Convergence groups

Convergence groups on compact metric spaces, from the definition by sequences: every element is elliptic, parabolic or loxodromic, loxodromic elements have attracting and repelling fixed points and a compact fundamental domain, and the stabilizer of a loxodromic fixed point is virtually infinite cyclic; convergence actions and minimal actions pass to equivariant quotients, a convergence group on a continuum acts as a convergence group on its dendrite quotient, the points of the completion of the cut points have supports in the continuum that control which of them a group element fixes, under conditions on the group the dendrite quotient is not a point, and when the group fixes a point of the dendrite quotient the cut points form the vertices of a tree on which the group acts. A group with a finite generating set and a one-ended Cayley graph has the fixed-point property for trees that these results assume.

Eight lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

  1. Convergence actions and their elements
  2. Quotients and minimal actions
  3. Supports and fixed points in the completion
  4. Loxodromic elements and the completion
  5. Trees and classes of finite intervals
  6. The dendrite quotient is not a point
  7. The tree of cut points
  8. The fixed-point property for groups with one end

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites