A description of $\operatorname{Aut}(dV_n)$ and $\operatorname{Out}(dV_n)$ using transducers
Luke Elliott · Groups Geometry and Dynamics · 2022
The groups dV_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 ( =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 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 \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 .