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Asymptotic-Möbius maps

Georg Grützner

Abstract

We introduce asymptotic-Möbius (AM) maps, a large-scale analogue of quasi-Möbius maps tailored to geometric group theory. AM-maps capture coarse cross-ratio behavior for configurations of points that lie far apart, providing a notion of "conformality at infinity" that is stable under quasi-isometries, compatible with scaling limits, and rigid enough to yield structural consequences absent from Pansu's notion of large-scale conformality. We establish basic properties of AM-maps, give several sources of examples, including quasi-isometries, sublinear bi-Lipschitz equivalences, snowflaking, and Assouad embeddings, and apply the theory to large-scale dimension and metric cotype. As applications we obtain dimension-monotonicity results for nilpotent groups and CAT(0) spaces, and new obstructions to the existence of AM-maps arising from metric cotype.

Asymptotic-Möbius maps

Abstract

We introduce asymptotic-Möbius (AM) maps, a large-scale analogue of quasi-Möbius maps tailored to geometric group theory. AM-maps capture coarse cross-ratio behavior for configurations of points that lie far apart, providing a notion of "conformality at infinity" that is stable under quasi-isometries, compatible with scaling limits, and rigid enough to yield structural consequences absent from Pansu's notion of large-scale conformality. We establish basic properties of AM-maps, give several sources of examples, including quasi-isometries, sublinear bi-Lipschitz equivalences, snowflaking, and Assouad embeddings, and apply the theory to large-scale dimension and metric cotype. As applications we obtain dimension-monotonicity results for nilpotent groups and CAT(0) spaces, and new obstructions to the existence of AM-maps arising from metric cotype.
Paper Structure (22 sections, 21 theorems, 126 equations)

This paper contains 22 sections, 21 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Results
  3. AM-maps
  4. Extended metrics
  5. Quasimetrics
  6. Quasi-Möbius maps
  7. Asymptotic Möbius maps
  8. Asymptotic cones and tangent cones
  9. Nonprinciple ultrafilters and ultrapowers
  10. Asymptotic cones
  11. Tangent cones
  12. Möbius equivalence of asymptotic cones and tangent cones
  13. AM-Theorem
  14. Proof the AM Theorem
  15. Examples
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $(\Gamma,|\cdot|)$ and $(\Gamma',|\cdot|)$ be finitely generated nilpotent groups equipped with some word norm and $f: \Gamma \rightarrow \Gamma'$ a regular AM-map, then $\operatorname{asdim}(\Gamma) \leq \operatorname{asdim}(\Gamma')$. If $f$ is an AM-map between simply connected nilpotent Lie

Theorems & Definitions (129)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3: extended metric topology
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 119 more