⭕Geometric Group Theory Unit 1 – Intro to Geometric Group Theory

Key Concepts and Definitions

• Geometric group theory studies finitely generated groups as geometric objects by considering their Cayley graphs and the spaces on which they act
• A group is a set equipped with a binary operation satisfying the group axioms: closure, associativity, identity, and inverses
• Generators of a group are a subset of elements that can generate the entire group through the group operation
• Relations in a group are equations that hold among the generators, often used to define the group in terms of a presentation
• The Cayley graph of a group is a directed graph where vertices represent group elements and edges represent multiplication by generators
• The word metric on a group is defined using the shortest path distance in the Cayley graph, making it a metric space
• Quasi-isometries are maps between metric spaces that preserve large-scale geometry, allowing for the study of coarse equivalence of groups
  • Quasi-isometries need not be continuous or bijective, but they preserve distances up to fixed additive and multiplicative constants

Fundamental Groups and Cayley Graphs

• The fundamental group of a topological space is the group of homotopy classes of loops based at a chosen point
• Fundamental groups encode information about the holes and loops in a space, providing a connection between topology and group theory
• The Cayley graph of a group can be seen as a geometric realization of the group, with the group acting freely and transitively on the graph
• The Cayley graph depends on the choice of generating set, but different Cayley graphs of the same group are quasi-isometric
• The growth function of a group measures the size of balls in the Cayley graph and is invariant under quasi-isometry
• Cayley graphs can be used to study properties such as ends, amenability, and hyperbolicity of groups
• The Švarc-Milnor lemma relates the geometry of a group to the geometry of a space on which it acts properly and cocompactly

Word Problems and Normal Forms

• The word problem for a group asks whether two words in the generators represent the same group element
• A solution to the word problem provides a normal form for group elements, a unique representative for each element
• The Dehn function of a group measures the complexity of the word problem by quantifying the area of minimal diagrams for relations
• Groups with a solvable word problem are called recursive or computable
• The conjugacy problem asks whether two group elements are conjugate and is generally harder than the word problem
• Automatic groups have a regular language of normal forms and a quadratic Dehn function, allowing for efficient computation
• Hyperbolic groups, which include free groups and fundamental groups of negatively curved manifolds, have a linear Dehn function and solvable word and conjugacy problems

Quasi-Isometries and Geometric Properties

• Quasi-isometries capture the large-scale geometry of metric spaces, ignoring small-scale details
• Two metric spaces are quasi-isometric if there exists a quasi-isometry between them
• Quasi-isometric groups share many geometric properties, such as growth type, ends, and hyperbolicity
• Gromov hyperbolicity is a geometric property of metric spaces that generalizes negative curvature and is preserved by quasi-isometries
• The boundary at infinity of a hyperbolic space is a compact space that encodes the asymptotic geometry and is invariant under quasi-isometries
• The growth type of a group (polynomial, exponential, intermediate) is a quasi-isometry invariant that reflects the volume growth of balls in the Cayley graph
• Amenable groups, which include finite groups, abelian groups, and solvable groups, admit a finitely additive, translation-invariant probability measure and are characterized by the Følner condition

Group Actions on Spaces

• A group action is a homomorphism from a group to the symmetries of a set, allowing the group to act on the set by permutations
• Group actions can be used to study the geometry and topology of spaces through the lens of symmetry
• The orbit of a point under a group action is the set of all points that can be reached by applying group elements
• The stabilizer of a point is the subgroup of elements that fix the point, and the orbit-stabilizer theorem relates the sizes of orbits and stabilizers
• A group action is free if the stabilizer of every point is trivial, and transitive if there is a single orbit
• Properly discontinuous actions, where each point has a neighborhood moved off itself by all but finitely many group elements, are important in geometric topology
• The quotient of a space by a properly discontinuous group action is a new space (an orbifold) that encodes the symmetries of the original space
• Cocompact actions, where the quotient space is compact, allow for the study of groups through the geometry of the spaces they act on

Examples and Applications

• Free groups, which have no relations among generators, act freely on trees and serve as building blocks for other groups
• Fundamental groups of surfaces act properly discontinuously and cocompactly on the hyperbolic plane, providing a link between geometry and topology
• Braid groups, which describe braids in three-dimensional space, have a geometric interpretation as fundamental groups of configuration spaces and have applications in knot theory and cryptography
• The mapping class group of a surface, consisting of isotopy classes of self-homeomorphisms, acts on the Teichmüller space of the surface and has important connections to 3-manifold topology and algebraic geometry
• Coxeter groups, generated by reflections, act on spherical, Euclidean, and hyperbolic spaces and have a rich combinatorial and geometric structure
• Hyperbolic 3-manifold groups, which are fundamental groups of hyperbolic 3-manifolds, have been extensively studied in the context of the geometrization conjecture and have applications to knot theory and number theory
• Geometric group theory has found applications in computer science, particularly in the study of algorithms and complexity for problems involving groups and their actions

Computational Techniques

• The Todd-Coxeter algorithm is a method for enumerating cosets of a subgroup and can be used to compute the index of a subgroup and to test for group isomorphism
• The Knuth-Bendix completion algorithm takes a set of rewrite rules (such as group relations) and attempts to find a confluent and terminating rewriting system, which can be used to solve the word problem
• Automatic structures for groups, consisting of a regular language of normal forms and fellow traveler property, allow for efficient computation in the group
• The Reidemeister-Schreier method is an algorithm for finding a presentation of a subgroup given a presentation of the larger group
• Geometric techniques, such as the study of van Kampen diagrams and the use of small cancellation theory, can be used to solve the word and conjugacy problems in certain classes of groups
• Probabilistic methods, such as the use of random walks on groups and the study of generic-case complexity, have been employed to analyze the behavior of group-theoretic algorithms
• Computational tools from algebraic topology, such as the computation of homology and cohomology groups, have been adapted to the study of groups and their properties

Open Problems and Current Research

• The Burnside problem asks whether a finitely generated group satisfying the identity $x^n = 1$ for a fixed exponent $n$ must be finite, and has been resolved negatively for large enough $n$
• The Hanna Neumann conjecture concerns the rank of the intersection of two finitely generated subgroups of a free group and has been proven in the strengthened form of the Strengthened Hanna Neumann conjecture
• 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, and has important implications in topology and geometry
• The Farrell-Jones conjecture states that certain assembly maps in algebraic K- and L-theory are isomorphisms for a wide class of groups, and has consequences for the classification of high-dimensional manifolds
• The study of asymptotic cones of groups, which are limit objects obtained by rescaling the metric on the group by larger and larger factors, has led to important insights into the large-scale geometry of groups
• The development of higher-dimensional versions of geometric group theory, such as CAT(0) cube complexes and acylindrical hyperbolicity, has opened up new avenues for the study of groups and their actions
• The application of geometric group theory to the study of rigidity phenomena, such as superrigidity and quasi-isometric rigidity, remains an active area of research with connections to ergodic theory and dynamical systems