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

TL;DR

This paper generalizes transducer-based descriptions of automorphism groups for a family of groups including Higman-Thompson and Brin-Thompson groups, extending previous work and revealing new structural embeddings.

Contribution

It extends the transducer representation of automorphisms to the groups dVn, providing new insights into their outer automorphism groups and their embeddings.

Findings

 Out(dV_2)   Out(V_2) \u007c S_d

Embedding of Out(dV_n) into Out(G_{n, n-1}) \u007c S_d

Extension of transducer descriptions to a broader class of groups

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 $\operatorname{Out}(G_{n, n-1})\wr S_d$.