广义小消去条件、非正曲率和图解约化
Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility
摘要：
我们提出了一个度量条件${\LARGEτ}'$，它描述了经典的小取消组的几何形状，也适用于其他已知类的组，如二维Artin组。我们证明了满足条件${\LARGEτ}'$的演示文稿在Sieradski和Gersten的意义下是图解可约的。特别是我们推导出的标准表示的阿廷群是非球面的当且仅当它是图解可约。证明了在一些额外的假设下，${\LARGEτ}'$ -群具有二次Dehn函数和可解的共轭问题.格林林格引理的精神，我们证明，如果一个演示$P=\langle X \mid R\rangle$组$G$满足条件${\LARGEτ}'-C'(\frac{1}{2})$，任何非平凡的字在自由群中产生的$X$表示平凡元素在$G$的长度至少是最短的关系子。我们还引入了一个严格的度量条件${\LARGEτ}'_{<}$，这意味着双曲。最后，我们调查这些条件的非度量和对偶变量，并研究了小抵消理论的调和平均版本。
英文摘要：
We present a metric condition ${\LARGEτ}'$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition ${\LARGEτ}'$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, ${\LARGEτ}'$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation $P=\langle X \mid R\rangle$ of group $G$ satisfies conditions ${\LARGEτ}'-C'(\frac{1}{2})$, the length of any nontrivial word in the free group generated by $X$ representing the trivial element in $G$ is at least that of the shortest relator. We also introduce a strict metric condition ${\LARGEτ}'_{<}$, which implies hyperbolicity. Finally we investigate non-metric and dual variants of these conditions, and study a harmonic mean version of small cancellation theory.
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