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

Harris Leung, Jeroen Schillewaert and Anne Thomas

Vol. 20 (2023), No. 2-3, 341–351
Abstract

We prove that if two finitely generated groups act by isometries 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.

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Keywords
Euclidean buildings, fixed point sets, distance
Mathematical Subject Classification
Primary: 20E42
Milestones
Received: 23 October 2022
Revised: 26 June 2023
Accepted: 17 July 2023
Published: 13 September 2023
Authors
Harris Leung
University of Auckland
Auckland
New Zealand
Jeroen Schillewaert
University of Auckland
Auckland
New Zealand
Anne Thomas
School of Mathematics and Statistics
University of Sydney
Sydney
Australia