Results on finite-energy functions on Cayley complexes, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Cocycles are gradients — Lemma 1.1. On the Cayley complex of a presentation, every 1-cochain with values in an abelian group whose integral around every cell vanishes is the gradient of a function, unique up to a constant.
• The space of finite-energy gradients — Lemma 2.1. The gradients of finite-energy functions form a closed subspace of the square-summable functions on the edges.
• Constant at infinity — Lemma 2.3. A finite-energy function that tends to a constant at infinity has its gradient in the closure of the gradients of finitely supported functions.
• Interaction of two cuts — Lemma 3.1. For two finite-energy functions, the number of pairs of equal or cell-sharing edges in their cuts at levels s and t has integral at most D times the product of the gradient norms; so it is finite for almost every pair of levels.
• A function with connected level sides — Theorem 4.1. If the finite-energy gradients are not all limits of gradients of finitely supported functions, there is a finite-energy function with values in [0,1], not tending to a constant at infinity, whose level sides at regular levels are connected and whose level cuts are dually connected.
• Good boxes exist — Lemma 2.1. Every neighbourhood of a point of the base cube contains a box around the point whose endpoint levels are regular and whose endpoint cuts interact in finitely many pairs.
• Finitely many components — Proposition 2.2. The vertex set of a good box has finitely many connected components.
• Localized functions — Lemma 3.1. The margin of a box times the indicator of one of its components has finite energy, and its cuts lie in shifted cuts of the functions defining the box.
• The compact space — Proposition 4.1. The closure of the profiles is a compact metrizable space in which the group is a dense set of isolated points and on which it acts by homeomorphisms.
• Square summable edge lengths — Proposition 4.2. In the weighted metric the lengths of the edges of the Cayley graph are square summable.
• The boundary is a continuum — Theorem 5.1. For a one-ended Cayley graph with finite-energy gradients beyond the closure of the finitely supported ones, the boundary is compact, connected and has more than one point.
• Paths near a boundary point — Theorem 6.2. Every neighbourhood of a boundary point contains a smaller one any two of whose vertices are joined by a graph path inside the larger.
• Virtually cyclic groups are not one-ended — Lemma 1.2. A finitely generated group with an infinite cyclic subgroup of finite index has a Cayley graph with at least two infinite components after removing a finite set.
• Translated cut lengths — Lemma 2.2. The integral over all levels of the total translated length near a level cut of a finite-energy function is finite and tends to zero as the translating element leaves finite sets.
• Collapse of translated cuts — Proposition 2.3. Along a subsequence of group elements converging to a boundary point, the translated ends of almost every level cut of finitely many coordinates converge uniformly to that point.
• Convergence — Theorem 3.1. If group elements converge to a boundary point a and their inverses to b, they map every closed set avoiding b into any neighbourhood of a, for almost all terms.
• Convergence group — Corollary 3.2. The group acts on the boundary as a convergence group, and the boundary is infinite.
• Parabolic comparison — Theorem 4.2. For a parabolic fixed point p, the distance between g and gp tends to zero as g leaves finite sets.
• Dense orbits — Theorem 5.2. If the group is torsion-free, every orbit in the boundary is dense.
• Thin walls — Lemma 1.1. For almost every level, the set of boundary points at which a level cut accumulates is covered by finitely many sets of arbitrarily small total diameter.
• Dimension of the boundary — Theorem 2.3. The finite-energy boundary of a one-ended group has covering dimension at most one.
• First Čech cohomology of the boundary — Theorem 3.2. If every 1-cochain on the Cayley complex with finitely supported coboundary can be corrected by a finitely supported cochain, the first Čech cohomology of the boundary vanishes.
• No circles; cut points — Corollary 3.3. Under these assumptions the boundary contains no simple closed curve, and it is locally connected only if it is a dendrite with cut points.
• A graph on the cut points — Proposition 2.1. On the cut points of the boundary there is a connected graph with finitely many orbits of edges whose edges join cut points with a common neighbour in the tree of cut points; it induces a connected graph on the cut points next to a fixed vertex of the tree.
• Paths of cut points near a point — Proposition 3.1. Two sequences of cut points converging to the same boundary point are joined by paths of cut points in the graph that converge to that point.
• A block of the boundary — Theorem 4.1. The closure of the set of cut points next to a vertex of the tree that is not a cut point is a retract of the boundary by a map that is locally constant outside it; it is a compact connected space with more than one point, without cut points, of dimension at most one, with vanishing first Čech cohomology when that of the boundary vanishes.
• A continuum without cut points and its graph — Theorem 2.1. For the boundary without cut points, and for a block of a boundary with cut points: a compact connected space without cut points of dimension at most one, a subgroup acting on it with the convergence property, and a connected graph with finitely many orbits whose vertices accumulate on the space, with paths near each point of the space.
• Component labels — Theorem 2.1. After deleting a set of vertices of the graph, each point of the continuum at which deleted vertices do not accumulate has a component of the remaining graph that contains all remaining vertices near it; this label is locally constant.
• A system of annuli — Theorem 3.2. There is an invariant set of pairs of disjoint closed sets in which only finitely many members have two sides of diameter above a given bound, and which contains chains of every length from a closed set to a neighbourhood of any point of the continuum outside it.
• Annular depth — Theorem 4.1. The largest length of a chain of annuli from a set to a point is invariant under the group, finite between group elements, and at least any given number near every point of the continuum outside a closed set.
• Translates that shrink to a point — Lemma 2.1. If the depths of group elements measured from a closed set tend to infinity, then along a subsequence the translates of the set by their inverses lie in arbitrarily small balls around one point of the continuum.
• A comparison of depths — Lemma 3.1. The depth from such a translate to a group element away from the limit point is at least the depth to the identity plus the depth from the identity to the element, minus a constant that depends only on the distance from the limit point.
• Deleted vertices after a translation — Lemma 5.1. In a compact set away from the limit point, the vertices whose translates are deleted at the next depth lie in one finite set for all large indices.
• Labels after a translation — Theorem 6.1. After translation by an element of large depth, any two points of the continuum other than the limit point have the same component label at the next depth, for all large indices.
• Majority labels — Lemma 2.1. Beyond one depth, for every group element of that depth two of the three points of the translated three-point set have the same component label at the next depth, and this majority label does not change under right multiplication by an element of a given finite set.
• One threshold for all larger depths — Theorem 4.1. Beyond one depth, two points of the continuum outside the closed set with equal component labels at a depth have equal labels at the next depth.
• Limit points of chains — Lemma 6.1. In a compact metric space, if finite chains of points have steps tending to zero and their first and last points converge, the set of their limit points is closed and connected and contains both limits.
• The continuum is locally connected — Theorem 7.1. A compact connected space without cut points with a group, a graph and the properties (S1) to (S7) is locally connected.
• The first Čech cohomology of the boundary does not vanish — Corollary 7.3. In both cases of the lesson on a continuum without cut points and its graph, the first Čech cohomology of the boundary with coefficients in the group with two elements is not zero.
• Finite-energy gradients are limits — Theorem 1.1. For a triangular presentation with a one-ended Cayley graph, without elements of finite order, with the cochain condition for the group with two elements, and such that the group fixes a point of the dendrite quotient of the boundary whenever the boundary has cut points, every finite-energy gradient is a limit of gradients of finitely supported functions.

Reading order

Read the lessons in order; the third uses the course on convergence groups, the fourth the course on continua, arcs and cut points, the fifth the lesson on the tree of cut points. The tenth lesson states the result of the course as one theorem and lists its hypotheses.