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Almost linear actions by finite groups on S2n−1

Hansjorg Geiges and Charles B Thomas

Geometry & Topology Monographs 2 (1999) 135–156

DOI: 10.2140/gtm.1999.2.135

arXiv: math.GT/9911250


A free action of a finite group on an odd-dimensional sphere is said to be almost linear if the action restricted to each cyclic or 2–hyperelementary subgroup is conjugate to a free linear action. We begin this survey paper by reviewing the status of almost linear actions on the 3–sphere. We then discuss almost linear actions on higher-dimensional spheres, paying special attention to the groups SL2(p), and relate such actions to surgery invariants. Finally, we discuss geometric structures on space forms or, more generally, on manifolds whose fundamental group has periodic cohomology. The geometric structures considered here are contact structures and Riemannian metrics with certain curvature properties.


almost linear action, surgery invariants, special linear group, contact structure, positive scalar curvature, positive Ricci curvature

Mathematical Subject Classification

Primary: 57S17

Secondary: 53C15, 57R65, 57R85, 57S25


Received: 12 January 1999
Revised: 10 June 1999
Published: 17 November 1999

Hansjorg Geiges
Mathematisch Instituut
Universiteit Leiden
Postbus 9512
2300 RA Leiden
The Netherlands
Charles B Thomas
University of Cambridge
16 Mill Lane
United Kingdom