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Norm growth for the Busemann cocycle

Type of publication Peer-reviewed
Publikationsform Original article (peer-reviewed)
Author Dumont Thibaut,
Project Cohomology and Bruhat-Tits Buildings
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Original article (peer-reviewed)

Journal Bulletin of the Belgian Mathematical Society - Simon Stevin
Volume (Issue) 25(4)
Page(s) 507 - 526
Title of proceedings Bulletin of the Belgian Mathematical Society - Simon Stevin
DOI 10.36045/bbms/1546570906

Abstract

Using explicit methods, we provide an upper bound to the norm of the Busemann cocycle of a locally finite regular tree X, emphasizing the symmetries of the cocycle. The latter takes value into a submodule of square summable functions on the edges of X, which corresponds the Steinberg representation for rank one groups acting on their Bruhat-Tits tree. The norm of the Busemann cocycle is asymptotically linear with respect to square root of the distance between any two vertices. Independently, Gournay and Jolissaint proved an exact formula for harmonic 1-cocycles covering the present case.
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