Existence of quasi-periodic triangular buildings (Q537867)

This paper, which is closely related to [the authors, Geom. Dedicata 130, 71--91 (2007; Zbl 1142.51011)], studies the topological space of euclidean buildings of type \(\tilde A_2\). More precisely, the space considered is the space \(\Lambda_q\) of all pointed buildings of type \(\tilde A_2\) and thickness \(q\) (up to isomorphism); two buildings being close if they share a large ball centered at the root. This space has a foliated structure: a leaf is the set of all isomorphic buildings (forgetting the root). In this paper, the authors are interested in \textit{quasi-periodic} buildings, i.e. such that every ball appears in an \(R\)-dense set of the building for some \(R\). These buildings correspond to (a leaf in) minimal foliated subspaces of \(\Lambda_q\). The authors first prove the existence of quasi-periodic buildings. Then they prove that in fact there are infinitely many minimal foliated subspaces. They also prove a measurable counterpart to these theorems: there are infinitely many quasi-invariant ergodic transverse measures on \(\Lambda_q\) not supported on a compact leaf.