Telescopic actions

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.

+higher dimensions

References in corpus (2)

• Combinatorial realisation of cycles and small covers
• Dirichlet fundamental domains and complex-projective varieties

Cited by in corpus (7)

• Rational curves and special metrics on twistor spaces
• On representation varieties of 3-manifold groups
• The diversity of symplectic Calabi-Yau six-manifolds
• Complex geometry of moment-angle manifolds
• Symmetries of hyperbolic 4-manifolds
• Dirichlet fundamental domains and complex-projective varieties
• Compact hyperbolic manifolds without spin structures