $\delta > 0$ such that every geodesic triangle is $\delta$-thin (each side is contained in the $\delta$-neighborhood of the other two sides)
• Hyperbolic groups have a linear Dehn function, solvable word problem, and finitely many cone types (local configurations in the Cayley graph)
• Examples of hyperbolic groups include free groups, fundamental groups of closed hyperbolic surfaces, and small cancellation groups
• The Gromov boundary of a hyperbolic group is a compact metric space that can be defined using equivalence classes of geodesic rays
  • It has a natural topology and a visual metric that encodes the asymptotic geometry of the group
• Hyperbolic groups satisfy the Tits alternative: every subgroup is either virtually cyclic (contains a cyclic subgroup of finite index) or contains a free subgroup of rank 2
• The geodesic flow on the unit tangent bundle of a closed hyperbolic manifold is Anosov (uniformly hyperbolic) and mixing (correlations decay exponentially)
  • It provides a connection between hyperbolic geometry and dynamical systems

Applications and Examples

• The fundamental group of a knot complement is a powerful invariant in knot theory and can be used to distinguish knots (trefoil knot, figure-eight knot)
  • It has a presentation given by the Wirtinger presentation, which relates to the diagram of the knot
• The mapping class group of a surface is the group of isotopy classes of self-homeomorphisms of the surface and is an important object in low-dimensional topology and geometric group theory
  • It is finitely presented and acts on the Teichmüller space of the surface, which is a contractible complex manifold that parameterizes hyperbolic structures on the surface
• Braid groups are fundamental groups of configuration spaces of points in the plane and have connections to knot theory, mapping class groups, and representation theory
  • They have a presentation given by the Artin presentation and can be viewed as mapping class groups of punctured disks
• Coxeter groups are groups generated by reflections in a real vector space and have a rich combinatorial and geometric structure
  • They act on the Davis complex, which is a CAT(0) cube complex (a metric space with non-positive curvature)
• Automatic groups are finitely generated groups that admit an automatic structure, which is a finite-state automaton that recognizes a regular language of normal forms for group elements
  • They have a quadratic Dehn function and include hyperbolic groups, braid groups, and mapping class groups

Advanced Topics and Open Problems

• The Baum–Connes conjecture relates the K-theory of the reduced C*-algebra of a group to the equivariant K-homology of the classifying space for proper actions of the group
  • It has important implications in topology, geometry, and functional analysis and has been proven for many classes of groups (hyperbolic groups, CAT(0) cubical groups)
• The Novikov conjecture states that the higher signatures of a compact oriented manifold are homotopy invariants and is a central problem in high-dimensional topology
  • It is implied by the Baum–Connes conjecture and has been proven for many classes of groups (hyperbolic groups, CAT(0) groups)
• The Farrell–Jones conjecture relates the K- and L-theory of the group ring of a group to the K- and L-theory of the virtually cyclic subgroups of the group
  • It has important applications in geometry and topology and has been proven for many classes of groups (hyperbolic groups, CAT(0) groups)
• The Geometric Group Theory problem list is a collection of open problems in the field, maintained by Mark Sapir, and includes questions on group presentations, growth, and quasi-isometries
• The Zimmer program aims to classify actions of lattices in Lie groups on compact manifolds and has connections to rigidity theory and dynamical systems
  • It has led to important results such as Zimmer's cocycle superrigidity theorem and Margulis' normal subgroup theorem
• The Burnside problem asks whether a finitely generated group with a fixed exponent (every element has finite order) is necessarily finite
  • It has been solved negatively in general but remains open for specific classes of groups (e.g., groups of exponent 5)
• The Hanna Neumann conjecture bounds the rank of the intersection of two finitely generated subgroups of a free group in terms of the ranks of the subgroups
  • It has been proven by Friedman and Mineyev using techniques from geometric group theory and algebraic topology