The pointed Gromov-Hausdorff limit of action on manifold

Question

The pointed Gromov - Hausdorff limit is a concept of the convergence of Riemannian manifolds :

For instance $$ (\lambda_i S^2(1) , p) \rightarrow_{G-H} ({\bf R}^2,O) = T_p S^2(1)$$ where $\lambda_i \rightarrow \infty$. Here $ \lambda_i S^2(1) $ means the metric $\lambda_i g_0$ where $g_0$ is a canonical metric on $S^2(1)$.

Here I want to know the following question :

(a) What is the limit of ${\bf Z}_2=\pi_1({\bf RP}^2)$-action on $S^2$ ?

(b) And ${\bf Z}_p = \pi_1( L_p) $-action on $S^3$ ? Here $L_p$ is a lens.

Thank you in advance.

Answer 1

(a) The antipodal action no the sphere is fixed-point-free and therefore does not preserve the distinguished point $p$ used in the construction of the G-H limit. Hence the action does not descend to the G-H limit.

(b) The fundamental group of the Lens space similarly acts freely on the 3-sphere, so the action does not descend in this case, either.