Abstract
We show that for any full and sufficiently transitive (i.e., flexible) group G of homeomorphisms of Cantor space, Aut(Aut(G)) = Aut(G). This class contains many generalisations of the Higman-Thompson g roups Gn,r, and the Rational group \({{\cal R}_2}\) of Grigorchuk, Nekrashevych, and Sushchanskiĭ. We also demonstrate that for generalisations Tn,r of R. Thompson’s group T, Aut(Aut(Tn,r)) = Aut(Tn,r). In the case of the groups Gn,r and Tn,r our results extend results of Brin and Guzmán for Thompson’s group T, and generalisations of Thompson’s group F.
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References
L. Bartholdi and S. Sidki, The automorphism tower of groups acting on rooted trees, Transactions of the American Mathematical Society 358 (2003), 329–358.
C. Bleak, P. Cameron, Y. Maissel, A. Navas and F. Olukoya, The further chameleon groups of Richard Thompson and Graham Higman: Automorphisms via dynamics for the Higman groups Gn,r, https://arxiv.org/abs/1605.09302.
M. G. Brin, The chameleon groups of Richard J. Thompson: automorphisms and dynamics, Institut de Hautes Études Scientifiques. Publications Mathématiques 84 (1996), 5–33.
M. G. Brin and F. Guzmán, Automorphisms of generalized Thompson groups, Journal of Algebra 203 (1998), 285–348.
W. Burnside, Theory of Groups of Finite Order, Cambridge Library Collection—Mathematics, Cambridge University Press, Cambridge, 2012.
C. Donoven and F. Olukoya, Conjugate subgroups and overgroups of Vn, International Journal of Algebra and Computation 30 (2020), 1129–1160.
R. I. Grigorchuk, V. V. Nekrashevych and V. I. Sushchanskiĭ, Automata, dynamical systems, and groups, Trudy Matematicheskogo Instituta Imeni V. A. Steklova 231 (2000), 134–214.
G. Higman, Finitely Presented Infinite Simple Groups, Notes on Pure Mathematics, Vol. 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974.
S. H. McCleary and M. Rubin, Locally moving groups and the reconstruction problem for chains and circles, https://arxiv.org/abs/math/0510122.
V. Nekrashevych, Self-similar Groups, Mathematical Surveys and Monographs, Vol. 117, American Mathematical Society, Providence, RI, 2005.
F. Olukoya, Automorphisms of the generalised Thompson’s group Tn,r, https://arxiv.org/abs/1908.03816.
C. E. Röver, Constructing finitely presented simple groups that contain Grigorchuk groups, Journal of Algebra 220 (1999), 284–313.
M. Rubin, Locally moving groups and reconstruction problems, in Ordered Groups and Infinite Permutation Groups, Mathematics and its Applications, vol. 354, Kluwer Academic, Dordrecht, 1996, pp. 121–157.
R. J. Thompson, Notes on three groups of homeomorphisms, unpublished.
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Olukoya, F. Automorphism towers of groups of homeomorphisms of Cantor space. Isr. J. Math. 244, 883–899 (2021). https://doi.org/10.1007/s11856-021-2196-z
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DOI: https://doi.org/10.1007/s11856-021-2196-z