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arXiv · 2005.09716

Coarse distinguishability of graphs with symmetric growth

Jesús Antonio Álvarez López·Ramón Barral Lijó·Hiraku Nozawa
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Abstract

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

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Jesús Antonio Álvarez López, Ramón Barral Lijó, Hiraku Nozawa. 2020-05-19. Coarse distinguishability of graphs with symmetric growth. https://arxiv.org/abs/2005.09716

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