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