Alphabeta MathHall–Mal’cev Coordinates and Bass–Guivarc’h Growth: Examples
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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
- Hall–Mal’cev Coordinates and Bass–Guivarc’h Growth
- 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
Example
Free abelian groups have degree equal to rank
Example
The discrete Heisenberg group has growth degree four
Example
The Heisenberg center is quadratically distorted
Example
UT₄(Z) has ranks three, two, one and growth degree ten
Example
Hirsch length and growth degree differ
Counterexample
Weight-one counting misses Heisenberg growth
Counterexample