A description of Aut ⁡ ( d V n ) \operatorname{Aut}(dV_n) Aut ( d V n ​ ) and Out ⁡ ( d V n ) \operatorname{Out}(dV_n) Out ( d V n ​ ) using transducers

[摘要] The groups dVndV_ndVn​ are an infinite family of groups, first introduced by C. Martínez-Pérez, F. Matucci and B. E. A. Nucinkis, which includes both the Higman–Thompson groups VnV_nVn​ (=1Vn=1V_n=1Vn​) and the Brin–Thompson groups nVnVnV (=nV2=nV_2=nV2​). A description of the groups Aut⁡(Gn,r)\operatorname{Aut}(G_{n, r})Aut(Gn,r​) (including the groups Gn,1=VnG_{n,1}=V_nGn,1​=Vn​) has previously been given by C. Bleak, P. Cameron, Y. Maissel, A. Navas, and F. Olukoya. Their description uses the transducer representations of homeomorphisms of Cantor space introduced in a paper of R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii, together with a theorem of M. Rubin. We generalise the transducers of the latter paper and make use of these transducers to give a description of Aut⁡(dVn)\operatorname{Aut}(dV_n)Aut(dVn​) which extends the description of Aut⁡(1Vn)\operatorname{Aut}(1V_n)Aut(1Vn​) given in the former paper. We make use of this description to show that Out⁡(dV2)≅Out⁡(V2)≀Sd\operatorname{Out}(dV_2) \cong \operatorname{Out}(V_2)\wr S_dOut(dV2​)≅Out(V2​)≀Sd​, and more generally give a natural embedding of Out⁡(dVn)\operatorname{Out}(dV_n)Out(dVn​) into Out⁡(Gn,n−1)≀Sd\operatorname{Out}(G_{n, n-1}) \wr S_dOut(Gn,n−1​)≀Sd​.

[学科分类] 神经科学

[关键词] Brin–Thompson groups;automorphism groups;transducers;homeomorphism groups;Rubin’s theorem