We study groups of reversible cellular automata, or CA groups, on groups. More generally, we consider automorphism groups of subshifts of finite type on groups. It is known that word problems of CA groups on virtually nilpotent groups are in co-NP, and can be co-NP-hard. We show that under the Gap Conjecture of Grigorchuk, their word problems are PSPACE-hard on all other groups. On free and surface groups, we show that they are indeed always in PSPACE. On a group with co-NEXPTIME word problem, CA groups themselves have co-NEXPTIME word problem, and on the lamplighter group (which itself has polynomial-time word problem) we show they can be co-NEXPTIME-hard. We show also two nonembeddability results: the group of cellular automata on a non-cyclic free group does not embed in the group of cellular automata on the integers (this solves a question of Barbieri, Carrasco-Vargas and Rivera-Burgos); and the group of cellular automata in dimension $D$ does not embed in a group of cellular automata in dimension $d$ if $D \geq 3d+2$ (this solves a question of Hochman).

翻译：我们研究群上的可逆元胞自动机群，或称CA群。更一般地，我们考虑群上有限型子移位的自同构群。已知在虚拟幂零群上的CA群的单词问题属于co-NP类，且可能是co-NP困难的。我们证明，在Grigorchuk的间隙猜想下，在所有其他群上它们的单词问题都是PSPACE困难的。在自由群和曲面群上，我们证明它们确实始终属于PSPACE类。在具有co-NEXPTIME单词问题的群上，CA群自身也具有co-NEXPTIME单词问题；而在点灯人群（其自身具有多项式时间单词问题）上，我们证明它们可以是co-NEXPTIME困难的。我们还展示两个不可嵌入性结果：非循环自由群上的元胞自动机群不能嵌入整数群上的元胞自动机群（这解决了Barbieri、Carrasco-Vargas和Rivera-Burgos提出的问题）；且若$D \geq 3d+2$，则$D$维元胞自动机群不能嵌入$d$维元胞自动机群（这解决了Hochman提出的问题）。