Results on convergence groups, 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.

• Convergence action — Definition 1.1. Every sequence of pairwise distinct elements has a subsequence that maps each closed set avoiding one point into any neighbourhood of another point, for almost all terms.
• Collapsing sequences — Lemma 1.2. Inverses collapse with the exchanged pair; the pair is unique on a space with at least three points; left and right translates collapse with the translated pairs.
• Elliptic, parabolic, loxodromic — Theorem 2.1. In a convergence group on a compact metric space with at least three points, the fixed points of an element of infinite order are the one or two points of any collapsing sequence of its powers.
• Parabolic dynamics — Proposition 2.3. All powers of a parabolic element with exponents tending to infinity in absolute value collapse to its fixed point.
• Loxodromic dynamics — Proposition 3.1. The positive powers of a loxodromic element collapse from its repelling to its attracting fixed point, without passing to a subsequence.
• Fundamental domain — Proposition 3.2. A power of a loxodromic element has a compact set whose translates cover the complement of the two fixed points.
• Finite index — Proposition 3.3. The cyclic group of a loxodromic element has finite index in the group of elements fixing both of its fixed points.
• One fixed point gives both — Proposition 3.4. An element that fixes one fixed point of a loxodromic element fixes the other.
• Point stabilizers — Corollary 3.6. The stabilizer of a point is virtually infinite cyclic and contains a loxodromic element, or consists of elliptic and parabolic elements.
• Quotients of convergence actions — Proposition 1.1. A continuous surjective equivariant map between compact metric spaces carries collapsing sequences to collapsing sequences; so the image of a convergence action is a convergence action.
• Fixed points in a quotient — Corollary 1.2. For an element of infinite order, the fixed points in the image are the images of its fixed points.
• Minimal actions — Lemma 2.1. An action is minimal exactly when all orbits are dense; minimality passes to equivariant images; a minimal action on more than one point has no fixed point.
• The action on the dendrite quotient — Theorem 4.1. A group of homeomorphisms of a compact connected metric space acts on the completion of an invariant set of cut points and on the dendrite quotient; a convergence action gives a convergence action on the dendrite quotient.
• Supports — Proposition 1.2. The support in the space of a point of the completion that is not a cut point consists of the points mapped to it and of the cut points adjacent to it.
• Supports meet in at most one point — Proposition 1.3. The supports of two different points of the completion are disjoint or meet in one cut point, which is one of the two points or adjacent to both.
• No fixed point — Theorem 3.1. A minimal action on a compact connected metric space fixes no point of the completion of a nonempty invariant set of cut points.
• Parabolic elements in the completion — Theorem 4.1. A parabolic element of a convergence group whose fixed point is not one of the chosen cut points fixes exactly one point of the completion, which is not a cut point.
• Loxodromic elements in the completion — Theorem 1.2. The points of the completion of an invariant set of cut points fixed by a loxodromic element form at most three points next to each other, or exactly two points without neighbours that lie between no two points and whose supports are the two fixed points of the element.
• Fixed-point property for trees — Definition 1.1. A group has the fixed-point property for trees with finite edge stabilizers if each of its actions on a tree without inversions and with finite edge stabilizers fixes a point.
• Contracting edges of a tree — Proposition 2.1. Identifying the ends of the edges in a given set turns a tree into a tree whose edges are the remaining edges; an action that respects the set passes to the quotient with the same edge stabilizers.
• A bipartite tree — Proposition 2.2. A group with the fixed-point property that acts on a tree with an invariant bipartition, such that at each point of one part at most one edge has an infinite stabilizer, fixes a point.
• No invariant class of finite intervals — Theorem 3.1. For a minimal convergence action of a group with the fixed-point property, and an invariant set of cut points whose stabilizers are finite or contain loxodromic elements, no class of the finite interval relation of the completion is invariant.
• No fixed point in the dendrite quotient — Theorem 1.1. A finitely generated convergence group with the fixed-point property for trees, in which infinite subgroups contain elements of infinite order, acting minimally on a compact connected metric space with a nonempty invariant set of cut points whose stabilizers are finite or contain loxodromic elements, fixes no point of the dendrite quotient; so the quotient has more than one point.
• A cut point that is not a parabolic fixed point — Corollary 1.2. Under the same assumptions on the group, if the space has a cut point whose stabilizer is finite or contains a loxodromic element, its dendrite quotient has more than one point.
• Entry points — Lemma 2.2. For two disjoint full subsets of a complete median pretree whose union is full, one of them has a point through which every interval from the other set enters it.
• The first invariant class — Theorem 1.3. A finitely generated convergence group with the fixed-point property for trees, in which infinite subgroups contain elements of infinite order, that fixes a point of the dendrite quotient formed with an invariant set of cut points, fixes a point of the completion of that set or preserves a class of its finite interval relation.
• Two fixed points far apart — Lemma 3.2. An element of infinite order of a convergence group that fixes two points of the completion with an infinite interval between them is loxodromic; the two points are its only fixed points and lie between no two points.
• The tree of cut points — Theorem 1.1. For a minimal convergence action on a compact connected metric space with cut points, of a finitely generated group with the fixed-point property for trees and elements of infinite order in infinite subgroups, that fixes a point of the dendrite quotient: every cut point is fixed by a parabolic element, and the cut points are vertices of a tree with an action without inversions, finitely many orbits, no proper invariant subtree and infinite edge stabilizers; a cut point separating two others lies on the path between them.
• Stabilizers of cut points — Proposition 2.1. Under the same assumptions the stabilizer of every cut point is infinite, contains no loxodromic element and contains a parabolic element.
• The centre of a bounded set — Proposition 2.1. A nonempty bounded set of points of a tree has a centre, one point or two adjacent points, defined by the set and the distance alone.
• Bounded orbits — Corollary 2.2. A group acting on a tree without inversions with a bounded orbit fixes a point.
• The fixed-point property for groups with one end — Theorem 5.1. A group with a finite generating set whose Cayley graph is one-ended fixes a point in each of its actions on a tree without inversions and with finite edge stabilizers.

Reading order

Eight lessons. The eighth can be read after the fifth.