Higman--Thompson groups $F_n$ all the way down

Abstract

We prove that for every $n\ge 2$ the Higman--Thompson group $F_n$ has a maximal subgroup of infinite index isomorphic to itself. In fact, we construct a chain of subgroups $F_n=H_0>H_1>H_2>\cdots$, all isomorphic to $F_n$ and with trivial intersection, such that for every $i$ the only subgroups of $F_n$ containing $H_i$ are $H_i,H_{i-1},\ldots,H_0=F_n$; in particular, each $H_{i+1}$ is maximal in $H_i$. We prove that for all $n\ge m\ge 2$, every closed maximal subgroup of $F_m$ isomorphic to $F_n$ arises from a homeomorphism between the $n$-ary and $m$-ary Cantor spaces given by a finite semi-synchronizing transducer--a variation of the synchronizing transducers of Bleak, Cameron, Maissel, Navas and Olukoya. We characterize the homeomorphisms of Cantor spaces conjugating $F_n$ into $F_m$ as the order-preserving or order-reversing rational homeomorphisms whose minimal transducer is semi-synchronizing. At the heart of the paper is a machinery bridging transducers and Stallings $2$-cores of subgroups, which reduces the conjugation of finitely generated closed subgroups by such homeomorphisms to an algorithmic procedure. As applications, we prove that Jones' ternary oriented subgroup $\vec F_3\le F_3$ is isomorphic to $F_4$, answering questions of Aiello, and that all known maximal subgroups of infinite index of Thompson's group $F$ which act minimally on $(0,1)$ are isomorphic to Higman--Thompson groups. That raises the problem of whether all maximal subgroups of infinite index of $F$ which act minimally on $(0,1)$ are isomorphic to Higman--Thompson groups. We briefly discuss related results regarding fast groups of homeomorphisms and maximal subgroups of Thompson groups.