Coarse distinguishability of graphs with symmetric growth

Let X be a connected, locally finite graph with symmetric growth. We prove that there is vertex coloring ϕ : → {0, 1} and some R ∈ ℝ such every automorphism f preserving -close to the identity map; this can seen as coarse geometric version of symmetry breaking. also infinite motion conjecture true for graphs where at least one stabilizer S x satisfies following condition: non-identity , sequence n lim d ( )) = ∞ .