The finite-energy boundary of a group

Finite-energy functions on the Cayley complex of a finitely presented group, and the boundary they give a one-ended group: a function with connected level sides, a compact metric space containing the group with square summable edge lengths, a boundary that is a continuum, paths near boundary points, the action of the group on the boundary as a convergence group with dense orbits, the dimension and first Čech cohomology of the boundary, a continuum without cut points inside the boundary, and the proof that this continuum is locally connected. The result: under named hypotheses on the group, every finite-energy gradient is a limit of gradients of finitely supported functions. This is the part of OpenAI's proof (2026) of the Singer conjecture in dimension four outside the middle degree that takes place on the group.

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

1. Finite-energy functions on a Cayley complex
2. The compact space and its boundary
3. The convergence action
4. Dimension and cohomology of the boundary
5. A block of the boundary
6. A continuum without cut points and its graph
7. Component labels and annular depth
8. Vertices of large depth
9. The continuum is locally connected
10. The theorem and its hypotheses

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Dendrites and cut points (Continua, arcs and cut points)
• Every path contains an arc (Continua, arcs and cut points)
• First Čech cohomology of a compact metric space (Continua, arcs and cut points)
• Paths in locally connected spaces (Continua, arcs and cut points)
• Convergence actions and their elements (Convergence groups)
• The fixed-point property for groups with one end (Convergence groups)
• The tree of cut points (Convergence groups)
• Covering dimension and finite trivializing covers · Index theory of elliptic operators (Index theory of elliptic operators — Hermite and Schwartz tools)
• The dendrite quotient of a continuum (Pretrees)
• Hilbert spaces (core course Functional Analysis), Tonelli's theorem and dominated convergence (core course Measure and Integration), free groups and presentations; the course proves the other facts it uses.