⭕Geometric Group Theory Unit 7 – CAT(0) Spaces and Groups

Definition and Basic Properties

• CAT(0) spaces are metric spaces that satisfy certain curvature conditions, generalizing non-positive curvature from Riemannian geometry
• Formally, a metric space $(X,d)$ is CAT(0) if for any geodesic triangle $\Delta(x,y,z)$ in $X$ and a comparison triangle $\overline{\Delta}(\overline{x},\overline{y},\overline{z})$ in Euclidean space, the CAT(0) inequality holds: $d(p,q) \leq d_E(\overline{p},\overline{q})$ for any points $p,q$ on $\Delta$ and their comparison points $\overline{p},\overline{q}$ on $\overline{\Delta}$
  • Intuitively, triangles in CAT(0) spaces are "thinner" than their Euclidean counterparts
• CAT(0) spaces are uniquely geodesic, meaning any two points are connected by a unique geodesic segment
• Convexity is a key property of CAT(0) spaces
  • Metric balls are convex
  • The distance function is convex along geodesics
• CAT(0) spaces are contractible, implying they have trivial topology (homotopy equivalent to a point)
• Complete CAT(0) spaces are often called Hadamard spaces, honoring the work of Jacques Hadamard on non-positively curved manifolds

Metric Space Structure

• As metric spaces, CAT(0) spaces inherit various properties and tools from metric geometry
• The metric induces a topology, allowing for the study of continuity, convergence, and compactness
• Completeness is an important property, ensuring the existence of limits and fixed points
  • Many results assume the CAT(0) space is complete (Hadamard space)
• The metric also induces a length structure, enabling the study of paths, curves, and geodesics
• Angles can be defined using the cosine law, generalizing the Euclidean notion
  • Alexandrov angles compare the angle between geodesics to the angle in a Euclidean comparison triangle
• The Gromov product, defined as $(x|y)_z = \frac{1}{2}(d(x,z) + d(y,z) - d(x,y))$, measures the "overlap" between geodesics from $z$ to $x$ and $y$, playing a crucial role in boundary theory and hyperbolic geometry

Curvature Conditions

• CAT(0) spaces are defined by a curvature condition comparing triangles to their Euclidean counterparts
• The CAT(0) inequality $d(p,q) \leq d_E(\overline{p},\overline{q})$ captures the idea of non-positive curvature
  • Triangles in CAT(0) spaces are "thinner" than Euclidean triangles
• An equivalent condition is the Bruhat-Tits non-positive curvature inequality involving the midpoint of a geodesic segment
• CAT(0) is a global condition, while the related notion of non-positive curvature (NPC) is local
  • NPC spaces are locally CAT(0) but may not be globally CAT(0) (branching may occur)
• The Cartan-Hadamard theorem states that complete, simply connected Riemannian manifolds of non-positive sectional curvature are CAT(0)
  • Provides a rich source of examples (symmetric spaces, hyperbolic spaces)
• Alexandrov's Patchwork Globalization theorem allows constructing CAT(0) spaces by gluing together non-positively curved spaces satisfying certain conditions

Geodesics and Triangles

• Geodesics are locally distance-minimizing paths generalizing straight lines in Euclidean space
• In CAT(0) spaces, geodesics are globally distance-minimizing, unique, and vary continuously with endpoints
• The CAT(0) condition is often phrased in terms of geodesic triangles and their comparison triangles in Euclidean space
  • For any points $p,q$ on a geodesic triangle $\Delta(x,y,z)$, the CAT(0) inequality compares their distance to the distance between corresponding points $\overline{p},\overline{q}$ on the Euclidean comparison triangle $\overline{\Delta}(\overline{x},\overline{y},\overline{z})$
• Geodesic triangles in CAT(0) spaces satisfy a number of properties generalizing the Euclidean case
  • The angle sum is at most $\pi$
  • The Alexandrov angle between geodesics is well-defined and satisfies triangle inequalities
  • Trigonometric formulas (sine law, cosine law) hold for geodesic triangles
• Geodesic triangles are key to understanding the geometry and topology of CAT(0) spaces
  • Used to prove contractibility, fixed point theorems, and boundary theory

Isometries and Group Actions

• Isometries are distance-preserving maps between metric spaces, forming a group under composition
• The isometry group $\text{Isom}(X)$ of a CAT(0) space $X$ captures its symmetries and rigidity properties
  • Often infinite-dimensional but shares properties with Lie groups (proper, locally compact)
• Group actions on CAT(0) spaces, particularly isometric actions, are a central object of study
  • An action $G \curvearrowright X$ is a homomorphism from $G$ to $\text{Isom}(X)$
• Proper actions (cocompact or properly discontinuous) are well-behaved and allow for a geometric study of the group $G$
  • Cocompact actions have compact quotient space $X/G$
  • Properly discontinuous actions have discrete orbits and well-defined quotient
• The Cartan Fixed Point Theorem guarantees a fixed point for any bounded isometric action on a complete CAT(0) space
  • Generalizes the Brouwer Fixed Point Theorem and has applications to group theory and topology
• The Flat Torus Theorem classifies isometric actions on CAT(0) spaces with a Euclidean subspace invariant under a $\mathbb{Z}^n$ subgroup
  • Key tool in geometric group theory and rigidity theory

Boundary Theory

• The visual boundary $\partial X$ of a CAT(0) space $X$ consists of equivalence classes of geodesic rays, capturing the asymptotic geometry
  • Geodesic rays are equivalent if they stay within bounded distance of each other
• The cone topology on $\overline{X} = X \cup \partial X$ generalizes the topology of the closed unit ball
  • Makes $\overline{X}$ compact if $X$ is proper (closed balls are compact)
• The Gromov product extends to the boundary, allowing for a metric on $\partial X$
  • The Gromov product at infinity is defined as $(x|y)_\infty = \sup\limits_{z \in X} \liminf\limits_{x' \to x, y' \to y} (x'|y')_z$
• Isometric actions on $X$ extend to actions on the boundary $\partial X$
  • Fixed points on the boundary are often easier to find and correspond to "directions moved" by the isometry
• The Tits metric $d_T$ on the boundary, defined using Alexandrov angles, captures the geometry of directions
  • CAT(0) boundaries with the Tits metric are themselves CAT(1) spaces
• Flat subspaces in $X$ correspond to spherical subbuildings in the Tits boundary $(\partial X, d_T)$
  • Key to the Flat Torus Theorem and rigidity results

Examples and Applications

• Euclidean spaces $\mathbb{R}^n$ and hyperbolic spaces $\mathbb{H}^n$ are primary examples of CAT(0) spaces
• Symmetric spaces and Bruhat-Tits buildings associated with semisimple Lie groups and algebraic groups are important examples in Lie theory and number theory
  • The visual boundary of a symmetric space identifies with the spherical building at infinity
• CAT(0) cube complexes, obtained by gluing Euclidean cubes along faces, are key examples in geometric group theory
  • Hyperplanes and half-spaces allow for a combinatorial study
  • Arise in the study of right-angled Artin groups, Coxeter groups, and small cancellation theory
• Gromov hyperbolic spaces are CAT(0) spaces with additional "negative curvature" properties
  • Boundary is a quasi-metric space, often fractal in nature
  • Include classical hyperbolic spaces, hyperbolic groups, and random groups
• The Weil-Petersson metric on Teichmüller space is CAT(0), connecting to the geometry of moduli spaces of Riemann surfaces
• Phylogenetic tree space, used in evolutionary biology, has a CAT(0) metric related to the space of probability distributions

Advanced Topics and Open Problems

• Isometry groups of CAT(0) spaces are often infinite-dimensional, but share properties with Lie groups
  • The Hilbert-Smith conjecture asks if locally compact groups acting effectively on manifolds must be Lie groups
• Quasi-isometries are maps between metric spaces that preserve large-scale geometry
  • Gromov hyperbolic groups are classified up to quasi-isometry
  • Quasi-isometric rigidity results aim to classify groups and spaces up to quasi-isometry
• The Rank Rigidity Theorem characterizes irreducible CAT(0) spaces of higher rank
  • Euclidean buildings or symmetric spaces associated with higher-rank semisimple groups
• Mostow's Strong Rigidity Theorem states that homotopy equivalent compact hyperbolic manifolds are isometric
  • Generalizations to CAT(0) spaces and buildings using the geometry of the boundary
• The Tits Alternative states that finitely generated linear groups either contain a non-abelian free subgroup or are virtually solvable
  • Versions exist for isometry groups of CAT(0) spaces and CAT(0) groups
• Boundaries of CAT(0) groups are often difficult to compute explicitly
  • Conjectured to be non-equivariantly homeomorphic to the Menger curve in many cases
• The existence of a cocompact CAT(0) space is conjectured to be equivalent to various analytic and geometric properties of groups (Novikov conjecture, Baum-Connes conjecture, Haagerup property)