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.
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D. Panov is a Royal Society University Research Fellow. A. Petrunin was partially supported by NSF Grant DMS 0905138.
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Panov, D., Petrunin, A. Telescopic actions. Geom. Funct. Anal. 22, 1814–1831 (2012). https://doi.org/10.1007/s00039-012-0194-3
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DOI: https://doi.org/10.1007/s00039-012-0194-3