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Finite groups acting on 3–manifolds and cyclic branched coverings of knots

Mattia Mecchia

Geometry & Topology Monographs 14 (2008) 393–416

DOI: 10.2140/gtm.2008.14.393

arXiv: 0904.1854


We are interested in finite groups acting orientation-preservingly on 3–manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic branched covering of a strongly invertible knot as well as by the isometry group of any hyperbolic 2–fold branched covering of a knot in S3. In the paper we give a characterization of nonsolvable groups of this type. Then we consider some possible applications to the study of cyclic branched coverings of knots and of hyperelliptic diffeomorphisms of 3–manifolds. In particular we analyze the basic case of two distinct knots with the same cyclic branched covering.

To the memory of Heiner Zieschang


3-manifold, finite group action, cyclic branched covering, knot in S³

Mathematical Subject Classification

Primary: 57M60

Secondary: 57M12, 57S17


Received: 14 July 2007
Revised: 2 March 2007
Accepted: 2 March 2007
Published: 29 April 2008

Mattia Mecchia
Università degli Studi di Trieste
Dipartimento di Matematica e Informatica
34100 Trieste