A Description of Aut(dVn) and Out(dVn) Using Transducers

Abstract

The groups $dV_n$ are an infinite family of groups, first introduced by C. Mart\'inez-P\'erez, F. Matucci and B. E. A. Nucinkis, which includes both the Higman-Thompson groups $V_n(=1V_n)$ and the Brin-Thompson groups $nV(=nV_2)$. A description of the groups $\operatorname{Aut}(G_{n, r})$ (including the groups $G_{n,1}=V_n$) 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 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 $\operatorname{Aut}(dV_n)$ which extends the description of $\operatorname{Aut}(1V_n)$ given in the former paper. We make use of this description to show that $\operatorname{Out}(dV_2) \cong \operatorname{Out}(V_2)\wr S_d$, and more generally give a natural embedding of $\operatorname{Out}(dV_n)$ into $\operatorname{Out}(G_{n, n-1})\wr S_d$.