## Abstract

A group action H on X is called "telescopic" if for any finitely presented group G, there exists a subgroup H' in H such that G is isomorphic to the fundamental group of X/H'. We 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: (1) Aitchison's theorem: Every finitely presented group G can appear 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; (2) Taubes' theorem: Every finitely presented group G can appear as the fundamental group of a compact complex 3-manifold.

Original language | English (US) |
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Pages (from-to) | 1814-1831 |

Number of pages | 18 |

Journal | Geometric and Functional Analysis |

Volume | 22 |

Issue number | 6 |

DOIs | |

State | Published - Dec 2012 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology