Combing Euclidean buildings

TL;DR

This paper introduces a new combing for Euclidean buildings that satisfies the fellow traveller property and proves that groups acting on these buildings have biautomatic structures, advancing understanding of their geometric group theory.

Contribution

It defines a specific combing for Euclidean buildings and demonstrates that certain groups acting on these buildings are biautomatic, a novel result in geometric group theory.

Findings

Defined a combing satisfying the fellow traveller property

Proved groups acting on Euclidean buildings are biautomatic

Applicable to buildings of types A_n, B_n, C_n

Abstract

For an arbitrary Euclidean building we define a certain combing, which satisfies the `fellow traveller property' and admits a recursive definition. Using this combing we prove that any group acting freely, cocompactly and by order preserving automorphisms on a Euclidean building of one of the types A_n,B_n,C_n admits a biautomatic structure.