10.3 Group actions on the boundary

Group actions on the boundary are a key concept in geometric group theory. They extend a hyperbolic group's action on its Cayley graph to the Gromov boundary, revealing crucial geometric and dynamical properties. This powerful tool connects a group's algebraic structure to its large-scale geometry.

The boundary, equipped with a visual metric, becomes a compact metrizable space. The group's action on this space is often minimal and proximal, with fixed points corresponding to hyperbolic element endpoints. Understanding these dynamics is essential for exploring the group's geometric and topological properties.

Hyperbolic group action on boundary

Gromov boundary and group action

Top images from around the web for Gromov boundary and group action

• 

  Cayley graph - Wikipedia View original

  Is this image relevant?

• 

  Cayley graph - Wikipedia View original

  Is this image relevant?

• 

  CayleyNestGraph | Wolfram Function Repository View original

  Is this image relevant?

• 

  Cayley graph - Wikipedia View original

  Is this image relevant?

• 

  Cayley graph - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Gromov boundary and group action

• 

  Cayley graph - Wikipedia View original

  Is this image relevant?

• 

  Cayley graph - Wikipedia View original

  Is this image relevant?

• 

  CayleyNestGraph | Wolfram Function Repository View original

  Is this image relevant?

• 

  Cayley graph - Wikipedia View original

  Is this image relevant?

• 

  Cayley graph - Wikipedia View original

  Is this image relevant?

1 of 3

• Gromov boundary of hyperbolic group comprises equivalence classes of geodesic rays in Cayley graph
• Group G acts on boundary ∂G through left multiplication on geodesic rays
• Action extends natural action of G on Cayley graph to boundary
• Continuous action preserves visual metric on boundary up to bounded multiplicative factor
• Crucial for studying geometric and dynamical properties of hyperbolic groups
• Key tool relates algebraic structure of G to its large-scale geometry
• Examples of hyperbolic groups with interesting boundary actions include:
  • Free groups (boundary is Cantor set)
  • Surface groups (boundary is circle)

Topology and metrics on the boundary

• Gromov boundary equipped with visual metric derived from hyperbolic geometry
• Visual metric measures "angle" between geodesic rays
• Topology on boundary induced by visual metric
• Compact metrizable space under this topology
• Homeomorphism type of boundary invariant under quasi-isometries
• Examples of boundary topologies:
  • Sphere at infinity for hyperbolic space
  • Cantor set for free groups

Minimality and proximality of action

Minimality of the action

• Every orbit dense in boundary demonstrates minimality
• Proof utilizes asymptotic behavior of geodesic rays
• Any two geodesic rays asymptotic to rays starting at identity
• Consequences for ergodic theory of hyperbolic groups
• Examples of minimal actions:
  • Action of $SL(2,\mathbb{Z})$ on circle
  • Action of surface group on its boundary circle

Proximality of the action

• Any two boundary points brought arbitrarily close by some group element defines proximality
• Proof employs contraction of neighborhoods by sequence of group elements
• North-south dynamics of hyperbolic elements crucial for establishing proximality
• Essential for characterizing topological dynamics of boundary action
• Examples of proximal actions:
  • Action of $SL(2,\mathbb{R})$ on real projective line
  • Action of Fuchsian group on circle

Fixed points of hyperbolic actions

Classification of fixed points

• Fixed points on boundary correspond to endpoints of hyperbolic element axes or parabolic fixed points
• Hyperbolic elements have two fixed points: attracting and repelling
• Elliptic elements (finite order) lack fixed points on boundary
• Parabolic elements, if present, have unique fixed point
• Set of all hyperbolic element fixed points dense in Gromov boundary
• Related to Nielsen-Thurston classification of surface homeomorphisms
• Examples of fixed point structures:
  • Fuchsian groups acting on circle (hyperbolic fixed points dense)
  • Kleinian groups acting on sphere (limit set contains fixed points)

Dynamics of individual elements

• Hyperbolic elements exhibit north-south dynamics on boundary
• Parabolic elements have horocyclic orbits converging to fixed point
• Elliptic elements rotate points around fixed point in group
• Understanding individual element dynamics crucial for global action properties
• Examples of element dynamics:
  • Hyperbolic isometries of hyperbolic plane
  • Parabolic elements in cusped hyperbolic manifolds

Dynamics of group actions on boundary

Limit sets and conical limit points

• Limit set comprises accumulation points of any orbit in boundary
• Non-elementary hyperbolic groups have entire Gromov boundary as limit set
• Conical limit points approached "radially" by orbits in group
• Conical limit set full measure for Patterson-Sullivan measure
• Examples of limit sets:
  • Cantor set for Schottky groups
  • Entire circle for cocompact Fuchsian groups

Measures and ergodic theory

• Action of hyperbolic group on boundary amenable
• Patterson-Sullivan measure on boundary quasi-invariant under group action
• Crucial for studying dynamics and ergodic properties
• Applications in rigidity theory and random walks on hyperbolic groups
• Examples of invariant measures:
  • Lebesgue measure for Fuchsian groups acting on circle
  • Patterson-Sullivan measure for general hyperbolic groups