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Descent in Buildings (AM-190)$
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Bernhard Mühlherr, Holger P. Petersson, and Richard M. Weiss

Print publication date: 2015

Print ISBN-13: 9780691166902

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691166902.001.0001

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Fixed Apartments

Fixed Apartments

Chapter:
(p.217) Chapter Twenty Five Fixed Apartments
Source:
Descent in Buildings (AM-190)
Author(s):

Bernhard M¨uhlherr

Holger P. Petersson

Richard M. Weiss

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691166902.003.0025

This chapter considers the conditions which force a group of order 2 acting on a building to stabilize various kinds of substructures. It shows that a group of order 2 acting on a spherical building must fix an apartment. The discussion begins with the assumption that (Δ‎, δ‎) is a building of type (W, S) and that τ‎ is an automorphism of Δ‎ of order 2. This is followed by the proposition that there exists a spherical residue R stabilized by Γ‎ such that dist(x, xτ‎) = diam(R) for all xR. The chapter also analyzes the case where either τ‎ stabilizes two opposite residues of Σ‎ or τ‎ maps every chamber of Σ‎ to its opposite in Σ‎.

Keywords:   building, substructure, spherical building, apartment, automorphism, residue, chamber

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