Hyperbolic buildings, conformal dimension and Mostow rigidity
Let \((p,q)\) be such that \(p\geqslant 5\) and \(q\rightsquigarrow 2\), and \(I_{pq}\) be the Tits building whose apartments are the hyperbolic planes of curvature \(-1\), the chambers are the regular hyperbolic \(p\)-gones with angles of measure \( \pi \)/2 and whose link at each vertex is a complete bipartite \(q\)-graph. A graph is bipartite if its set of vertices can be partitioned into two disjoint subsets such that no two vertices in the same subset lie on a common edge. \(I_{pq}\) is a CAT(\(-1\)) space, which means that the geodesic triangles are thinner then their comparison triangles in the hyperbolic plane. Therefore it has a boundary at infinity. In this paper the author obtains a numerical expression that gives the exact conformal Pansu dimension of the boundary of the building. This was known in such an exact way only for homogeneous manifolds of negative curvature. The second theorem in this article shows that in this type of buildings the conformal dimension reflects the combinatorial structure of the building, and does not privilege any CAT(\(-1\)) structure, as is the case in symmetric non-compact spaces of rank one. In the second part of this theorem it is proved using Patterson-Sullivan measures, that if \(G\) is a cocompact lattice of \(I_{pq}\) such that \(q\geqslant 3\), and \(X\) is a CAT(\(-1\))-space on which \(G\) acts by isometries in a proper discontinuous way, then the conformal Pansu dimension of the boundary of the building is strictly smaller then the Hausdorff dimension of the boundary of \(X\). The third theorem shows that if \(q\geqslant 3\) and \(G\), \(G'\) are two isometric cocompact lattices from \(I_{pq}\) and \(I_{p'q' \text{ }}\), respectively, then \((p, q)=(p',q')\), and the isomorphism is the conjugate of an element of Isom (\(I_{pq}\)).To prove this last theorem the author uses an ergodic argument by \textit{D. Sullivan} [Bull. Am. Math. Soc., New Ser. 6, 57-73 (1982; Zbl 0489.58027)].
scientific article; zbMATH DE number 1013772