We show that conjugacy of reversible cellular automata is undecidable, whether the conjugacy is to be performed by another reversible cellular automaton or by a general homeomorphism. This gives rise to a new family of f.g. groups with undecidable conjugacy problems, whose descriptions arguably do not involve any type of computation. For many automorphism groups of subshifts, as well as the group of asynchronous transducers and the homeomorphism group of the Cantor set, our result implies the existence of two elements such that every f.g. subgroup containing both has undecidable conjugacy problem. We say that conjugacy in these groups is eventually locally undecidable. We also prove that the Brin-Thompson group $2V$ and groups of reversible Turing machines have undecidable conjugacy problems, and show that the word problems of the automorphism group and the topological full group of every full shift are eventually locally co-NP-complete.

中文翻译：

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可逆元胞自动机的共轭

我们表明可逆元胞自动机的共轭是不可判定的，无论是由另一个可逆元胞自动机还是由一般同胚来执行共轭。这产生了具有不可判定共轭问题的 fg 群的新家族，其描述可以说不涉及任何类型的计算。对于许多子移的自同构群，以及异步变换器群和康托集的同胚群，我们的结果意味着存在两个元素，使得包含这两个元素的每个 fg 子群都有不可判定的共轭问题。我们说这些群中的共轭最终是局部不可判定的。我们还证明了 Brin-Thompson 群 $2V$ 和可逆图灵机群具有不可判定的共轭问题，