群乘积刚性与Higman-Thompson重结合群
Group-product rigidity and Higman-Thompson reassociation groups
AI总结：
该论文研究群上迭代运算的形式分类，确定满足特定双射条件的运算形式与对应$r$元树，还明确了保持迭代运算形式的Higman-Thompson群$F_r$的子群。
AI中文摘要：
固定元数$r\ge 2$、群$G$以及至少含两个内部顶点的完全有序$r$元树$T$。对任意运算$\u03c9:G^r\to G$，令$\u03c9_T$表示按$T$迭代$\u03c9$得到的运算。我们对所有满足存在双射$F_T:G\to G$使得$\u03c9_T(x_1,\ldots,x_n)=F_T(x_1\cdots x_n)$的$\u03c9$进行分类，证明$\u03c9$必为以下形式之一：$ax_1\cdots x_r$、$x_1\cdots x_rb$、$d\\,\psi(x_1\cdots x_r)$，其中$a,b\in G$，$d\in Z(G)$，$\u03c8\in\operatorname{Aut}(G)$，且参数的显式条件由$T$确定。接下来我们反向处理该问题：固定上述三种形式之一的运算$\u03c9$，确定所有满足存在双射$F_T:G\to G$使得$\u03c9_T(x_1,\ldots,x_n)=F_T(x_1\cdots x_n)$的完全有序$r$元树$T$，答案由$aZ(G)$或$bZ(G)$在$G/Z(G)$中的阶，或$\u03c8$在$\u03c6$中的阶决定。我们还针对上述三种解形式，研究剩余结合度：对两个叶数相同的完全有序$r$元树$S$和$T$，精确确定$\u03c9_S=\u03c9_T$的条件。$r$元树对编码括号化的变化，模同时扩张后对应Higman-Thompson群$F_r$的元素；保持迭代运算形式的括号化变化构成$F_r$的一个子群，我们将其明确确定。
英文摘要：
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.
