Groups, Geometry, and Dynamics

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Volume 5, Issue 1, 2011, pp. 177–188
DOI: 10.4171/GGD/121

Published online: 2011-01-19

(Non)-completeness of ℝ-buildings and fixed point theorems

Koen Struyve[1]

(1) Universiteit Gent, Belgium

We prove two generalizations of results of Bruhat and Tits involving metrical completeness and $\mathbb{R}$-buildings. Firstly, we give a generalization of the Bruhat–Tits fixed point theorem also valid for non-complete $\mathbb{R}$-buildings with the added condition that the group is finitely generated. Secondly, we generalize a criterion which reduces the problem of completeness to the wall trees of the $\mathbb{R}$-building. This criterion was proved by Bruhat and Tits for $\mathbb{R}$-buildings arising from root group data with valuation.

Keywords: Euclidean buildings, fixed point theorems, metric completeness

Struyve Koen: (Non)-completeness of ℝ-buildings and fixed point theorems. Groups Geom. Dyn. 5 (2011), 177-188. doi: 10.4171/GGD/121