⭕Geometric Group Theory Unit 4 – Quasi-Isometries: Geometric Properties

What's the deal with Quasi-Isometries?

• Quasi-isometries provide a way to compare metric spaces that are not necessarily isometric but share similar large-scale geometry
• Allow for a more flexible notion of equivalence between metric spaces, capturing the idea that two spaces are "roughly the same" from a geometric perspective
• Useful in studying the geometry of finitely generated groups, as they are invariant under the choice of generating set (up to quasi-isometry)
• Play a crucial role in geometric group theory, enabling the classification of groups based on their large-scale geometric properties
• Quasi-isometries have applications beyond group theory, such as in the study of manifolds, graphs, and other geometric objects
  • For example, quasi-isometries can be used to compare the geometry of two Riemannian manifolds that are not necessarily isometric but have similar large-scale features
• Provide a bridge between the worlds of metric geometry and group theory, allowing for the transfer of ideas and techniques between these two domains

Key Concepts and Definitions

• A map $f: (X, d_X) \to (Y, d_Y)$ between metric spaces is a quasi-isometry if there exist constants $A \geq 1$ and $B \geq 0$ such that:
  • $\frac{1}{A} d_X(x_1, x_2) - B \leq d_Y(f(x_1), f(x_2)) \leq A d_X(x_1, x_2) + B$ for all $x_1, x_2 \in X$
  • For every $y \in Y$, there exists an $x \in X$ such that $d_Y(y, f(x)) \leq B$
• The constants $A$ and $B$ are called the quasi-isometry constants
• Two metric spaces $(X, d_X)$ and $(Y, d_Y)$ are quasi-isometric if there exists a quasi-isometry $f: X \to Y$
• A quasi-geodesic in a metric space $(X, d)$ is a quasi-isometric embedding of an interval $I \subseteq \mathbb{R}$ into $X$
  • Quasi-geodesics are a generalization of geodesics, allowing for some bounded deviation from the shortest path
• The Gromov product of two points $x, y \in X$ with respect to a basepoint $p \in X$ is defined as $(x|y)_p = \frac{1}{2}(d(p, x) + d(p, y) - d(x, y))$
  • The Gromov product measures how close $x$ and $y$ are to each other relative to their distances from $p$

Quasi-Isometry vs. Other Geometric Notions

• Quasi-isometries are weaker than isometries, as they allow for bounded distortion of distances and the existence of "holes" in the image
  • Isometries preserve distances exactly, while quasi-isometries allow for bounded multiplicative and additive errors
• Stronger than bi-Lipschitz equivalence, which requires the map to be bijective and the distortion of distances to be bounded by a multiplicative constant
• Incomparable with uniform continuity, as quasi-isometries may not be uniformly continuous, and uniformly continuous maps may not be quasi-isometries
• Quasi-isometric embedding is a weaker notion than quasi-isometry, as it does not require the map to have a quasi-dense image
• Quasi-isometries induce a well-defined equivalence relation on the class of metric spaces, called quasi-isometry equivalence
  • Quasi-isometry equivalence is coarser than isometry equivalence, allowing for a wider range of spaces to be considered equivalent

Properties of Quasi-Isometries

• Quasi-isometries are invertible up to bounded distance, meaning that if $f: X \to Y$ is a quasi-isometry, then there exists a quasi-isometry $g: Y \to X$ such that $d_X(x, g(f(x)))$ and $d_Y(y, f(g(y)))$ are bounded for all $x \in X$ and $y \in Y$
• The composition of two quasi-isometries is again a quasi-isometry, with quasi-isometry constants depending on the constants of the individual maps
• Quasi-isometries preserve the large-scale geometry of metric spaces, such as the growth rate of balls, the existence of quasi-geodesics, and the boundary at infinity
• Quasi-isometries between proper geodesic metric spaces induce homeomorphisms between their Gromov boundaries
• Quasi-isometries preserve the property of a metric space being Gromov hyperbolic, a key concept in geometric group theory
  • A metric space is Gromov hyperbolic if there exists a $\delta \geq 0$ such that for any three points $x, y, z$, the Gromov product $(x|y)_z$ is bounded below by the minimum of $(x|z)_y$ and $(y|z)_x$, up to an additive error of $\delta$

Examples and Applications

• The Cayley graph of a finitely generated group, equipped with the word metric, is quasi-isometric to the group itself with any left-invariant proper metric
  • This allows for the study of the large-scale geometry of groups using their Cayley graphs
• Fundamental groups of compact Riemannian manifolds are quasi-isometric to the universal cover of the manifold, equipped with the lifted Riemannian metric
• Quasi-isometries between Cayley graphs of groups can be used to define quasi-isometric invariants, such as growth rate, ends, and Dehn functions
• In coarse geometry, quasi-isometries are used to define coarse equivalence between metric spaces, which captures the idea of two spaces having the same large-scale geometry
• Quasi-isometries play a role in the study of negatively curved manifolds and spaces, such as hyperbolic groups and CAT(0) spaces
  • For example, the fundamental group of a closed hyperbolic manifold is quasi-isometric to the hyperbolic space of the same dimension

Techniques for Proving Quasi-Isometry

• Constructing explicit quasi-isometries between metric spaces, such as by using the Milnor-Švarc lemma for groups acting geometrically on metric spaces
• Proving that a map between metric spaces satisfies the quasi-isometry inequalities directly, using the definitions and properties of the spaces involved
• Using the concept of quasi-geodesics to show that a map is a quasi-isometric embedding, and then proving that the image is quasi-dense to establish a quasi-isometry
• Applying known quasi-isometries and their properties, such as the quasi-isometry between a group and its Cayley graph, to deduce new quasi-isometries
• Utilizing the connection between quasi-isometries and other geometric notions, such as Gromov hyperbolicity or the Gromov boundary, to indirectly establish quasi-isometry

Quasi-Isometric Invariants

• Growth rate of a metric space, which measures how the volume of balls grows as a function of their radius
  • Two quasi-isometric metric spaces have equivalent growth rates, up to a multiplicative constant
• The number of ends of a metric space, which captures the large-scale connectedness of the space
  • Quasi-isometries preserve the number of ends of a metric space
• The Dehn function of a finitely presented group, which quantifies the difficulty of filling loops in the group's Cayley complex
  • Quasi-isometric groups have equivalent Dehn functions, up to a multiplicative constant
• The asymptotic dimension of a metric space, a coarse analogue of topological dimension
  • Quasi-isometries preserve the asymptotic dimension of a metric space
• The Gromov boundary of a hyperbolic metric space, which encodes the structure of the space at infinity
  • Quasi-isometries between hyperbolic metric spaces induce homeomorphisms between their Gromov boundaries

Challenges and Open Problems

• Classifying finitely generated groups up to quasi-isometry, a major goal in geometric group theory
  • While some classes of groups, such as hyperbolic groups and abelian groups, have been well-studied, the quasi-isometric classification of many other classes remains open
• Understanding the quasi-isometric invariants of important classes of metric spaces, such as Riemannian manifolds, graphs, and buildings
• Developing new techniques for proving quasi-isometry between metric spaces, particularly in cases where explicit constructions are difficult or impossible
• Exploring the connections between quasi-isometries and other areas of mathematics, such as coarse geometry, topology, and analysis
• Investigating the role of quasi-isometries in the study of infinite-dimensional spaces, such as Banach spaces and Hilbert spaces, where the geometry is less well-understood than in the finite-dimensional case