Finite Displacement Colors and Virtual Nilpotence
Abstract
An isometric group action with finitely many displacement colors has a virtually nilpotent small-displacement generated subgroup when equal colors differ by a nilpotent subgroup.
Theorem 1.1 (A general isometric-action bridge).
Proof. Machine-checked in Lean as D5/S3/Combinatorics/GroupActions/FiniteColorVirtualNilpotence.result (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. FrenzyMath and upstream contributors (2026). Poincare-Conjecture library prerequisites for hyperbolic rigidity. URL: https://github.com/frenzymath/Poincare-Conjecture/tree/432c38f2aa5a30efb13871292d17b4a3309a496a.
Commentary.
The universes u and v are independent. G has its chosen group structure, X its chosen pseudometric, and the same chosen action of G on X is isometric for every element. Fix p in X and a nilpotent subgroup H of G. Let m be any natural number, B the subtype of all g in G with dist(p,g acting on p) at most one, and c a function from B to Fin(m). The predicate colorDiff(c,H) means that for every v,w in B with c(v)=c(w), the original group element v inverse times w belongs to H. The predicate nilpotent(H) concerns H with its induced group structure. All distances use the same given pseudometric.
The symbol Gamma(G,X,p,m) denotes the subgroup generated by all g whose displacement at p is strictly less than one divided by m plus one, where the natural number m is cast to the reals before adding one. VirtNil(Gamma) means that Gamma has a subgroup that is nilpotent and has finite index. Neither H nor that subgroup is required to be normal. The generator set may be infinite. Finite index, finiteness of the coset space, and virtual nilpotence of Gamma are conclusions, not hypotheses.
In an infinite transitive action, every finite set containing the base point has an escape by some generator from a symmetric generating set. Otherwise closure induction and transitivity would force that finite set to contain every point. Inductively, one obtains n plus one distinct points reached by generator words of length at most n for every n. Therefore a collision among every m plus one words of length at most m implies a finite action space.
Set epsilon to one divided by m plus one and Gamma to the small-displacement generated subgroup. Isometry and the triangle inequality bound a product’s displacement by the sum of its letters’ displacements; the inverse has the same displacement. Thus each Gamma word of length at most m lies in B. Pigeonhole on its m colors makes two of m plus one words differ by H. For K equal to H viewed as a subgroup of Gamma, those words act equally on the identity coset of Gamma modulo K. The transitive coset action does not need K to be normal. The preceding growth argument makes this coset space finite, proving K has finite index.
Swapping the two nested subgroup memberships identifies K with Gamma viewed as a subgroup of H. A subgroup of the given nilpotent group H is nilpotent, so the actual multiplicative equivalence transfers nilpotence to K. This completes the general implication. It does not prove that any particular hyperbolic deck representation supplies the required coloring and subgroup, or establish Mostow-Prasad rigidity. Exact original H3 integration is a separate mathematical obligation.
References
- Truth anchor:
D5/S3/Combinatorics/GroupActions/FiniteColorVirtualNilpotence.result