Distinguishing extension numbers for $\mathbf R^n$ and $S^n$

Abstract: In the setting of a group $\Gamma$ acting faithfully on a set $X$, a $k$-coloring $c: X\rightarrow {1, 2, ..., k}$ is called $\Gamma$-distinguishing if the only element of $\Gamma$ that fixes $c$ is the identity element. The distinguishing number $D_\Gamma(X)$ is the minimum value of $k$ such that a $\Gamma$-distinguishing $k$-coloring of $X$ exists. Now, fixing $k= D_\Gamma(X)$, a subset $W\subset X$ with trivial pointwise stabilizer satisfies the precoloring extension property $P(W)$ if every precoloring $c: X-W\rightarrow {1, ..., k}$ can be extended to a $\Gamma$-distinguishing $k$-coloring of $X$. The distinguishing extension number $\text{ext}_D(X, \Gamma)$ is then defined to be the minimum $n$ such that for all applicable $W\subset X$, $|W|\geq n$ implies that $P(W)$ holds. In this paper, we compute $\text{ext}_D(X, \Gamma)$ in two particular instances: when $X = S^1$ is the unit circle and $\Gamma = \text{Isom}(S^1) = O(2)$ is its isometry group, and when $X = V(C_n)$ is the set of vertices of the cycle of order $n$ and $\Gamma = \text{Aut}(C_n) = D_n$, the dihedral group of a regular $n$-gon. This resolves two conjectures of Ferrara, Gethner, Hartke, Stolee, and Wenger. In the case of $X=\mathbf R^2$, we prove that $\text{ext}_D(\mathbf R^2, SE(2))<\infty$, which is consistent with (but does not resolve) another conjecture of Ferrara et al. On the other hand, we also prove that for all $n\geq 3$, $\text{ext}_D(S^{n-1}, O(n)) = \infty$, and for all $n\geq 3$, $\text{ext}_D(\mathbf R^n, E(n))=\infty$, disproving two other conjectures from the same authors.