Asymptotic-M\"obius maps

TL;DR

This paper introduces asymptotic-M"obius maps as a new large-scale conformality concept in geometric group theory, demonstrating their properties, examples, and applications to dimension and metric cotype.

Contribution

It defines asymptotic-M"obius maps, explores their properties, provides examples, and applies them to large-scale geometric problems, offering new insights and obstructions.

Findings

AM-maps are stable under quasi-isometries and compatible with scaling limits.

Dimension-monotonicity results for nilpotent groups and CAT(0) spaces.

New obstructions to the existence of AM-maps from metric cotype.

Abstract

We introduce asymptotic-M\"obius (AM) maps, a large-scale analogue of quasi-M\"obius 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.