Group-product rigidity and Higman-Thompson reassociation groups

Abstract

Fix an arity $r\ge 2$, a group $G$, and a full ordered $r$-ary tree $T$ with at least two internal vertices. For an arbitrary operation $ω:G^r\to G$, let $ω_T$ denote the operation obtained by iterating $ω$ according to $T$. We classify all $ω$ for which there exists a bijection $F_T:G\to G$ such that $ω_T(x_1,\ldots,x_n)=F_T(x_1\cdots x_n)$. We prove that $ω$ must have one of the forms $ax_1\cdots x_r$, $x_1\cdots x_rb$, $d\,ψ(x_1\cdots x_r)$, where $a,b\in G$, $d\in Z(G)$, and $ψ\in\operatorname{Aut}(G)$, with explicit conditions on the parameters determined by $T$. We next reverse the problem. Fix an operation $ω$ of one of these three forms and determine every full ordered $r$-ary tree $T$ for which there exists a bijection $F_T:G\to G$ satisfying $ω_T(x_1,\ldots,x_n)=F_T(x_1\cdots x_n)$. The answer is governed by the order of $aZ(G)$ or $bZ(G)$ in $G/Z(G)$, or by the order of $ψ$ in $\operatorname{Aut}(G)$. We also ask, for each of the three solution forms above, how much associativity remains. More precisely, for two full ordered $r$-ary trees $S$ and $T$ with the same number of leaves, we determine exactly when $ω_S=ω_T$. Pairs of $r$-ary trees encode changes of parenthesization, and modulo simultaneous expansion they represent elements of the Higman--Thompson group $F_r$. The changes of parenthesization that preserve the iterated operation form a subgroup of $F_r$, which we determine explicitly.