1 Quasi - isometry rigidity for hyperbolic build - ings

Recall that the quasi-isometry group QI(X) of a metric space X is the set of equivalence classes of quasi-isometries f : X → X, where two quasiisometries f1, f2 are equivalent iff supx d(f1(x), f2(x)) <∞ (here we consider G as a metric space with a word metric). One approach to this question, which has been the most successful one, is to find an ‘optimal’ space X quasi-isometric to G and show that the natural map Isom(X) → QI(X) is an isomorphism. This statement was proved in various settings, started by Pansu (for quaternionic and Cayley hyperbolic spaces), followed by results of Kleiner and Leeb (for products of irreducible affine buildings and symmetric spaces of dimension ≥ 2), Kapovich and Schwartz (for universal covers of compact locally symmetric spaces of dimension ≥ 3). Bourdon and Pajot showed this quasi-isometry rigidity for some buildings ([3]). The class of Tits buildings they study are called right-angled Fuchsian buildings: their apartments are hyperbolic planes, their chambers are regular hyperbolic p-gons with right angles. These buildings are CAT (−1)-spaces. By studying the quasi-conformal structure of the boundary at infinity of those buildings, Bourdon and Pajot showed the following quasi-isometry rigidity: Theorem 1. Let ∆,∆′ be two right-angled Fuchsian buildings. Any quasiisometry F : ∆→ ∆′ lies within bounded distance from an isometry.