Alphabeta MathHall–Mal’cev Coordinates and Bass–Guivarc’h Growth
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✓ 18 results · all verified · 18 also independently AI-judged
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1 · Prerequisites
- Binary Operations, Monoids, Groups and Subgroups
- Cayley Graphs, Word Metrics and Quasi-Isometry
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Free Groups and Presentations
- Graphs, Walks and Connectivity
- Group Homomorphisms and the Isomorphism Theorems
- Linear Independence, Bases and Dimension
- Metric Spaces
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Roots, Rational Powers, and Classical Inequalities
- Suprema and Infima
- The ZFC Axioms and the Basic Set Constructions
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Lemma
Integer abelian structure and rank by finite reduction
Lemma
Commutator product identities in the fixed convention
Lemma
The three-subgroup containment for normal subgroups
Lemma
Lower-central commutators add weights
Lemma
Finite generation of lower-central factors
Lemma
Subgroups of finitely generated nilpotent groups are finitely generated
Lemma
Finite torsion and the torsion-free quotient
Lemma
Upper-central factors of a torsion-free nilpotent group
Lemma
Integral coordinates from a central cyclic refinement
Definition
Bass–Guivarc’h dimension and nilpotent Hirsch length
Definition
Lower-central generators, residue coordinates and weighted length
Lemma
Finite normal quotients preserve lower-central ranks
Lemma
Finite normal quotients preserve ball growth
Lemma
Finite lower-central coordinate systems with torsion accounted for
Lemma
Finite collection alphabets include commutators and torsion carries
Lemma
Weighted collection with finite-order carries
Lemma
Power compression in the last lower-central term
Theorem
Both bounds for last-term weighted distortion
Lemma
Coordinate boxes and word balls have matching size
Theorem