Telescopic actions

TL;DR

The paper introduces the concept of telescopic group actions on CAT[-1] spaces, providing new constructions and proofs that any finitely presented group can be realized as the fundamental group of certain 3-manifolds and complex 3-manifolds.

Contribution

It constructs examples of telescopic actions on hyperbolic spaces and offers new proofs of existing theorems about fundamental groups of 3-manifolds.

Findings

Existence of telescopic actions on hyperbolic spaces

Any finitely presented group can be realized as a fundamental group of specific 3-manifolds

New proofs of Aitchison's and Taubes' theorems

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.