Distances between fixed-point sets in 2-dimensional Euclidean buildings are realised

TL;DR

This paper proves that in 2-dimensional Euclidean buildings, the distance between fixed-point sets of two finitely generated groups acting on the space is always realized, using CAT(0) geometry and ultrapower techniques.

Contribution

It establishes that fixed-point set distances are always realized in Euclidean buildings, combining geometric and ultrapower methods for the first time.

Findings

Distance between fixed-point sets is always realized.

Uses CAT(0) space properties and ultrapower techniques.

Applicable to finitely generated group actions.

Abstract

We prove that if two finitely generated groups act on a metrically complete 2-dimensional Euclidean building, then the distance between their fixed-point sets is realised. Our proof uses the geometry of Euclidean buildings, which we view as CAT(0) spaces, and properties of ultrapowers of Euclidean buildings.