关于逆闭有限收敛减长重写系统表示的群
On groups presented by inverse-closed finite convergent length-reducing rewriting systems
摘要：
我们表明，由逆封闭的有限收敛的长度减少重写系统的组的特征在于一个惊人的几何性质：他们的凯莱图是测地和边长的非退化三角形一致有界。这导致了一个新的代数结果：群是普通的（同构于许多有限群和$\mathbb Z$的副本的自由积）当且仅当群的非平凡有限阶元素集合上的某个关系在有界集上是传递的。我们用它来证明，如果一个组提出了一个逆封闭的有限收敛的长度减少重写系统是不是平原是在$\mathsf{NP}$。一个“是”的答案将反驳马德莱纳和奥托从1987年开始的一个长期猜想。我们还证明了由逆闭有限收敛减长重写系统给出的平面群的同构问题在$\mathsf{PSPACE}$中。
英文摘要：
We show that groups presented by inverse-closed finite convergent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to the free product of finitely many finite groups and copies of $\mathbb Z$) if and only if a certain relation on the set of non-trivial finite-order elements of the group is transitive on a bounded set. We use this to prove that deciding if a group presented by an inverse-closed finite convergent length-reducing rewriting system is not plain is in $\mathsf{NP}$. A "yes" answer would disprove a longstanding conjecture of Madlener and Otto from 1987. We also prove that the isomorphism problem for plain groups presented by inverse-closed finite convergent length-reducing rewriting systems is in $\mathsf{PSPACE}$.
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