On the rigidity of discrete isometry groups of negatively curved spaces.

*(English)*Zbl 0908.57009The authors study ergodic properties of discrete groups of isometries of \(\text{CAT}(-1)\)-spaces. The main goal of the paper is to generalize Mostow’s rigidity theorem for locally symmetric spaces and Sullivan’s rigidity theorem for hyperbolic manifolds to the groups mentioned in the title. Recall that a discrete group of isometries of a \(\text{CAT}(-1)\)-space \(X\) is called group of divergent type (or simply divergent group) if its Poincaré series diverges at the critical exponent. The authors use a generalization of the crossratio due to J.-P. Otal [Rev. Mat. Iberoam. 8, No. 3, 441-456 (1992; Zbl 0777.53042)] to define a Möbius map of the ideal boundary \(\partial X\) of \(X\). Namely this is a map of \(\partial X\) which preserves the crossratio. Here is one of the main results of the paper:

Theorem A. Let \(X_1, X_2\) be locally complete \(\text{CAT}(-1)\) metric spaces. Let \(\Gamma_1\) and \(\Gamma_2\) be discrete groups of isometries of \(X_1\), \(X_2\), having the same critical exponent. Suppose \(\Gamma_2\) is a divergence group. Let \(\widehat{\phi}:\partial X_1\mapsto \partial X_2\) be a Borel map, equivariant for some morphism \(\Gamma_1\mapsto \Gamma_2\), which is not singular with respect to Patterson-Sullivan measures. Then \(\widehat{\phi}\) is Möbius on the limit set of \(\Gamma_1\).

If both \(X_i\;(i=1,2)\) are rank one symmetric spaces, C. B. Yue [Invent. Math. 125, No. 1, 75-102 (1996; Zbl 0853.58076)] proved a somewhat stronger result, namely the above theorem is true in this case without requiring that the critical exponents should be equal. The authors provide examples of non-trivial applications of their theorem to \(\text{CAT}(-1)\) spaces different than rank one symmetric spaces and trees. Another important result of the paper is related to J. P. Otal’s rigidity theorem for marked length spectra of closed surfaces [Ann. Math., II. Ser. 131, No. 1, 151-162 (1990; Zbl 0699.58018)]. The authors obtain an analogous result for negatively curved closed surfaces with conical singularities of angle at least \(2\pi\).

Theorem A. Let \(X_1, X_2\) be locally complete \(\text{CAT}(-1)\) metric spaces. Let \(\Gamma_1\) and \(\Gamma_2\) be discrete groups of isometries of \(X_1\), \(X_2\), having the same critical exponent. Suppose \(\Gamma_2\) is a divergence group. Let \(\widehat{\phi}:\partial X_1\mapsto \partial X_2\) be a Borel map, equivariant for some morphism \(\Gamma_1\mapsto \Gamma_2\), which is not singular with respect to Patterson-Sullivan measures. Then \(\widehat{\phi}\) is Möbius on the limit set of \(\Gamma_1\).

If both \(X_i\;(i=1,2)\) are rank one symmetric spaces, C. B. Yue [Invent. Math. 125, No. 1, 75-102 (1996; Zbl 0853.58076)] proved a somewhat stronger result, namely the above theorem is true in this case without requiring that the critical exponents should be equal. The authors provide examples of non-trivial applications of their theorem to \(\text{CAT}(-1)\) spaces different than rank one symmetric spaces and trees. Another important result of the paper is related to J. P. Otal’s rigidity theorem for marked length spectra of closed surfaces [Ann. Math., II. Ser. 131, No. 1, 151-162 (1990; Zbl 0699.58018)]. The authors obtain an analogous result for negatively curved closed surfaces with conical singularities of angle at least \(2\pi\).

Reviewer: L.Potyagailo (Villeneuve d’Ascq)

##### MSC:

57S30 | Discontinuous groups of transformations |

53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

51E24 | Buildings and the geometry of diagrams |