An Example of 2-dimensional Hyperbolic Group Which Can't Act on 2-dimensional Negatively Curved Complexes

It is a long standing open problem wether or not any word hyperbolic group admits a discrete faithful cocompact isometric action on a space of negative curvature. The goal of this note is to show that the answer is negative if one restricts to the class of groups of isometries of 2-dimensional CAT(0)-complexes. Namely, we will prove the following: Theorem 0.1 There exists a word-hyperbolic group G which acts discretely and eeectively by isometries with compact quotient on a contractible 2-dimensional complex P of non-positive curvature so that G doesn't admit such action on any negatively curved 2-dimensional polyhedron. By negatively (or nonpositively) curved space we mean CAT(?1) (or CAT(0)) space in the sense of comparison theorems. We shall need a definition of the angle in CAT(0)-space X which we take from 1]. Suppose c(t); c 0 (t) are geodesics emanating from a point x 2 X. Let f(t) = arccos(2 t 2 (d(c(t); c 0 (t)) 2 ? 1) Then the angle between c; c 0 at x is deened to be lim t!0 f(t) This deenition coincides with the usual one in the case of Riemann manifolds. Angle comparison theorem states that for CAT(?k) space angles of geodesic 1