The group (F4 x F2) rtimes F_infinity, which is not finitely generated, is shown to admit an infinite minimal sofic shift, answering Question 7.18(ii) of Doucha, Melleray and Tsankov.
The Higman--Thompson groups $V_n$ are $(2,2,2)$-generated
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abstract
We provide a family of generating sets $S_{\alpha}$ of the Higman--Thompson groups $V_n$ that are parametrized by certain sequences $\alpha$ of elements in $V_n$. These generating sets consist of $3$ involutions $\sigma$, $\tau$, and $s_{\alpha}$, where the latter involution is inspired by the class of spinal elements in the theory of branch groups. In particular this shows the existence of generating sets of $V_n$ that consist of $3$ involutions.
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Minimal sofic shift on a group that is not finitely-generated
The group (F4 x F2) rtimes F_infinity, which is not finitely generated, is shown to admit an infinite minimal sofic shift, answering Question 7.18(ii) of Doucha, Melleray and Tsankov.