All the manifolds I ever run into have finitely many connected components (for example any compact manifold, or the manifold of real or complex points of any linear algebraic group, or even the manifold of real or complex points of any algebraic variety in general).
Whenever these manifolds (with finitely many connected components) are equipped with a Riemannian metric it seems that the isometry group always has finite component group. For example, all surfaces.
Is it true that the isometry group of every connected (or even with finitely many connected components) Riemannian manifold has finite component group?
Best Answer
Here is a simple example. Take the cylinder with boundary $ C = S^1 \times [0,1] $ Then leave the two boundary circles identical (isometric) but slightly distort the interior of the cylinder (say by adding a bump somewhere) so that the new cylinder with boundary $ C' $ has no isometries. Then glue infinitely many copies of $ C' $ end to end. The resulting manifold $ M $ is a topological cylinder (without boundary) with isometry group exactly $ \mathbb{Z} $. This example can be isometrically embedded in $ \mathbb{R}^3 $ by putting a unit circle around every integer on the z axis (all circles parallel to xy plane) and smoothly connecting all the consecutive circles in the exact same way (again adding a bump or something to break any non translational symmetry).
A more general example which addresses Moishe Kohan's comment above is to instead look at an infinite genus surface. It turns out every countable group is the isometry group of a complete hyperbolic manifold (of infinite genus if the group is infinite and finite genus if the group is finite). The construction is done by reproducing the Cayley graph of the group by gluing together hyperbolic pairs of pants. See this nice little 3 page article [hyperbolic surfaces with prescribed infinite symmetry group, Daniel Allcock]
https://www.google.com/url?sa=t&source=web&rct=j&url=https://web.ma.utexas.edu/users/allcock/research/symmsurf.pdf&ved=2ahUKEwiRt8v_65_1AhWKmnIEHV3jBK4QFnoECAQQAQ&usg=AOvVaw0Em67laczcWGF3e-p948j5