2025/01/24 by Barbieri, Sebastián, Carrasco-Vargas, Nicanor, Rivera-Burgos, Paola
#20B27 #37B10 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2501.14463
We study the automorphism group Aut(X) of a non-trivial strongly irreducible subshift X on an arbitrary infinite group G and generalize classical results of Ryan, Kim and Roush. We generalize Ryan's theorem by showing that the center of Aut(X) is generated by shifts by elements of the center of G modded out by the kernel of the shift action. We generalize Kim and Roush's theorem by showing that if the free group Fk of rank k≥ 1 embeds into G, then the automorphism group of any full Fk-shift embeds into Aut(X). If X is an SFT, or more generally, if X satisfies the strong topological Markov property, then we can weaken the conditions on G. In this case we show that the automorphism group of any full ℤ-shift embeds into Aut(X) provided G is not locally finite, and that the automorphism group of any full Fk-shift embeds into Aut(X) whenever G is nonamenable. Our results rely on a new marker lemma which is valid for any nonempty strongly irreducible subshift on an infinite group. We remark that our results are new even for G=ℤ as they do not require the subshift to be an SFT.