7.2 Geometric and algebraic properties of CAT(0) groups

CAT(0) groups are a key class in geometric group theory, acting geometrically on spaces with non-positive curvature. They exhibit fascinating properties like biautomaticity, solvable word problems, and finite asymptotic dimension. These traits make them crucial for studying group geometry and combinatorics.

The Flat Torus Theorem and visual boundary are essential tools for understanding CAT(0) groups. Fixed point theorems reveal deep connections to amenable groups and solvable subgroups. These properties have far-reaching implications for rigidity phenomena and the structure of lattices in Lie groups.

Groups Acting on CAT(0) Spaces

Geometric Actions and CAT(0) Groups

• CAT(0) groups form a significant class of groups in geometric group theory acting geometrically on CAT(0) spaces
• Geometric group action on a metric space requires proper and cocompact properties
  • Proper action ensures finite set of group elements g for compact subset K where gK intersects K
• CAT(0) spaces exhibit geodesic metric spaces with global non-positive curvature condition
  • Triangles in CAT(0) spaces appear "thinner" than Euclidean counterparts
• Švarc-Milnor lemma establishes quasi-isometry between the group and the CAT(0) space it acts on
• Examples of CAT(0) groups include fundamental groups of non-positively curved manifolds, Coxeter groups, and right-angled Artin groups

Flat Torus Theorem and Visual Boundary

• Flat Torus Theorem states for group G acting geometrically on CAT(0) space X, free abelian subgroup of G of rank n acts cocompactly on n-dimensional flat in X
• Visual boundary of CAT(0) space defined as equivalence classes of geodesic rays
  • Plays crucial role in understanding asymptotic geometry of CAT(0) groups
• Center of CAT(0) group acts trivially on visual boundary of CAT(0) space

CAT(0) Groups and Biautomatic Structures

Biautomatic Structures and Language Theory

• Biautomatic structures serve as formal language theoretic tools for studying group geometry and combinatorics
• Groups classified as biautomatic admit regular language of normal forms with fellow-traveler properties in forward and backward directions
• CAT(0) groups possess biautomatic structure, connecting geometric and language-theoretic properties
• Biautomaticity of CAT(0) groups implies:
  • Solvable word and conjugacy problems
  • Geodesics form a regular language
• Language of geodesics in CAT(0) groups satisfies fellow-traveler property, key component of biautomatic structures

Combings and Applications

• Combings represent systems of paths in Cayley graph of a group
  • Play crucial role in establishing biautomaticity of CAT(0) groups
• Connection between CAT(0) groups and biautomatic structures enables application of formal language theory techniques to study geometric properties
• Examples of applications:
  • Analyzing growth rates of groups
  • Studying subgroup distortion

Fixed Points of Group Actions

Fixed Point Theorems

• Fixed Point Theorem for CAT(0) spaces states any group action by isometries on complete CAT(0) space with bounded orbit has fixed point
• Amenable groups acting by isometries on complete CAT(0) space always have fixed point or stabilize flat in space
• Bruhat-Tits Fixed Point Theorem generalizes CAT(0) Fixed Point Theorem to actions on affine buildings
  • Affine buildings represent important examples of CAT(0) spaces (Bruhat-Tits buildings)

Isometry Classification and Solvable Subgroup Theorem

• Semisimple isometries of CAT(0) spaces classified into:
  • Elliptic (with fixed point)
  • Hyperbolic (with axis)
  • Analogous to classification of isometries in hyperbolic geometry
• Solvable Subgroup Theorem states any solvable subgroup of group acting properly by semisimple isometries on CAT(0) space virtually abelian
• Fixed point properties of group actions on CAT(0) spaces apply to:
  • Study of rigidity phenomena (Mostow rigidity)
  • Structure of lattices in Lie groups (Margulis superrigidity)

Algebraic Consequences of CAT(0) Condition

Finiteness Properties and Dimension

• CAT(0) groups possess finite asymptotic dimension
  • Implies various analytic and geometric properties (Novikov conjecture)
• CAT(0) groups classified as semihyperbolic
  • Results in quadratic Dehn function and finite presentation
• CAT(0) groups satisfy cohomological finiteness properties
  • Classified as type F∞, admitting classifying spaces with finite skeleta in all dimensions

Algebraic Structure and Decision Problems

• Centralizer of infinite order element in CAT(0) group always non-trivial
• CAT(0) groups satisfy Tits Alternative:
  • Any subgroup either contains free group on two generators or virtually abelian
• Word problem and conjugacy problem solvable in CAT(0) groups
  • Consequence of biautomaticity
• Flat Closing Conjecture, if proven true, would imply CAT(0) groups virtually torsion-free
  • Represents major open problem in field