Geometry & Topology Monographs 2 (1999), Proceedings of the Kirbyfest, paper no. 8, pages 135-156.

Almost Linear Actions by Finite Groups on S^{2n-1}

Hansjorg Geiges and Charles B Thomas

Abstract. 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 SL_2(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.

Keywords. Almost linear action, surgery invariants, special linear group, contact structure, positive scalar curvature, positive Ricci curvature

AMS subject classification. Primary: 57S17. Secondary: 57S25, 57R65, 53C15, 57R85.

E-print: arXiv:math.GT/9911250

Submitted: 12 January 1999. (Revised: 10 June 1999.) Published: 17 November 1999.

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Hansjorg Geiges and Charles B Thomas

Mathematisch Instituut, Universiteit Leiden
Postbus 9512, 2300 RA Leiden, The Netherlands

DPMMS, University of Cambridge
16 Mill Lane, Cambridge CB2 1SB, UK


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