A description of $\mathrm{Aut}(d{V}_n)$ and $\mathrm{Out}(d{V}_n)$ using transducers

Abstract

The groups $d{V}_n$ 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 $V_n$ ($=1V_n$) and the Brin–Thompson groups $nV$ ($=n{V}_2$). A description of the groups $\mathrm{Aut}(G_{n,r})$ (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 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 $\mathrm{Aut}(d{V}_n)$ which extends the description of $\mathrm{Aut}(1V_n)$ given in the former paper. We make use of this description to show that $\mathrm{Out}(d{V}_2)\cong \mathrm{Out}(V_2)\wr S_d$, and more generally give a natural embedding of $\mathrm{Out}(d{V}_n)$ into $\mathrm{Out}(G_{n,n-1})\wr S_d$.