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

2020/09/11 by Elliott, Luke
#20B27 #20E36 #37B05 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2009.05450

Abstract

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(=nV₂). A description of the groups 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 Aut(dV_n) which extends the description of Aut(1V_n) given in the former paper. We make use of this description to show that Out(dV₂) ≅ Out(V₂)\wr S_d, and more generally give a natural embedding of Out(dV_n) into Out(G_{n, n-1})\wr S_d.

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