Finite combinatorial dimension of buildings

Prove that buildings have finite combinatorial dimension, thereby extending the fixed-point result from Euclidean and hyperbolic buildings to all locally finite buildings.

Background

The paper's main fixed-point theorem applies to locally finite graphs with stable intervals and finite combinatorial dimension. It consequently covers several families of Euclidean buildings, but not arbitrary locally finite buildings.

The authors identify finite combinatorial dimension for all buildings as a specific conjecture. They state that proving it would extend their result to all locally finite buildings, not merely the Euclidean or hyperbolic cases.

References

In particular, we pose the following conjecture. Establishing it would extend Corollary~\ref{cor:fixcor}(1) to all (not only Euclidean or hyperbolic) locally finite buildings.

— Locally elliptic actions, torsion groups, and nonpositively curved spaces  (2110.12431 - Haettel et al., 2021) in Section 1, Introduction, immediately after the question on finite combinatorial dimension

What is the combinatorial dimension of the standard Cayley graph of a Coxeter group?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, final list of questions on combinatorial dimension, item 3

What is the combinatorial dimension of a Euclidean buiding?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, final list of questions on combinatorial dimension, item 4