We show that Gromov’s monsters arising from i.i.d. random labellings of expanders (that we call random Gromov’s monsters) have linear divergence along a subsequence, so that in particular they do not contain Morse quasigeodesics, and they are not quasi-isometric to Gromov’s monsters arising from graphical small cancellation labellings of expanders.Moreover, by further studying the divergence function, we show that there are uncountably many quasi-isometry classes of random Gromov’s monsters.

中文翻译：

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随机格罗莫夫怪物的发散和准等距类

我们证明了由扩展器的 iid 随机标记产生的格罗莫夫怪物（我们称为随机格罗莫夫怪物）沿子序列具有线性发散，因此特别是它们不包含莫尔斯准测地线，并且它们与格罗莫夫怪物不是准等距的此外，通过进一步研究散度函数，我们证明了随机格罗莫夫怪物的准等距类数不胜数。