Alphabeta MathAsymptotic Cones and the Sublinear Triangle Criterion
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1 · Prerequisites
Abstract Algebra
Group Theory
Real Analysis
Set Theory
2 · Logical flowchart
3 · Items
Definition
Rescaled ultralimits and asymptotic cones
Lemma
Free tail ultrafilters and bounded real ultralimit calculus
Lemma
The rescaled ultradistance defines a metric
Definition
Oriented geodesic rays, lines, parameters and tails
Lemma
Limits of geodesic segments, rays and lines
Definition
Real trees, tripod triangles, slimness and minsize
Lemma
Triangle extrema and the tripod and branch rules for real trees
Lemma
Tree cones force uniform control of sides with a common endpoint
Theorem
Tree cones at all basepoints and scales imply uniform slimness
Lemma
Sublinear minsize identifies every cone segment with a limit segment
Theorem
Sublinear triangle minsize implies hyperbolicity
Definition
Bounded-edge coarse fillings of loops and triangles
Definition
Singular planar labelled relator diagrams and their outer walks
Lemma
Finite polygonal disk parametrizations and boundary surgery
Lemma
Relator expressions admit singular planar diagrams with controlled incidence
Proposition
A recursive Dehn function yields a solution to the word problem
Lemma
Controlled coarse triangulation of singular planar diagrams
Lemma
Algebraic relator area controls coarse filling area
Lemma
Polygonal boundary crossing forces coverage by affine triangles
Lemma
Coarse triangle minsize is bounded by square root of area
Theorem
Linear algebraic relator area implies slim Cayley triangles
Lemma
Point wedges preserve common triangle minsize bounds
Lemma
Filling constants give a uniform slimness bound
Sources
- Drutu–Kapovich, Geometric Group Theory — §10.4 opening construction; §10.6 opening cone definition, PDF p.372
- Drutu–Kapovich, Geometric Group Theory — §10.1 Lemma 10.25 and real ultralimit calculus; local rational reciprocal supplement
- Drutu–Kapovich, Geometric Group Theory — §10.4 initial pseudometric and quotient construction
- Drutu–Kapovich, Geometric Group Theory — §1.2 interval geodesics; §10.4 limit geodesics
- Drutu–Kapovich, Geometric Group Theory — §10.4 Lemmas 10.48 and 10.51, PDF pp.366–367
- Drutu–Kapovich, Geometric Group Theory — §9.7.4 Definitions 9.101–9.102; §11.21 Definition 11.175
- Frigerio–Sisto, Characterizing hyperbolic spaces and real trees — §3 Lemma 11, PDF pp.7–8; local interval extrema and branch proofs
- Drutu–Kapovich, Geometric Group Theory — §11.20 Lemma 11.168(a), PDF p.443
- Drutu–Kapovich, Geometric Group Theory — §11.20 Proposition 11.167(a), full proof PDF pp.443–445
- Drutu–Kapovich, Geometric Group Theory — §11.21 Lemma 11.177, PDF p.449
- Drutu–Kapovich, Geometric Group Theory — §11.21 Proposition 11.176, reverse implication, PDF pp.448–449
- Drutu–Kapovich, Geometric Group Theory — §9.7.4 Definitions 9.101–9.102 and coarse filling conventions
- Bridson, The geometry of the word problem — §4.1 and §4.2 Definition 4.2.1, PDF pp.20–22
- Erickson, Simple Polygons — §1.2 complete separation proof; §1.4 Lemma 1.4/Theorem 1.5; §1.6 Theorem 1.10, PDF pp.4–9,13–14
- Bridson, The geometry of the word problem — §4.2 Lemma 4.2.3, Remark 4.2.5, Lemma 4.2.6 and final existence proof, PDF pp.22–24; local PL and occurrence supplement
- Erickson, Planar Graphs (2023) — Abstract graphs through Rotation systems; distinct germs and boundary successors
- John Meier, Groups, Graphs and Trees
- Dexter Chua after H. Wilton, Topics in Geometric Group Theory
- Drutu–Kapovich, Geometric Group Theory — §7.10.2 canonical enlargement after Definition 7.98, PDF pp.261–262; local compatible 4d/8/4k subdivision count
- Drutu–Kapovich, Geometric Group Theory — §9.7.4, Definition 9.101 and Proposition 9.103 (PDF pp. 349–351); explicit local conversion and count
- Drutu–Kapovich, Geometric Group Theory — §9.7.4, proof of Propositions 9.103–9.104, PDF pp. 350–352; the crossing and finite-area argument is supplied here
- Drutu–Kapovich, Geometric Group Theory — §9.7.4, Propositions 9.103–9.104, PDF pp. 350–352; corrected affine-edge barrier $h=m/2-r$
- Drutu–Kapovich, Geometric Group Theory — §9.7.4, Theorem 9.100 and Proposition 9.103; §11.20, Proposition 11.167(a)
- Frigerio–Sisto, Characterizing hyperbolic spaces and real trees — Lemma 11 (PDF pp. 7–8), sublinear criterion context; local wedge argument for uniformity
- Frigerio–Sisto, Characterizing hyperbolic spaces and real trees — Lemma 11 (PDF pp. 7–8); point-wedge contradiction supplied locally to make uniformity precise