Norm growth for the Busemann cocycle.

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 to the Steinberg representation for rank one groups acting on their Bruhat-Tits tree. The normof the Busemann cocycle is asymptotically linear with respect to square root of the distance between any two vertices. Independently, Gournay and Jolissaint [10] proved an exact formula for harmonic 1-cocycles covering the present case. [ABSTRACT FROM AUTHOR]