On the displacement function of isometries of Euclidean buildings

TL;DR

This paper investigates the displacement function of isometries in Euclidean buildings, providing bounds based on the distance to the minimal set, which enhances understanding of geometric group actions.

Contribution

It introduces a lower bound for the displacement function of isometries in Euclidean buildings related to the minimal set, a novel geometric insight.

Findings

Established a lower bound for displacement functions

Connected displacement to minimal set distance

Improved understanding of isometry dynamics in Euclidean buildings

Abstract

In this note we study the displacement function $d_g(x):=d(gx,x)$ of an isometry $g$ of a Euclidean building. We give a lower bound for $d_g(x)$ depending on the distance from $x$ to the minimal set of $g$.