Alphabeta MathAsymptotic Cones and the Sublinear Triangle Criterion: Examples
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1 · Prerequisites
Group Theory
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
Cones of a line and of real trees
Example
A Euclidean right triangle has minsize proportional to its scale
Example
Scaling distinguishes sublinear minsize from a fixed perimeter cutoff
Sources
- Drutu–Kapovich, Geometric Group Theory — §10.6 (cones) and §11.21, Proposition 11.176 (PDF pp. 448–449), with the explicit line calculation
- Drutu–Kapovich, Geometric Group Theory — §11.21, Definition 11.175 and Proposition 11.176, comparison with the Euclidean-plane example; explicit computation here
- Drutu–Kapovich, Geometric Group Theory — §11.20, Lemma 11.168(a), PDF pp. 443–445; explicit scaling calculation and independent Euclidean witness