自由可解群中的严格死端元
Strict dead end elements in free soluble groups
摘要：
设$G$是由有限集$A$生成的群。如果对于任何$a_1,a_2, ..., a_k\in A^{\pm1}$有$|g|>|ga_1|>|ga_1a_2|>...>|ga_1a_2... a_k|$，使得词$a_1a_2... a_k$是自由不可约的，则元素$g\in G$是深度$k$（相对于$A$）的严格死端。(Here $|g|$是从$g$到$G$的凯莱图中的恒等式的距离。）我们证明了在$d\ge2$度的自由可解群中存在深度为$k=k(d)$的严格死元，它相对于$d$呈指数增长。
英文摘要：
Let $G$ be a group generated by a finite set $A$. An element $g\in G$ is a strict dead end of depth $k$ (with respect to $A$) if $|g|>|ga_1|>|ga_1a_2|>...>|ga_1a_2... a_k|$ for any $a_1,a_2, ..., a_k\in A^{\pm1}$ such that the word $a_1a_2... a_k$ is freely irreducible. (Here $|g|$ is the distance from $g$ to the identity in the Cayley graph of $G$.) We show that in finitely generated free soluble groups of degree $d\ge2$ there exist strict dead elements of depth $k=k(d)$, which grows exponentially with respect to $d$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
