Coarse Equivalences of Euclidean Buildings

TL;DR

This paper establishes rigidity results for Euclidean buildings, showing that coarse equivalences preserve their structure at infinity and, under certain conditions, imply isometries that are close to the original mappings.

Contribution

It proves that coarse equivalences between Euclidean buildings preserve their spherical boundaries and, in specific cases, are essentially isometries close to the original maps.

Findings

Coarse equivalences preserve spherical buildings at infinity.

Euclidean buildings with irreducible factors of dimension ≥ 2 are isometric under coarse equivalence.

Unique isometries exist close to coarse equivalences when no Euclidean cone factors are present.

Abstract

We prove the following rigidity results. Coarse equivalences between Euclidean buildings preserve spherical buildings at infinity. If all irreducible factors have dimension at least two, then coarsely equivalent Euclidean buildings are isometric (up to scaling factors). If in addition none of the irreducible factors is a Euclidean cone, then the isometry is unique and has finite distance from the coarse equivalence.