On the rigidity of discrete isometry groups of negatively curved spaces

Abstract

Abstract. We prove an ergodic rigidity theorem for discrete isometry groups of CAT(−1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2-polyhedra, hyperbolic Bruhat-Tits buildings and rank one symmetric spaces. We prove that two negatively curved Riemannian metrics, with conical singularities of angles at least 2π, on a closed surface, with boundary map absolutely continuous with respect to the Patterson-Sullivan measures, are isometric. For that, we generalize J.-P. Otal's result to prove that a negatively curved Riemannian metric, with conical singularities of angles at least 2π, on a closed surface, is determined, up to isometry, by its marked length spectrum.Mathematics Subject Classification (1991). 57S30, 53C45, 20H10, 51M10, 51E24.

Spectral rigidity of automorphic orbits in free groups

“…Isom.X 1 / and 2 W G ! Isom.X 2 / where X 1 ; X 2 are allowed to be higher dimensional, but there are more significant restrictions on the geometry of X 1 ; X 2 than in the results about MLSRC for surfaces (see, for example, Croke, Eberlein and Kleiner [19], Hersonsky and Paulin [34], Kim [47;48;49] and Dal'Bo and Kim [25]). However, the original version of MLSRC is still mostly open (except for rather special classes of metrics) in dimensions bigger than two.…”

Prescribing the behaviour of geodesics in negative curvature

“…x t is a geodesic ray with x 0 2 H which converges to , the Hamenstädt distance (defined in [26;29,Appendix]) of a and b in…”

Graphs, spectral triples and Dirac zeta functions

“…for Re(s) > δ the series converges and for Re(s) < δ, the series diverges. If δ / ∈ {0, ∞}, then Γ is called a divergence group with respect to its action on X (see for instance [13]). In our case Γ acts as divergence group, see Remark 5.5.…”

“…(see [13]). This converges weakly to a probability measure on X ∪ ∂X with support on the boundary ∂X and hence turns ∂X into a measure space.…”

“…By well-definedness, ifφ is equivariant for some group-action, then so is F . In the proof of the main theorem we will use the following theorem (see [13], Theorem A):…”