We show that every locally compact group which acts faithfully on a connected three-manifold is a Lie group. By known reductions, it suffices to show that there is no faithful action of ℤp (the p-adic integers) on a connected three-manifold. If ℤp acts faithfully on M3, we find an interesting ℤp-invariant open set U ⊆ M with H2(U)=ℤ and analyze the incompressible surfaces in U representing a generator of H2(U). It turns out that there must be one such incompressible surface, say F, whose isotopy class is fixed by ℤp. An analysis of the resulting homomorphism ℤp→MCG(F) gives the desired contradiction. The approach is local on M.
All Science Journal Classification (ASJC) codes
- Applied Mathematics