## Abstract

The recent paper

*The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups G*of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterizes the automorphisms of the Higman-Thompson groups Gn,r. his characterization is as the specific subgroup of the rational group R_{n,r}_{n,r}of Grigorchuk, Nekrashevych and Suchanski{\u i}'s consisting of those elements which have the additional property of being bi-synchronizing. This article extends the arguments of BCMNO to characterize the automorphism group of T_{n,r}as a subgroup of Aut(G_{n,r}). We naturally also study the outer automorphism groups Out(T_{n,r}) . We show that each group Out(T_{n,r})_{ }can be realized a subgroup of Out(T_{n,n−1}). Extending results of Brin and Guzman, we also show that the groups Out(T_{n,r}), for n>2, are all infinite and contain an isomorphic copy of Thompson's group F. Our techniques for studying the groups Out(T_{n,r}) work equally well for Out(G_{n,r}) and we are able to prove some results for both families of groups. In particular, for X ∈ {T,G}, we show that the groups Out(X_{n,r}) fit in a lattice where Out(X_{n,1}) ⊴ Out(X_{n,r}) for all 1 ≤ r ≤n−1 and Out(X_{n,r}) ⊴ Out(X_{n,n−1}). This gives a partial answer to a question in BCMNO concerning the normal subgroup structureof Out(G_{n,n−1}). Furthermore, we deduce that for 1 ≤ j,d ≤ n−1 such that d = gcd (j,n−1), Out(X_{n,j}) = Out(X_{n,d}) extending a result of BCMNO for the groups G_{n,r}to the groups T_{n,r.}We give a negative answer to the question in BCMNO which asks whether or not Out(G_{n,r})_{ }≅ Out(G_{n,s}) if and only if gcd (n−1,r) = gcd (n−1,s). Lastly, we show that the groups T_{n,r}have the R∞ property. This extends a result of Burillo, Matucci and Ventura and, independently, Gonçalves and Sankaran, for Thompson's group T.Original language | English |
---|---|

Pages (from-to) | 86-135 |

Number of pages | 50 |

Journal | Transactions of the London Mathematical Society |

Volume | 9 |

Issue number | 1 |

DOIs | |

Publication status | Published - 15 Aug 2022 |

## Fingerprint

Dive into the research topics of 'Automorphisms of the generalised Thompson's group T_{n,r}'. Together they form a unique fingerprint.