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 $c:X\to \{1,2,...,k\}$ 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 $c:X-W\to \{1,...,k\}$ 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 $W\subset X$, $|W|\ge n$ 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 $n\ge 3$, ${\text{ext}}_D(S^{n-1},O(n))=\infty$, and for all $n\ge 3$, ${\text{ext}}_D({\mathbf{R}}^n,E(n))=\infty$, disproving two other conjectures from the same authors.

22 pages