Results on Thompson's group F, the Hilbert ball map and amenability, 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.

• Thompson's group F — Proposition 1.3. The increasing piecewise linear homeomorphisms of [0,1] with dyadic breakpoints and slopes in 2^Z form a countable group under composition; they map dyadic rationals to dyadic rationals (Lemma 1.2).
• Nesting of dyadic intervals — Lemma 2.2. Dyadic intervals with intersecting interiors are nested; a dyadic partition whose cells are no longer than a dyadic interval I respects I.
• Normalized restrictions — Lemma 2.4. I.J is a dyadic interval with s_(I.J) = s_I o s_J; T_I is a dyadic partition, with fewer cells than T if I is not [0,1]; if T respects I and I.J, then T_I respects J and (T_I)_J = T_(I.J).
• Images of uniform partitions — Lemma 3.1. For g in F, g T^(n) is a dyadic partition of mesh at most 2^(c-n) for all large n; finitely many elements and finitely many dyadic intervals admit a common level at which every image respects every interval.
• Moving pairs of intervals — Lemma 4.1. For internal dyadic intervals I < J and I' < J' some h in F maps I affinely onto I' and J affinely onto J'.
• Covariance — Lemma 4.2. If h o s_I = s_(I') and g T^(n), (hg) T^(n) are dyadic partitions respecting I and I', then ((hg) T^(n))_(I') = (g T^(n))_I.
• Curves in a real Hilbert space — Lemma 1.1. Taylor bound for a curve whose derivative is K-Lipschitz, the mean value inequality, and the derivative of the squared distance from a point to a curve.
• A separated curve with a bounded tail — Lemma 2.3. The curve gamma(t) = a(t) E(theta(t)) in L^2([0,1];R^2) is injective, and points whose parameters differ by at least d are at least sigma(d) > 0 apart, although gamma([1, infinity)) lies in the ball of radius 5/16.
• The tube around the curve — Lemma 3.1. In a tube of fixed radius every point has a unique nearest curve point; the nearest parameter is 16-Lipschitz, the residual is orthogonal to the tangent, and orthogonal decompositions determine the nearest point.
• Rotations carrying one unit vector to another — Lemma 4.1. For unit vectors u, v with <u,v> >= 0, R = 1 + A + A^2/(1+<u,v>) with Ah = <u,h>v - <v,h>u is orthogonal, Ru = v, and R depends Lipschitz-continuously on (u,v).
• A Lipschitz self-map of the Hilbert ball moving every point (after Benyamini-Sternfeld) — Theorem 5.1. On the closed unit ball B of L^2([0,1];R^2) there is a Lipschitz map f: B -> B with ||f(x)|| = 1 and ||f(x) - x|| >= 1/2 for every x.
• Finite translation — Lemma 1.2. For nonempty finite A, h in the group and phi with values in [-1,1], the averages over A of phi(h.) and of phi differ by at most |hA symmetric difference A|/|A|; equality can occur.
• The recursion along the group — Lemma 3.2. At an admissible level, the color of each parent interval is f applied to the average of the colors of its descendants.
• Transport of colors — Lemma 3.3. An element of F mapping a separated pair I < J affinely onto the first two parents carries the colors of I and J at g to the colors of those parents at hg.
• A uniform boundary bound — Proposition 4.1. For one explicit finite S in F, every nonempty finite A has max over h in S of |hA symmetric difference A|/|A| >= (delta^2/L^2 - 4/D)/(4(1 - 1/D)) > 0.
• Thompson's group F is not amenable (OpenAI) — Theorem 5.1. Thompson's group F has no left invariant mean on its bounded functions; equivalently it has no Folner sequence.

Reading order

Read the lessons in order. The first develops Thompson's group F through dyadic partitions: normalized restrictions, images of uniform partitions, moving pairs of separated intervals and covariance. The second constructs a Lipschitz map of the closed unit ball of a Hilbert space into its unit sphere that moves every point by at least 1/2. The third colors dyadic partitions with this map and proves that F is not amenable.