Visual limits of maximal flats in symmetric spaces and Euclidean buildings

TL;DR

This paper investigates the geometric limits of maximal flats in symmetric spaces and Euclidean buildings, revealing new behaviors and restrictions in their degenerations, especially in low-rank cases.

Contribution

It reduces the problem of visual limits of flats to limits of apartments in the boundary building and classifies these degenerations in specific low-rank cases.

Findings

Complete classification of apartment degenerations in rank 1 and certain classical groups.

Identification of surprising algebraic restrictions on limits.

Illustration of remarkable behaviors in small-rank symmetric spaces.

Abstract

Let X be a symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let B denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building B: this defines a natural, geometric compactification of the space of maximal flats of X. We then completely determine the possible degenerations of apartments when X is of rank 1, associated to a classical group of rank 2 or to PGL(4). In particular, we exhibit remarkable behaviours of visual limits of maximal flats in various symmetric spaces of small rank and surprising algebraic restrictions that occur.