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Description

A group action H on X is called ”telescopic” if for any finitely pre- sented group G, there exists a subgroup H’ in H such that G is isomorphic to the fundamental group of X/H’.\\r\\nWe construct examples of telescopic actions on some CAT[-1] spaces, in particular on 3 and 4-dimensional hyperbolic spaces. As applications we give new proofs of the following statements:\\r\\n(1) Aitchison’s theorem: Every finitely presented group G can ap- pear as the fundamental group of M/J, where M is a compact 3-manifold and J is an involution which has only isolated fixed points;\\r\\n(2) Taubes’ theorem: Every finitely presented group G can appear as the fundamental group of a compact complex 3-manifold.